TOPOLOGY
and
APPLICATIONS
International Topological Conference
Dedicated to P.S.Alexandroff's 100th Birthday
Moscow
May 27-31,1996

TOPOLOGY and APPLICATIONS
International Topological Conference
Dedicated to P.S.Alexandroff's 100th Birthday
Moscow
May 27- 31,1996
Φ
PHASIS
Moscow

UDC 515.1 -|- 514.7 The book is published under financial support
τ>Λτ*τ* °f Russian Foundation for Basic Research,
JJ Project No 96-01-10048
Topology and Applications
International Topological Conference,
Dedicated to P. S. AlexandrofTs 100th Birthday
Moscow, May 27-31, 1996
Moscow: PHASIS, 1996
The papers appear as they were presented by the authors
without editing
The picture of P. S. Alexandroff is due to Constantin P. Radimov
and reproduced here by permission of Maria P. Radimova
ISBN 5-7036-0017-0
© PHASIS, 1996

International Topological Conference
Dedicated to P.S. AlexandrofTs 100th Birthday
Moscow, May 27-31, 1996
Organized by
Lomonosov Moscow State University
and
Steklov Mathematical Institute
Russian Academy of Sciences
Supported by
Russian Foundation for Basic Research
Organizing Committee
A.V.Arhangel'skff (Lomonosov Moscow State University)
R.Engelking (Warsaw University, Poland)
V.V.Fedorchuk (Lomonosov Moscow State University), Vice-Chairman
A.T.Fomenko (Lomonosov Moscow State University), Vice-Chairman
S.D.Iliadis (University of Patras, Greece)
A.A.Maltsev (International Institute for System Research, Austria)
A.S.Mishchenko (Lomonosov Moscow State University)
S.P.Novikov (Lomonosov Moscow State University), Chairman
B.A.Pasynkov (Lomonosov Moscow State University)
V.I.Ponomarev (Lomonosov Moscow State University)
M. V.Shamolin (Lomonosov Moscow State University)
E.V.Shchepin (Steklov Mathematical Institute), Vice-Chairman
Yu.M.Smirnov (Lomonosov Moscow State University)
V.M.Tikhomirov (Lomonosov Moscow State University)
J.E.Vaughan (University of North Carolina, USA)
A.N. Yakivchik (Lomonosov Moscow State University), Secretary
H.Zieschang (Ruhr University of Bochum, Germany)

Cpntents
Yu.M. Smirnov. On P.S. Alexandroff's main results XI
G. Fret, U. Stammbach. Correspondence between Aleksandrov
and Hopf 1926-1971 xxni
Plenary Lectures 1
S. C. Ferry. Homology manifolds and the topological
characterization of manifolds 3
A.N. Dranishnikov. Cohomological dimension theory
and applications 11
S. V. Matveev. Algorithmic classification of sufficiently large
3-manifolds 17
A. Bak. Foundations of algebraic homotopy theory and dimension
theory 25
A.V. Bolsinov. Fomenko's invariants in the theory of integrable
Hamiltonian systems 27
B.N. Apanasov. Geometry and topology of geometrically finite
negatively curved and Carnot-Caratheodory manifolds 35
Session 1. General and Geometric Topology '43
S.M. Ageev. Absolute extensors of spaces with countable filtration 45
F.D. Ancelj C.R. Guilbault. Topological and geometric structures
on compact contractible manifolds 47
S.A. Antonian. Extension properties of the orbit space 49
A. Bella, I.V. Yaschenko. Embeddings preserving character
and cardinal invariants 51
R. Cauty. Locally connected subgroups of the Hubert space
and ANR-spaces 53
M.M. Cioban. Isomorphism problems for Banach spaces
of measurable functions on compact spaces 55
P. Fabel. The Banach-Mazur compactum Q(2) is an absolute retract 57
V. V. Fedorchuk. On completeness of spaces of r-additive measures 59
P.M. Gartside. Permutation groups 61
A.A. Gryzlov. Some points of compactifications 63

VII
Υ. Hattori, Η. Ohta. Dugundji extension theorems for ordered
spaces and their products 65
V.M. Karaulov. On the monotonicity of dimension dim
of continuous mappings 67
U.H. Karimov, D. Repovs. On #n-bubble in n-dimensional
compacta '. 69
A. P. Kombarov. Weak norrgality, exp (X) and powers 71
S.A. Logunov. Butterfly points in separable spaces with π-weight ω\ 73
V.L Malykhin. Resolvability of A-, CA- and PCA-sets in compacta 75
Μ. Κ Matveev. Covers by stars of discrete subspaces 77
A.V. Mironov. Some contribution to the theory of partially ordered
locally compact groups 81
Yu.N. Mukhin. On compact elements of locally compact groups .... 83
S.E. Nokhrin. Local compactness of C\{X) 85
A. Okuyama. Note on hereditary normality of product spaces 87
D.S. Ohezin. Descriptive theory and discontinuity points of Baire I
functions 89
A. V. Osipov. F-closed spaces 91
S.A. Peregudov. On some covering properties 93
E.G. Pytkeev. On σ-additive covers of /f-analytic spaces 95
D. Repovs. Cell-like mappings and their applications in Geometric
Topology 97
E.A. Reznichenko. On stratifiable subspaces of spaces of continuous
functions with the compact-open topology 99
L.B. Shapiro. On simultaneous extension of continuous partial
functions 101
E. V. Shchepin. On selections of multivalued mappings 103
A.B. Skopenkov. On the deleted product criterion for embeddability
of manifolds in Em : 105
A.P. Sostak, J. Steprans. On some compactness-type properties
defined by special open covers 107
V.V. Uspenskii. On the dimension of the Higson corona Ill
V.L Varankina. On semirings of continuous nonnegative functions
on F-spaces 113
EM. Vechtomov. On the lattice of subalgebras of a ring
of continuous functions 115
N.V. Velichko. On sequential completeness of C\{X) 117
A.Yu. Zoubov. Partial coverings and fibrewise uniformities 119

VIII
Session 2. Algebraic Topology 121
A.M. Aslanyan. Asymptotically flat solutions of Bogomolny
equations with solvable gauge group 123
D. V. Berzin. The orbits of the coadjoint representation for Lie
groups which have some special structure 125
V.M. Buchstaber, N. Ray. Double cobordism, flag manifolds,
and quantum doubles 127
Yu.T. Lisica. Strong excision property for coherent homology 131
M.J. Chasco, E. Martin-Peinador. Reflexivity of convergence
Abelian groups 133
T.L. Melekhina. On the orbit topology for со-adjoint representation
of tensor extensions of Lie groups 137
A.S. Mishchenko. Metric approach to constructing Fredholm
representations 141
N. Mramor-Kosta. Parametrized Borsuk-Ulam theorems for Banach
bundles 143
Yu.V. Muranov. Splitting along one-sided submanifolds 145
H. Nencka. Generalization of the Markov theorem
and Cantorian-like braid groups 147
A.Yu. Neronov, A.M. Boyarskii. General Relativity
as a construction of the formal theory of Lie pseudogroups 149
V. Yu. Ovsienko. Space of linear differential operators as a module
over the Lie algebra of vector fields 153
N. Ray. Umbral and Schubert calculi 155
N.N. Saveliev. Invariants of homology 4-cobordisms from gauge
theory 157
K. Shimomura. The homotopy of 1,2-local finite spectra 159
V. V. Trofimov. On the topology of the path space for a symplectic
manifold 161
A.V. Zarelua. Equivariant exterior algebra of finite groups 163
Session 3· Applications of Topology and Geometry 165
S.A. Bogatyi. The Knaster problem, the Borsuk theorem and cyclic
systems 167
Yu.G. Borisovich. Multivalued vector fields with Fredholm
and monotone components 171
G. Burdet, P. Combe, H. Nencka. Statistical manifolds, a-geodesics
and λ-Jacobi fields 173
A.S. Denisiuk. On a problem of integral geometry in spaces
of constant curvature 177

IX
N.P. Dolbilin, Μ.A. Stan'ko, M.I. Stogrin. Extremality
of the Bricard Octahedra 179
V. V. Filippov. Basic topological structures of the theory of ordinary
differential equations 181
A. Gray. Costa's minimal surface 183
B.S. Klebanov. Convergence in the space of solution spaces
of ordinary differential equations and its applications 185
B.S. Kruglikov. Exact smooth classification of divergence-free vector
fields on surfaces of small genus 187
A.T. Lipkovski. Serret's curves 191
M.A. MalakhaUtsev. On cohomology of a sheaf over foliation
with tangential (X, G)-structure 193
J. Malesic. The standard Cantor set is Lipschitz ambient
homogeneous on the plane 195
O.E. Orel Orbital classification of integrable Hamiltonian systems
with two degrees of freedom in a neighborhood of equilibrium 199
R. V. Plykin. On topological distinction of Waza continua
in the theory of smooth dynamical systems 201
Z.G. Psiola. Realization of geodesic flow within the monopolistic
framework 203
P. V. Semenov. Local paraconvexity and local selection theorem . . . 205
M. V. Shamolin. Relative structural stability and relative structural
instability of different degrees in Topological Dynamics 207
E. V. Shchepin. Hausdorff dimension and dynamics
of diffeomorphisms 209
V. V. Shurygin. On the bigraduated cohomology of manifolds
over local algebras and its applications 211
Yu.M. Smirnov. Minimal topologies on acting groups 213
M. V. Sokolov. Summands of the Turaev-Viro invariants 215
D. Y. Suh. Entire rational approximation of G-maps 217
A.Yu. Volovikov. Cohomological sphere bundles and parametrized
Borsuk-Ulam theorems 219
V.G. Zvyagin. On some variant of the degree theory
and its applications to problems of Hydrodynamics 223

Dedicated to the memory
of Pavel Sergeevich Alexandroff

TOPOLOGY and APPLICATIONS
International Topological Conference
Dedicated to P.S.Alexandroff's 100th Birthday
Moscow, May 27-31, 1996
On P.S. AlexandrofTs main results*
Yuri M. Smirnov
Professor,
Chair of Higher Geometry and Topology,
Lomonosov Moscow State University
Pavel Sergeevich has written about 82 original mathematical works (the
total number with new editions exceeds 170). With a few exceptions they are
devoted to topology, set theory and the theory of functions in Luzin's sense.
Of course, not all of them are of equal value, but each of them features his
always interesting creative thought, starting with "On the sets complement
to A-sets" [6] and concluding with "Principal points in the development of
set-theoretic topology" [125] (with V.V. Fedorchuk).
Pavel Sergeevich obtained his first significant mathematical result in
1915, being still a student at N.N. Luzin's seminar and answering one of his
questions: any uncountable Borel set contains a perfect subset and therefore
has the cardinality of continuum [6]. The tool for the proof was a special
operation on sets introduced by Pavel Sergeevich and named "A-operation"
in his honor, which had an essential influence upon the development of
set-theoretic topology. It greatly impressed Lebesgue, young Souslin and
Luzin himself.
The following theorem on absolute Gs-sets proved later is relevant to
this result. It states: Лп the class of separable spaces, every absolute Gs-set
is homeomorphic to a complete metric space [7].
After that, as one knows, Pavel Sergeevich had a rather long pause in
his mathematical activity since 1917 when N.N. Luzin had suggested him
to solve the continuum problem which was insoluble, as we know today, by
the methods available at that time. After a strong nervous overstrain (or
stress) he decided to give up mathematics, leave Moscow and turn to a quite
different — literary and theatrical — activity.
He returned to Moscow and the Moscow University only in 1920. The
same year, when passing his Master degree exams, he got acquainted with
Moscow: PHASIS, 1996
Pages XI-XXI
* Translated from Russian by Andrew N. Yakivchik.

XII
Pavel Samuilovich Urysohn, and they became friends forever. He was a
very extraordinary man and a well-educated mathematician. Conversations
with him gradually revealed a new research area — topology, to which Pavel
Sergeevich devoted all his life. Since the summer of 1922, Pavel Sergeevich
had felt a new wave of enthusiasm and passion for mathematics. Together
with Pavel Samuilovich he proved a metrizability theorem [3] which many
years later engaged such eminent set-theoretic topologists as R. Bing and
K. Nagata. At the same time Pavel Sergeevich and Pavel Samuilovich wrote
their famous work "Memoir on compact topological spaces" [1, 35].
First of all, a new axiomatics of a topological space (with the help of
open sets) which is most commonly used nowadays was given there. But,
certainly, the most important was the creation of a sufficiently extensive
general theory of compact (called bicompact by Pavel Sergeevich, to
emphasize the role of two types of compactness involved implicitly in this notion)
and locally compact spaces. They examined and "introduced into practice" a
whole series of important properties of compact and locally compact spaces
and constructed many interesting examples.
In this work they also proved that the compactness property is equivalent,
in the class of Hausdorff spaces, to the property of absolute closedness, and
the local compactness property is equivalent, in the class of compacta, to the
property of absolute openness. Besides, in this memoir Pavel Sergeevich
introduced the well-known one-point Alexandroff 's compactification that
further led him to the theory of compactifications (compact extensions of
topological spaces) [66] and gave a strong impulse for A.N. Tychonoff 's theorem
on the compactness of the product of compact spaces and for the notion of a
completely regular, or Tychonoff, space.
One should add, that in 1924, long before Dieudonne, in connection
with the metrization problem always being of interest for Pavel Sergeevich,
he defined the notion of a locally finite cover and showed that every open
cover of a separable metric space can be refined by a locally finite open cover
(paracompactness!).
A little bit earlier, in 1923, Pavel Sergeevich and Pavel Samuilovich
left abroad where by delivering public lectures on Relativity Theory (the
poster has been preserved) they earned recently invented convertible "pure
gold" Russian currency. They were the first Soviet mathematicians who
came abroad from Soviet Russia. They worked under the auspices of the
world-famous Gottingen University where they became familiar with
German mathematical school of that time and with its remarkable
representatives such as Klein, Hubert, Landau, Kurant, Emmy Noether and others.

XIII
This was the reason for Pavel Sergeevich's interest in algebra and algebraic
methods in topology.
In the spring of 1924, they decided to organize their topological seminar
in the Moscow University, but held only one first session together. Actually
this seminar started its work only in September, after Pavel Samuilovich's
tragic death. This seminar still continues its great work nowadays, after
Pavel Sergeevich's death, and is called the Alexandroff seminar.
Already during their visits to Gottingen (there were some of them) Pavel
Sergeevich and Pavel Samuilovich began studying combinatorial topology in
connection with some works of Poincare. Among many its aspects, they were
interested in the theory of approximations. In the French town of Batz at
the Atlantic coast where they spent vacation in the summer of 1924, Pavel
Sergeevich introduced the notion of a nerve of a cover, the main one —
by words of Pavel Sergeevich — in all his works on topology, being of a
great value nowadays. He understood that nerves of infinitely developing
covers of a compactum approximate this compact arbitrarily exactly and
permit to reduce the study of its topology to the study of the combinatorial
structure of any sequence of finite simplicial complexes approximating the
given compactum. In 1925 he introduced these ideas in the work [17] which
was very highly estimated by Brouwer.
The first application of the concept of a nerve of a cover was the
theorem on ε-translations of compact subsets of Euclidean spaces to polyhedra
proved by him in 1926 [12]: for any compactum X from Kn there exists a
map f : X -» Ρ, Ρ С Rn, such that the distance d(x, fx) < ε for each χ in
X. These were remarkable discoveries: the former gave a possibility to
construct the so-called spectral homology groups for a rather wide class of spaces,
and the latter formed a basis for the famous Nobeling-Pontrjagin theorem
on embedding η-dimensional compacta into the (2ra + 1)-dimensional
Euclidean space.
In the twenties Pavel Sergeevich often visited Germany, Holland and
France, communicated with Brouwer, Neugebauer and Hopf. In 1927,
together with Hopf, he came to the United States where they established close
contacts with topological schools of Veblen, Alexander and Lefschetz. As a
result, Pavel Sergeevich began to study homological dimension theory. Its
main idea is that if in a compactum X there is a non-trivial cycle of
dimension η — 1, homological to zero on this compactum, then the dimension of
the compactum X is not less than n.
Pavel Sergeevich considered the theorem on essential maps to spheres
to be the key to this theory. The matter is that earlier definitions of dimen-

XIV
sion, given by Brouwer, Menger and Urysohn, provide actually only upper
estimates for dimension of compacta whereas Pavel Sergeevich's theorem on
essential maps of compacta to spheres permits to estimate the least possible
dimension of mapped compacta [41]. As a result, in this period of
communication with Pavel Sergeevich, already by correspondence, Hopf came to his
famous classification theorem.
At the same time, Pavel Sergeevich worked on the position problem and
the problem of the "shape" of closed sets of arbitrary dimension in
Euclidean spaces. The work [34] with the similar title appeared in 1929 and
was, in essence, devoted to duality laws. The dimension of a compactum X
is characterized there by homological properties of the complementary open
set En \ X. Moreover, as it is commonly accepted, the duality for compacta
in Euclidean spaces is actually proved for the first time. But it is not
formulated — and could not have been formulated — in its natural form, as
there were neither Pontrjagin characters nor adequate algebraic formalism.
But nevertheless under Pavel Sergeevich's influence quite different duality
theorems were obtained by Kolmogorov.
These results led Pavel Sergeevich's student — L.S. Pontrjagin — to his
famous duality theorem and enabled him to solve a complicated problem
on the dimension of the product of compacta. Both of these problems were
posed by Pavel Sergeevich. Later, in 1947, Pavel Sergeevich generalized
Pontrjagin's duality theorem to arbitrary (not only closed) subsets of
Euclidean spaces [79]. Further strong generalizations of duality theorems were
obtained by another Pavel Sergeevich's student — K.A. Sitnikov.
In 1928 Kurant, being one of Springer's editors, suggested that Hopf
and Pavel Sergeevich should write a book on topology. They agreed. This
was a very great work! The work on this book, mainly by correspondence,
continued till 1935 when the first volume "Topology" [53] (which happened
to be the last of the three planned volumes and was dedicated to Brouwer)
was published. It served a handbook of topology for many generations of
mathematicians. It consisted of four parts: 1) foundations of set-theoretic
topology; 2) topology of complexes; 3) theorems on topological invariance
and relevant notions; 4) linkages in a Euclidean space and mappings of
polyhedra.
In the first chapter, written by Pavel Sergeevich alone, besides necessary
and already known facts, the new important concept of a continuous
decomposition leading to theorems on factorization of compacta is contained. The
rest of the chapters include both Pavel Sergeevich's results already described
and results of Alexander, Brouwer, Jordan, and, of course, Hopf (but with-

XV
out his classification theorem). Before that book there were many other
books on topology (by Schoenflies, Hausdorff, Frechet, Menger, Kuratowski,
Dehn-Heegaard, Veblen, Lefschetz, Seifert-Threlfall), but, as the authors
of "Topology" said, "none of these books considers topology as a whole, each
of them represents only one branch of this science".
The next book was "An introduction to the theory of functions in a real
variable" [64], actually a textbook, written together with Kolmogorov in
1938. It was followed by "Combinatorial topology" [84] in 1947. In this
great work he describes the status of algebraic topology up to 1941,
starting with necessary facts from set-theoretic topology up to Alexander's and
Pontrjagin's duality theorems and Hopf's classification theorem. I
remember this book very well, because I participated directly in its creation, in
preparing the final text and some drawings — mainly the drawings were
made by Kolmogorov.
After that he thoroughly remade his book "An introduction to Group
Theory", 2nd edition [93] and wrote a new book "An introduction to the
general set theory and theory of functions" [88], which was, in essence,
published anew in 1977 and entitled "An introduction to Set Theory and
General Topology" [124]. The latter book filled the gap existing at that
time and up to-day does remain a remarkable modern handbook on General
Topology (before that, the only book on General Topology in Russian was
a widely known excellent book by Hausdorff, but it was almost unavailable
and, of course, a bit obsolete).
The last Pavel Sergeevich's book "An introduction to dimension theory"
[123] was written together with B.A. Pasynkov. It contains the latest results
of that time: the theorems of Dowker, Katetov, Sitnikov and, certainly,
Pasynkov. This work is still of great importance for topologists.
Beside that, Pavel Sergeevich wrote many surveys, among which we
mention only the most remarkable ones, that have influenced the development
of topology greatly. Firstly, a plenary lecture at the АН-Union
Mathematical Congress of 1934 "Algebraic methods in topology" [47], secondly — a
plenary lecture at the Prague Topological Symposium of 1961 "On some
results concerning topological spaces and their continuous mappings" [117],
and, thirdly -^ an article in "Uspekhi Matematicheskikh Nauk" (1964) "On
some principal trends in General Topology" [119].
Pavel Sergeevich Alexandroff was not only an outstanding
mathematician, but also a wonderful personality. He was a brilliant speaker, with

XVI
excellent knowledge of literature, music, theatre. He was an active
public figure, but primarily the university's one. For many years he was the
President of Moscow Mathematical Society, the Head of the Department
of Mathematics of the Faculty of Mechanics and Mathematics, he delivered
various lectures with enthusiasm and his usual emotional-power.
Many years have passed since 1982 when Pavel Secgeevich Alexandroff
left us. I am sure, however, that many modern topologists, and the young
ones as well, still are being influenced by his great creative power.
List of P.S. Alexandroff's principal scientific works1
1. On compact topological spaces, Bull. Acad. Pol. Sci. (A) (1923), 5-8 (in
French) — with P.S. Urysohn.
2. On local properties of sets and the notion of compactness, Bull. Acad. Pol.
Sci. (A) (1923), 9-12 (in French).
3. A necessary and sufficient condition for a class (L) being a class (D), Cont.
R. Acad. Sci. 177 (1923), 1274-1276 (in French) — with P.S. Urysohn.
4. On the topological invariance of the sets complement to sets (A), Mat. Sbornik
31 (1924), 310-318 (in French).
5. Integration in M. Denjoy's sense considered as investigation of primitive
functions, Mat. Sbornik 31 (1924), 465-476 (in French).
6. On the sets complement to Л-sets, Fund. Math. 5 (1924), 160-165 (in French).
7. On the first class sets and abstract spaces, Cont. R. Acad. Sci. 178 (1924),
185-187 (in French).
8. On the theory of topological spaces, Math. Ann. 92 (1924), 258-266 (in
German) — with P.S. Urysohn.
9. On the structure of bicompact topological spaces, Math. Ann. 92 (1924),
267-274 (in German).
10. On the metrization of compact topological spaces in the small, Math. Ann.
92 (1924), 294-301 (in German).
11. On the basis of η-dimensional set-theoretic topology, Math. Ann. 94 (1925),
296-308 (in German).
12. On the dimension of closed sets, Cont. R. Acad. Sci. 183 (1926), 640-643 (in
French).
13. On Cantorian manifolds and generalizations of the Phragmen-Brouwer
theorem, Cont. R. Acad. Sci. 183 (1926), 722-724 (in French).
14. Additional notes to "Memoir on Cantorian manifolds", from P.S. Urysohn's
posthumous records, Fund. Math. 8 (1926), 352-359 (in French).
^he list is taken from the books "Mathematics in the USSR within the forty years.
1917-1957" (Moscow, 1959) and "Mathematics in the USSR. 1958-1967" (Moscow, 1969),
with the exception of the most recent works [122]-[125].

XVII
15. On continuous mappings of compact spaces, Proc. Amst. Akad. 28 (1926),
997 (in German).
16. On principal trends of modern topology, in: Proc. All-Russian Math. Congr.,
1927, 64-89 (in Russian).
17. Simplicial approximations in General Topology, Math. Ann. 96 (1927),
489-511 (in German).
18. On combinatorial properties of generalized curves, Math. Ann. 96 (1927),
512-554 (in German).
19. On continuous mappings of compact spaces, Math. Ann. 96 (1927), 555-571
(in German).
20. On the duality between connectedness indices of a closed set and its
complement space, Gott. Nachr. (1927), 323-329 (in German).
21. A definition of Betti numbers for arbitrary closed sets, Cont. R. Acad. Sci.
184 (1927), 317-320 (in French).
22. On a decomposition of a space into closed sets, Cont. R. Acad. Sci. 184 (1927),
425-427 (in French).
23. On a new generalization of the Phragmen-Brouwer theorem, Cont. R. Acad.
Sci. 184 (1927), 575-578 (in French).
24. A sketch of main courses of Urysohn's dimension theory, Math. Ann. 98
(1928), 31-63 (in German).
25. On zero-dimensional punctiform sets, Math. Ann. 98 (1928), 86-106 (in
German) — with P.S. Urysohn.
26. The proof of the theorem "every closed set of positive dimension is
topologically contained in a locally connected continuum of the same
dimension", Fund. Math. 11 (1928), 141-144 (in German) — with
L.A. Tumarkin.
27. On a space with vanishing first Brouwer number, Proc. Amst. Akad. 31 (1928)
(in German) — with P.S. Urysohn.
28. On the general concept of dimension and its relationship with the elementary
geometric representation·, Math. Ann. 98 (1928), 617-636 (in German).
29. On the generalized Phragmen-Brouwer theorem, Fund. Math. 11 (1928),
222-227 (in German)'.
30. On the general dimension problem, Gott. Nachr. (1928), 25-44 (in German).
31. On the homeomorphism of closed sets, Cont. R. Acad. Sci. 186 (1928),
1340-1342 (in French).
32. On the boundaries of connected domains in the η-dimensional space, Cont.
R. Acad. Sci. 186 (1928), 1696-1698 (in French).
33. On finitely connected continuous curves, Fund. Math. 13 (1929), 34-41 (in
German).
34. Observations on the shape and position of closed sets of arbitrary dimension,
Ann. Math. 30 (1929), 101-187 (in German).
35. Memoir on compact topological spaces, Verh. Kon. Akad. Wet. 14(1) (1929)
(in French) — with P.S. Urysohn.
36. On closed Cantorian manifolds, Gott. Nachr. (1930), 211-219 (in German).

XVIII
37. On the dimension theory, Cont. R. Acad. Sci. 190 (1930), 1102-1104 (in
French).
38. Geometric analysis of dimension of closed sets, Cont. R. Acad. Sci. 191 (1930),
475-477 (in French).
39. The simplest concepts of Topology, Berlin, 1932 (in German).
40. On the notion of dimension of closed sets, J. Math. Pur. et Appl. 11 (1932),
283-298 (in French).
41. Dimension theory. Applications to the geometry of closed sets, Math. Ann.
106 (1932), 161-238 (in German).
42. On Urysohn's constants, Fund. Math. 20 (1933), 140-150 (in German).
43. On a theorem by K. Borsuk, Monatsh. fur Math, und Phys. 40 (1933), 127
(in German).
44. Betti numbers and ε-mappings, Fund. Math. 22 (1934), 17-20 (in German).
45. On local properties of closed sets, Cont. R. Acad. Sci. 198 (1934), 227-229
(in French).
46. On Betti groups at a point, Cont. R. Acad. Sci. 198 (1934), 315-317 (in
French).
47. Algebraic methods in topology, in: Proc. 2nd АН-Union Math. Congr., v. 1,
Leningrad, 1934, 89-108 (in Russian).
48. On the simplest notions of modern topology, Moscow-Leningrad, 1935 (in
Russian) — with V.A. Efremovich.
49. The Л-sets and topological convergence, Fund. Math. 25 (1935), 561-567 (in
German).
50. On discrete spaces, Cont. R. Acad. Sci. 200 (1935), 1649-1651 (in French).
51. On sequences of topological spaces, Cont. R. Acad. Sci. 200 (1935), 1708-1711
(in French).
52. On local properties of closed sets, Ann. Math. 36 (1935), 1-35.
53. Topology, v. 1, Berlin, 1935 (in German) — with H. Hopf.
54. On some questions on the topology of closed sets, in: Proc. 2nd Ail-Union
Math. Congr., v. 2, Leningrad, 1935, 123 (in Russian).
55. η-dimensional generalized manifolds, Cont. R. Acad. Sci. 202 (1936),
1327-1329 (in French) — with L.S. Pontrjagin.
56. Finite covers of topological spaces, Fund. Math. 26 (1936), 267-271 (in
German) — with A.N. Kolmogorov.
57. A sketch of basic notions in topology, Moscow-Leningrad, 1936 (in Russian)
— with V.A. Efremovich.
58. On countably multiple open mappings, Dokl. AN SSSR 4 (1936), 283-288 (in
Russian).
59. Some problems in set-theoretic topology, Mat. Sbornik 1 (43) (1936), 619-634
(in German).
60. On the theory of topological spaces, Dokl. AN SSSR 2 (1936), 51-54 (in
Russian).
61. Discrete spaces, Mat. Sbornik 2 (44) (1937), 501-520 (in German).

XIX
62. On Brouwer's notion of dimension, Сотр. Math. 4 (1937), 239-255 (in
German) — with L.S. Pontrjagin and H. Hopf.
63. On the homology theory of compacta, Сотр. Math. 4 (1937), 256-270 (in
German).
64. An introduction to the theory of functions in a real variable, 3rd ed.,
Moscow-Leningrad, 1938 (in Russian)4— with A.N. Kolmogorov.
65. Conditions for metrizability of topological spaces and the symmetry axiom,
Mat. Sbornik 3 (45) (1938), 663-672; 8 (50) (1940), 519 (in Russian) — with
V.V. Niemytzki.
66. On bicompact extensions of topological spaces, Mat. Sbornik 5 (47) (1939),
403-424 (in Russian).
67. On the dimension of bicompact spaces, Dokl. AN SSSR 26 (1940), 627-630
(in Russian).
68. Betti groups and homology rings of locally bicompact spaces Dokl. AN SSSR
26 (1940), 631-634 (in Russian).
69. General homology theory, Uchen. Zap. Mosk. Univ. 45 (1940), 3-60 (in
Russian).
70. Deriving Alexander-Pontrjagin's duality law from Kolmogorov's duality law,
Soobshch. Gruz. Fil. AN 1 (1940), 401-410 (in Russian).
71. The addition theorem in dimension theory for bicompact spaces, Soobshch.
Gruz. Fil. AN 2 (1941), 1-6 (in Russian).
72. Basic homology constructions for general projection spectra, Soobshch. Gruz.
Fil. AN 2 (1941), 213-219 (in Russian).
73. The duality law for projection spectra and locally bicompact spaces,
Soobshch. Gruz. Fil. AN 2 (1941), 315-319 (in Russian).
74. General combinatorial topology, Trans. Amer. Math. Soc. 49 (1941), 41-105.
75. On reducible sets, Izv. AN SSSR, Ser. Mat. 5 (1941), 217-224 (in Russian)
— with I.V. Proskuryakov.
76. On homological situation properties of complexes and closed sets, Izv. AN
SSSR, Ser. Mat. 6 (1942), 227-282 (in Russian).
77. On homological situation properties of complexes and closed sets, Trans.
Amer. Math. Soc. 54 (1943), 286-339.
78. On the approximation of bicompact spaces by finite ones, Uspekhi. Mat. Nauk
1(5-6/15-16) (1946), 234 (in Russian).
79. The general duality law for non-closed sets, Uspekhi Mat. Nauk 2(4/20)
(1947), 166-167 (in Russian).
80. Elementary duality domains, Uspekhi Mat. Nauk 2(4/20) (1947), 168-169 (in
Russian).
81. On the notion of a space in topology, Uspekhi Mat. Nauk 2(1/17) (1947),
5-57 (in Russian).
82. The general duality law for non-closed sets in the η-dimensional space, Dokl.
AN SSSR 57 (1947), 107-110 (in Russian).
83. Homological relations in duality domains, Dokl. AN SSSR 57 (1947), 211-214
(in Russian).

XX
84. Combinatorial topology, Moscow-Leningrad, 1947 (in Russian).
85. Duality theorems in combinatorial topology, in: Jub. Collect, to 30th Anniv.
of October Revol., Moscow-Leningrad, 1947, 134-180 (in Russian).
86. Main duality theorems for non-closed sets in the η-dimensional space, Mat.
Sbornik 21 (63) (1947), 161-231 (in Russian).
87. On the dimension of normal spaces, Proc. Royal Soc. 189 (1947), 11-39.
88. An introduction to the general set theory and theory of functions, Moscow-
Leningrad, 1948 (in Russian).
89. On the so-called quasi-uniform convergence, Uspekhi Mat. Nauk 3(1/23)
(1948), 213-215 (in Russian).
90. On the dimension of closed sets, Uspekhi Mat. Nauk 4(6/34) (1949), 17-88
(in Russian).
91. On ordered systems of closed and open sets, Uspekhi Mat. Nauk 5(2/36)
(1950), 178-179 (in Russian).
92. On continuous mappings of closed manifolds, Dokl. AN SSSR 71 (1950),
821-823 (in Russian) — with K.A. Sitnikov.
93. An introduction to Group Theory, 2nd ed., Moscow, 1951 (in Russian).
94. On components of maximal bicompact extensions, Uchen. Zap. Mosk. Univ.
148(Mat. 4) (1951), 216-218 (in Russian).
95. Pontrjagin's topological duality law, Uchen. Zap. Mosk. Univ., 163(Mat. 6)
(1952), 3-29 (in Russian).
96. On the combinatorial topology of non-closed sets, Mat. Sbornik 33 (75)
(1953), 241-260 (in Russian).
97. On some corollaries to Sitnikov's second duality law, Dokl. AN SSSR 96
(1954), 885-887 (in Russian).
98. On the notion of a space in topology, Acta Math. Hung. 5 (1954), 43-60 (in
Russian).
99. On the homeomorphism of punctiform sets, Dokl. AN SSSR 97 (1954),
757-760 (in Russian).
100. From the set-theoretic topology of twenties, Lecture at Amst. Intern. Math.
Conf., 1954 (in German).
101. On some new achievements in the combinatorial topology of non-closed sets,
Fund. Math. 41 (1954), 68-88 (in Russian).
102. On the homeomorphism of punctiform sets, Trudy Mosk. Mat. Obshch. 4
(1955), 405-420 (in Russian).
103. Non-dualizability of Betti groups based on finite covers, Dokl. AN SSSR 105
(1955), 5-6; 107 (1956), 357 (in Russian).
104. Topological duality theorems, Part 1. Closed sets, Trudy Mat. Inst. AN 48,
Moscow, 1955 (in Russian).
105. On Cantorian manifolds, Uspekhi Mat. Nauk 11(5/71) (1956), 233-234 (in
Russian).
106. The combinatorial topology of non-closed sets, in: Proc. 3rd All-Union Math.
Congr., v. 2, Moscow, 1956, 49-51 (in Russian) — with K.A. Sitnikov.

XXI
107. On two Yu. Smirnov's theorems in the theory of bicompact extensions, Fund.
Math. 43 (1956), 394-398 (in Russian).
108. Continua (Vp) — a strengthening of Cantorian manifolds, Monatsh. fur Math.
61(1) (1957), 67-76 (in German).
109. An elementary proof that the identity mapping on a simplex is essential,
Uspekhi Mat. Nauk 12(5/77) (1957), 175-179 (in Russian) — with
B.A. Pasynkov.
110. On bicompact extensions of topological spaces, Dokl. AN SSSR 121(4) (1958),
575-578 (in Russian) — with V.I. Ponomarev.
111. On bicompact extensions of topological spaces, Vestn. Mosk. Univ., Ser. Mat.,
No 5 (1959), 93-108 (in Russian) — with V.I. Ponomarev.
112. Topological duality theorems, Part 2. Non-closed sets, Trudy Mat. Inst. AN
54, Moscow, 1959 (in Russian).
113. On the metrization of topological spaces, Bull. Acad. Pol. Sci., Ser. Math.
8(3) (1960), 135-140 (in Russian).
114. On some classes of η-dimensional spaces, Sib. Mat. Zh. 1(1) (1960), 3—13 (in
Russian) — with V.I. Ponomarev.
115. On completely regular spaces and their bicompact extensions, Vestn. Mosk.
Univ., Ser. Mat., No 2 (1962), 37-43 (in Russian) — with V.I. Ponomarev.
116. On dyadic bicompacta, Fund. Math. 50(4) (1962), 419-429 (in Russian) —
with V.I. Ponomarev.
117. On some results concerning topological spaces and their continuous mappings,
in: General Topology and its Relations to Modern Analysis and Algebra,
Prague, 1962, 41-54.
118. Projection spectra and canonical covers, Uspekhi Mat. Nauk 18(5) (1963),
125-132 (in Russian) — with V.I. Ponomarev.
119. On some principal trends in General Topology, Uspekhi Mat. Nauk 19(6)
(1964), 3-46; 20(1) (1965), 253-254 (in Russian).
120. On Ponomarev's theory of absolutes of topological spaces, Dokl. AN SSSR
161(2) (1965), 263-266 (in Russian).
121. On the theory of projection spectra, in: Abstr. Intern. Math. Congr., Moscow,
1966, 15 (in Russian) — with V.I. Zaitsev.
122. On the main theorem of homological dimension theory, Dokl. AN SSSR
180(3) (1968), 519-522 (in Russian).
123. An introduction to dimension theory, Moscow, 1973 (in Russian) — with
B.A. Pasynkov.
124. An introduction to Set Theory and General Topology, Moscow, 1977 (in
Russian).
125. Principal points in the development of set-theoretic topology, Uspekhi Mat.
Nauk 33(3) (1978), 3-48 (in Russian) — with V.V. Fedorchuk.

TOPOLOGY and APPLICATIONS Moscow: PHASIS, 1996
International Topological Conference Pages XXIII-XXXVIII
Dedicated to P.S.Alexandroff's 100th Birthday
Moscow, May 27-31, 1996
Correspondence between Aleksandrov and Hopf
1926-1971 *
Gunther Prei
Universite Laval, Quebec, PQ, Canada
Urs Stammbach
ETH-Zentrum, Zurich, Switzerland
1. Introduction
The scientific estate of Heinz Hopf is in the possession of the Wissen-
schaftshistorische Sammlungen of the ΕΤΗ Library in Zurich. This includes
amongst other objects 132 letters which Pavel Sergeevich Aleksandrov wrote
to Heinz Hopf and which carry the signature Hs. 621:15-146.
Thanks to the mediation of Professor A. Shiryaev, copies of Hopf's
letters to Aleksandrov were acquired by the ΕΤΗ Library in the summer of
1995. These carry the signature Hs. 160 and consist of 50 letters. They
had been taken as lost for a long time, but were found by Shiryaev in the
attic of a dacha in which Aleksandrov had lived from 1935 and which,
after his death came first into the possession of Kolmogorov and afterwards
of Shiryaev.
Aleksandrov had a nearly perfect mastery of both oral and written
German and with only two exceptions, all the letters are written in this
language. Aleksandrov had learnt German as a child from a German governess
and perfected his knowledge during stays in Germany. Only occasionally do
his letters to Hopf contain short remarks in Russian — part of an exchange
between him and Hopf's wife, Anja Mickwitz, who had learnt Russian at
the teacher's training college in St. Petersburg.
* The present paper originated from the initiative of Albert N. Shiryaev who asked
Professor U. Stammbach and Professor G. Frei to write a paper on the extensive, almost half
a century long, correspondence between P.S. AlexandrofF and H. Hopf.
Publisher's remark: We preserve here the spelling of Pavel Sergeevich's name preferred
by the authors of the present paper.

XXIV
Unfortunately, the correspondence between Aleksandrov and Hopf is no
longer extant in its entirety. Many letters were apparently not kept but are
referred to in those we have. Nevertheless, the existing correspondence is
large enough to present us with a lively impression of the personalities of
these two great mathematicians, and of their respect and friendship for one
another. Indeed, many letters bubble over with charm and humor. Apart
from this, they also contain a considerable amount of mathematics which
provides important source material in understanding the development of
algebraic topology. With only a short time at our disposal, we regrettably
have to renounce dealing with the mathematical contents of the letters in
this paper.
2. The beginning of the friendship — 1926
The correspondence between Aleksandrov and Hopf covers a period
from 1926 to 1970 with an interruption during the war from 1940 to 1946.
At times the exchange was very intensive with four or five letters in a single
month, while at others there were long pauses of which Aleksandrov
complained bitterly. He was, without a doubt, the keener writer and the more
communicative of the two.
Aleksandrov and Hopf met for the first time in the summer of 1926 in
Gottingen. Hopf was 32 and Aleksandrov 30 years old. Aleksandrov talks of
this meeting in his article Einige Erinnerungen an Heinz Hopfx (cf. [3]):
My first meeting with Heinz Hopf took place in the summer of 1926.
After his final exams in Berlin under the supervision of Erhard
Schmidt, Hopf had arrived in Gottingen and this summer term
completed his stay of one year.
In his memories of Heinz Hopf (cf. also [2]) Aleksandrov goes on:
Our acquaintance developed into close friendship during this
summer. We both formed part of the group of mathematicians around
Courant and Emmy Noether and thus belonged to this
unforgettable society with its musical evenings, boat trips with Courant, its
"topological walks" under the direction of Noether, and last, but
not least, its frequent and regular swimming parties and
entertainment which took place at the baths of the university.
In his memories Hopf himself describes his meeting with Aleksandrov as
the one most important event of his stay in Gottingen (cf. [6]):
1 Some Memories of Heinz Hopf

XXV
This meeting soon led to close friendship; topology and
mathematics were not our only topics of conversation; it was a very happy
and also merry time which was not confined to Gottingen but
continued during our many mutual travels. At the time I met him,
Aleksandrov was already one of the great men in pure set theoretic
topology, but he was also on the point of introducing the concept of
"nerves", which was to bring down the wall separating set theoretic
and algebraic topology.
Aleksandrov's connection to Gottingen had already begun in the summer
of 1923, when he first came to Gottingen together with his friend Pavel
Samuilovich Urysohn (1898-1924) ? From that time onwards, until 1928,
Aleksandrov was a regular visitor to Gottingen. He spent the academic
year 1925/26 in Blaricum in Holland and returned yet again to Gottingen
the following summer of 1926 where he first met Hopf. Their common
interest in the work of Poincare and Brouwer immediately led to a strong
attachment. As a consequence, Hopf was led into the inner circle around
Emmy Noether and Courant.
Aleksandrov continues in his memories of Hopf (cf. [3]):
The friendly relationship which existed between Hopf and myself
soon extended to include Otto Neugebauer. Thus a triad was
formed consisting of Hopf, Neugebauer and myself, the so-called
(two-dimensional) "Simplex" that maintained close and friendly
relations to both Courant and Emmy Noether.
The Simplex mentioned here came to be something of a connecting
thread in their deep friendship. The members of the group had nicknames
for one another and among themselves used a special, sometimes quite
ribald, language which was not afraid to call a spade a spade. Nearly all
subsequent letters make reference to these nicknames and evoke memories
of the happy times they spent in one another's company.
At the end of this eventful summer term Aleksandrov, Hopf and
Neugebauer left for France. This is what Aleksandrov had to say about this long
holiday in his memories on Hopf (cf. [3]):
First of all we went to Brittany to Bourg de Batz, a small village
on the south coast of Brittany where Urysohn had lost his life two
years previously. After staying a few days, we left for the
Pyrenees and there we undertook a long hike on foot which eventually
brought us to Collioure. Collioure is (or at least was, in those days)
2 Urysohn drowned in Brittany near Bourg de Batz.

XXVI
a tiny fishing village on the coast near the French-Spanish border.
There was a small and unpretentious, but to all extents and
purposes clean, guesthouse which was named Hotel Bougnol after the
proprietor. The clientele was mostly made up of young artists from
Paris, many of whom later became well known. The modest dining
room of the house was embellished with pictures which departing
guests had given to their hostess as tokens of their gratitude.
Despite the simplicity of the house, which bordered on the primitive,
these paintings — together with the pleasant atmosphere of the
house — and not least, the personality of Madame Bougnol, who
combined affectionate hospitality with dignity and a fine feeling for
propriety, left us all with the best of memories of our stay. [... ]
From Collioure our journey continued through Marseille to Ajac-
cio and ended after a sojourn of about 8 days near Ajaccio with a
wonderful round trip of the Corsican coast (with a short excursion
into Corte, the ancient capital of the island). The week spent near
Ajaccio was marvellous. At no small distance from the shore there
was an isolated rock in the sea which was only generally visited by
sea-gulls. Our swimming skills were such that we swam out to this
rock every day and after a certain time, swam back again. We had
no watches and so only an intuitive idea of how long we stayed on
the rock. But in all, one of these excursions lasted about 4 hours,
from 10 o'clock in the morning until 2 o'clock in the afternoon.
Despite their duration these swims were not dangerous because, apart
from the fact that we were all three good swimmers, the weather
was settled, the sea quiet and calm and — perhaps most important
of all — there were three of us, so there were always two who could
help the third in an emergency. Altogether, Corsica was the climax
of this, in all respects, wonderful journey.
3. Contents of the letters from 1926-1928
After this journey Hopf returned to Berlin where he obtained his final
qualifications as an academic lecturer with the submission of the Habilitations-
schrift consisting of two papers on mapping classes and vector fields on
га-dimensional manifolds.
In the first of the letters in our possession, Aleksandrov congratulates
Hopf (7.11.26):
Sibi congratulentur Docentes universitatis Berolinensis talem
tantumque Hopfum sibi adjungisse.
Hopf was now qualified to read as a lecturer. Already in the winter term
of 1926 he announced a four-hour lecture course on topology. He regularly

XXVII
sent copies of the manuscript of these lectures to Aleksandrov in Moscow,
where they were avidly greeted with great interest by Aleksandrov and his
colleagues.
After having visited Brouwer in Bourg de Batz in mid-August,
Aleksandrov had returned to Moscow in October 1926 and resumed his extensive
course of lectures there at two universities. He too sent his own lectural
manuscripts on topology and Euclidean spaces to Hopf in Berlin.
Aleksandrov tells in his letters of how every Wednesday he held advanced tutorial
seminars on topology in his private apartment from 7 p.m. to 8.30 p.m. and
during which the works of Brouwer and Menger and the recently published
work of Hopf were discussed. It was within this context that Aleksandrov
first mentioned the name of a young student, Pontryagin.
It was only incidentally that Aleksandrov gave any account of his private
life: his marriage was dissolved and he resigned from his position at the
II University of Moscow in order to take on a well-paid full professorship in
Smolensk. During this term of 1926/27 Aleksandrov and Hopf submitted
an application for a Rockefeller grant which would enable them to pass the
winter term of 1927/28 in Princeton. Aleksandrov, whose experience in
these affairs was greater than Hopf's, concerned himself with obtaining the
necessary letters of recommendation. He had already personally spoken to
Lefschetz about this matter in the summer of 1926 in Paris. In his detailed
letter of 23.12.26 Aleksandrov expresses his fear that Brouwer might be
opposed to a simultaneous stay of himself and Hopf at Princeton and that
therefore Brouwer must not find out that they were planning to visit at the
same time. Brouwer had at one time in the past critically remarked that
Aleksandrov was taking advantage of poor Hopf because Hopf went over
Aleksandrov's work with regard to the German. Paradoxically, Brouwer had
himself proposed this, but had apparently forgotten that he had done so.
In direct preparation for the journey to the United States Aleksandrov
was busy learning English from his sister. He had a great aptitude for
languages and already spoke nearly perfect German. He could also converse
fluently in French and Dutch. By the end of the summer term 1927 which
he again spent in Gottingen, he was already able to talk effortlessly to
Newman in English when they met at a reception. As can be deduced from
the exchange of letters, the "vertices" of the "two-dimensional Simplex"
met fairly often during this summer term. Journeys were undertaken by
Hopf to Gottingen as well as by Aleksandrov and Neugebauer to Berlin. A
pleasure trip for the entire Simplex was planned to southern Bavaria for the
Whitsun weekend.

XXVIII
The meetings not only served as venues for the exchange of
mathematical thoughts but also brought welcome opportunities to cultivate common
interests. For Aleksandrov, his visits to Berlin were welcome opportunities
to go to concerts. He was very interested in music — one of his two brothers
was a well-known concert violinist — and he possessed a wide knowledge
of music. He managed to keep himself informed of Berlin's concert life in
far away Moscow with the help of the Vossische Zeitung, and in his
letters he was able to draw Hopf's attention to outstanding performances.
In later years, when Hopf was working and living in Zurich, Aleksandrov
often asked him to procure certain gramophone records and send them to
Moscow. Hopf's own cultural interests tended more towards the theatre and
literature. Nevertheless, the two friends shared much in common, above all
long swimming and ski excursions and mountain hikes. When in Gottingen,
a visit to the baths, which were surveyed by bath-master Klie, was
mandatory and lots of members of the Gottingen circle gathered together there.
Aleksandrov gives a detailed account of these bathing parties in his
memories (cf. [3]):
The Kliesche baths were hot just used by students but were also
visited by many members of the university's teaching staff, amongst
others Hubert, Courant, Emmy Noether, Prandtl, Friedrichs, Deur-
ing, Hans Lewy, Neugebauer and many others. From the non-local
mathematicians, Jakob Nielsen, Harald Bohr, van der Waerden,
von Neumann, Andre Weil must also be mentioned. Brouwer also
visited during the summer of 1926, which he spent in Gottingen.
The baths were decidedly a male domain; womanhood was only
represented by Miss Emmy Noether and Mrs Nina Courant. Both
ladies made daily use of their exceptional right of entry regardless
of the weather.
Towards the end of the summer of 1927 Aleksandrov, Hopf and
Neugebauer, as in the previous year, left on their travels, this time to the
Dauphine, to Cassis near Marseille and from there to Portofino on the
Italian riviera.
In mid-September Aleksandrov and Hopf left for Princeton to commence
a stay that had been made possible by a Rockefeller grant. A year later in his
final report to the Rockefeller Foundation, Hopf gave an account of the visit
which lasted from 1.10.27 until the 1.6.28. He wrote that he had attended
the lectures of professors Lefschetz and Alexander on "Analysis situs" and
had been asked by the Princeton mathematicians to give a number of talks

XXIX
himself about his own work and other topological works from J. Nielsen. He
continues in this report (cf. [5] 620:47):
However, I did not judge these incidents to be the most important
occurrences during my stay at Princeton. These I saw rather in
the informal discussions with the professors Alexander, Lefschetz
and Veblen as well as with Professor P. Aleksandrov from Moscow,
with whom I was together every day in Princeton and with whom I
was able to discuss thoroughly and immediately all new impressions
and thoughts.
In his memories in 1966 Hopf gives a somewhat more personal account
of this stay in Princeton (cf. [6]):
At the time Princeton was still an idyllic little university town.
The famous institute had not yet been founded, not even the "Fine
Hall" existed and in the "French restaurant" Aleksandrov and I
were the only foreign regulars (and as such, and because of the
prohibition, were served wine on Sundays in coffee cups). But at
the university there were lectures from O. Veblen, S. Lefschetz and
J.W. Alexander with all of whom we had interesting discussions.
Lefschetz was probably the most important for us — on the one
hand because he was Aleksandrov's ally in his struggle for the use
of algebraic methods in set theoretical topology and on the other,
because my work on fixpoints was closely related to his own basic
work.
In many of his later letters Aleksandrov reminds Hopf in a nostalgic way
of their time in Princeton, especially of the Christmas they had spent there
and celebrated with a small fir tree and goose liver which Hopf's parents
had sent from Breslau, and of their subsequent trip to Florida.
In the summer of 1928 Aleksandrov and Hopf were both back in
Gottingen. They each gave lectures on their special fields and together
they organized tutorial evenings on various questions of topology. It was
around this time that Courant suggested they write a book on topology for
the Grundlehren series of Springer. The two agreed with this suggestion and
thus took a heavy burden upon themselves which would keep them
intensively occupied for the next seven years. They envisaged a comprehensive
work in which the entire field of set theoretic and algebraic topology would
be introduced. They quickly came to realize that one volume would not be
sufficient and therefore planned a second, and eventually a third. However,
only the first volume was ever published, and that not until 1935. The
difficulties arising from the times which led to the outbreak of the Second

XXX
World War colluded to delay the project. A further reason for the delays
are to be found in the changes to which algebraic topology was subjected
in these years.
In October 1928 Hopf married Anja Mickwitz (1891-1967) whom he had
met in Berlin in 1927 and to whom he had become engaged before his visit
to Princeton. Anja Mickwitz came from a German-Baltic pastor's family
and had concluded her training as a teacher at the college in St. Petersburg,
where she also learnt Russian. The marriage was alluded to in a letter; Alek-
sandrov wrote: "The beneficial effect of KT's3 marriage is already visible.
On photographs he already looks less audacious than in his bachelor days".
Following their wedding, the Hopfs spent a few weeks in a holiday house
which belonged to Hopfs parents in Hain in the Riesengebirge region.
At Aleksandrov's invitation, Emmy Noether and Otto Neugebauer spent
some time in Moscow the following year. Emmy Noether gave an advanced
lecture on algebra. The course was in German, and it is thus perhaps not
surprising that the number of attendants dwindled from 70 to 15 after the
first lectures. Aleksandrov writes about visits to the Egyptian museum
where Neugebauer impressed him by his profound knowledge of the ancient
Egyptian culture. On 3.5.29 he also writes of a theatre visit with Emmy
Noether and remarks that the latter "... does not care very much for art,
for hardly had the curtain descended than he (sic) was already talking again
about mathematics".
4. From 1929-1941
In his memories of Heinz Hopf, Aleksandrov has this to say about the
following period (cf. [3]):
In the years between 1928 and 1932 we met frequently — so that
during this period we really settled down to work, as it were —
the last time in autumn 1932, which for the main part we spent
together in Zurich. Then however there was an interruption until
the end of August 1935. Unfortunately, we did not meet one single
time in between. Although we corresponded regularly during this
period, here also there were interruptions, lost or late deliveries and
such like.
In the summer of 1929 Aleksandrov was not able to travel to Germany for
the first time in many years. To be sure, neither did he stay in Moscow but
3KT for "kleines Tier" (little animal), Hopfs common nickname within the Simplex.

XXXI
instead undertook a long journey to the Caucasian coast on the Black Sea.
In later years he repeated such journeys regularly and told Hopf about them
in his letters. As one example of them, the first is described here in some
detail. Aleksandrov left Moscow on 15th June together with Kolmogorov.
They started out on a 22-day rowing excursion on the Volga which took
them from Yaroslavl' to Samara. The journey then continued by steam
boat to the Astrakhan estuary. From Astrakhan they travelled by steamer
over the Caspian Sea to Baku and then on to Armenia. Here Aleksandrov
spent three weeks in a monastery dating from the 9th century at an altitude
of 1900 m on the Gotschka Lake and from there climbed the 4000 m high
Mount Alag near Mount Ararat. Finally, he spent a month with his mother
and sister on the Black Sea coast.
Aleksandrov was back in Gottingen in the autumn of 1929 where he
lectured during the winter term. After giving talks in Hamburg in
February 1930 he boarded ship for New York in order to spend some time in
Princeton with Alexander and Veblen.
1930 was to be a decisive year for Hopf s destiny. In December 1929
he was offered an assistant professorship at Princeton University, and in
the following year calls for two vocational appointments reached him —
one from the University of Freiburg i. B. as successor to the emeritus Paul
Heffter, and the other from the Swiss Federal Institute of Technology (ΕΤΗ)
in Zurich as successor to Hermann Weyl. Weyl himself had been called to
succeed to Hubert's chair in Gottingen in the autumn of 1930 (cf. [4]). Hopf
accepted the offer from ΕΤΗ and took up the appointment of Professor of
Hohere Mathematik on 1st April 1931.
Both Aleksandrov and Hopf were working very hard on the manuscript
of their topology book, and in his letter of 18.2.31 Aleksandrov writes that
the book "will be finished by the end of June". This was a huge
miscalculation and work on the manuscript dragged on for many years. Aleksandrov
was in Zurich for a good part of summer 1932. Sadly, he fell ill with phlebitis
which restricted his freedom of movement considerably, but he was able to
join the Hopfs for a few weeks on the shores of Lago Maggiore in the southern
part of Switzerland and he returned to Zurich in September for the
International Congress of Mathematicians. From there he went to Gottingen
where he stayed with Emmy Noether and he wrote letters to "both Hopfs"
in Zurich thanking them for their hospitality. He writes, amongst other
things, about the lectures Weyl was giving on topology and comments: "I
fear he is becoming somewhat old-fashioned". He also mentions an
invitation to Landau's and highly praises the good food and, especially, Nina

XXXII
Courant's violin recital "it was really exceptionally good". With regard
to the projected book he notes that Courant was "frightfully angry" at the
overdue deadline and that he had "seldom seen [Courant] so enraged". From
Gottingen Aleksandrov returned to Moscow by way of Berlin and Warsaw.
The work on the book continued at a painstaking pace. Time and time
again — mostly as a result of Aleksandrov's intervention — whole chapters
had to be revised, even newly drafted and the material newly arranged.
When however, in 1934, Kuratowski's book on set theoretic topology was
published and the book from Seifert and Threlfall on algebraic topology was
nearing its completion, Aleksandrov pushed for a speedy winding-up of the
first volume of their work. The bulk of the manuscript was sent to Springer
in autumn 1934. In the event there were still many occasions during 1935
when the authors put in more changes.
Aleksandrov and Kolmogorov had invited the leading topologists of the
day to a congress which took place from the 4th to the 10th of September in
Moscow. Aleksandrov carefully planned and arranged the journey for Hopf
and his wife. He had hoped that Hopf would stay on after the congress and
give a course of lectures, however Hopf preferred another plan. Immediately
after the end of the congress Aleksandrov, Kolmogorov and Hopf and his
wife left for the Crimea. Their route took them by train to Moscow, on to
Sevastopol and then further east to Gaspra, a place about 30 km from Yalta.
Here by the shores of the Black Sea, in a convalescent home in Baty Liman,
was a place where Aleksandrov regularly passed part of his vacations, often
together with Kolmogorov. Hopf and Aleksandrov got down to work on the
preface and on the final corrections of their book which among themselves
they called the "Unicum".
The visit lasted until mid-October. Unhappily, Aleksandrov was ill for
part of the time and sometimes even bed-bound and he often had to leave
the swimming parties and walking excursions up to Hopf and Kolmogorov.
The latter was an extremely capable walker and climber and had conquered,
amongst other peaks, the 5,047 m Kazbek summit in the Caucasus.
In 1935 Markoff in Moscow and Stiefel in Zurich proof-read the galleys
of the book. Several times Hopf found himself forced to remonstrate with
Aleksandrov for making, corrections which he said were too many and too
big. This is what he had to say in a letter of 3.11.35:
What the deuce! Confound it! Devil take you! Now one has to sit
and rack one's head and put everything in order!

XXXIII
The editors at Springer were also worried, indeed thrown into confusion
by the many changes to the manuscript that were being sent in. In the end,
however, Hopf was able to report: "Today on the 18.12.35 the first copy of
"Unicum" arrived with the morning mail. It looks splendid! I found the
first mistake straight away."
The conference in Moscow steered the development of topology in a new
direction and furnished many new impulses. Hopf tells us about this in his
memories (cf. [6]):
The year of 1935 was very important for the developments in
topology for several reasons. The first International Conference on
Topology took place in Moscow in September. The completely
independent talks given at this conference by J.W. Alexander, I. Gordon
and A.N. Kolmogorov can be seen as the beginning of cohomology
theory — of which Lefschetz had already been the precursor with
his "pseudo-cycles" in 1930. What completely surprised me — and
probably numerous other topologists — was not the cohomology
groups — these were nothing other than character groups of
homology groups — as much as the fact that one can, for an arbitrary
complex and for a more general space, define a multiplication
between them, in other words the cohomology ring, which generalizes
the intersection ring of a manifold. We had thought that this was
only possible in manifolds owing to local Euclidicity.
Coming back to the book which they had concluded in Crimea, Alek-
sandrov writes in his memories of Heinz Hopf (cf. Щ):
This marked the end of our work on this book and also the end of
the seven year period 1928-1935 which for both Hopf and myself
was mainly devoted to our collaboration writing the book. Sadly,
our trip to Crimea was also the beginning of a long period which
was to stretch over 15 years when we would not see each other one
single time: 5 pre-war years, 5 war years and 5 post-war years.
Shortly after the conference in Moscow, Aleksandrov and Kolmogorov
bought an apartment in a spacious house which Aleksandrov refers to in his
letters as "Castle Muckenau"4. The house had seven big rooms, kitchen and
bathroom. Later, in 1938, Aleksandrov and Kolmogorov became sole
proprietors. Aleksandrov looked after the yard and garden. At the same time,
the two of them worked closely together on mathematical issues.
Aleksandrov was suffering more and more with his eyes; his myopia became steadily
4 It was in the attic of this selfsame house that Shiryaev found Hopf's letters.

XXXIV
worse and he writes more than once of the "serious prognosis" of his
doctors. It was made clear to him that he must desist from writing by hand,
which was a great strain on the eyes, and he asked Hopf in Zurich to send
him a typewriter with Cyrillic keys. The import into the Soviet Union
proved to be extremely difficult and finally the Moscow Academy of Science
had to intervene. The typewriter did however arrive eventually and served
Aleksandrov to the end of his life.
The plans for further volumes of their topology book were not
abandoned but never got beyond the stages of preparatory work. Political
circumstances played their part in this, for as Aleksandrov says in his
letter of 21.11.38, alluding to the forced resignation of Otto Neugebauer as
editor-in-chief of Zentralblatt:
Naturally, there can be no question now of a book of mine being
published by Springer, especially after Neugebauer's resignation as
editor of Zentralblatt.
The political situation in Germany was seldom mentioned in the
letters. Through their direct contacts in Gottingen, both were fully aware of
how serious the situation had become after the NSDAP's rise to power in
January 1933. But in their correspondence they hardly touched upon the
subject, and if so, only in a very general manner; an analysis of the situation
is not to be found in the letters.5 They both seem to have underestimated
the gravity of the events at first, as did so many others. Hopf writes in a
letter of 30.10.35 concerning, amongst other things, the Nuremberg Laws:
What we hear from Germany is not gratifying: butter rationing
and anti-semitism, and anti-semitism really seems [...] to be on
the rise amongst the population and not merely within legislation.
[...] The laws, by the way, are not as severe as we believed them to
be a few weeks ago; existing marriages will remain untouched.
Both Hopf in Zurich and Aleksandrov in Moscow tried in the event to
help Jewish emigrants from Germany by finding and sometimes even
creating jobs for them. On 16.12.35 Hopf wrote anxiously to Aleksandrov:
"The problem with the German emigrants is becoming increasingly
difficult". Aleksandrov used all his influence in Moscow to obtain a position for
Emmy Noether, but the obstacles could not be overcome in time, and Emmy
Noether was finally driven to taking up a position at Bryn Mawr College in
the United States. Thanks to Hopf's efforts, many Jewish emigrants found
5 It is possible that part of the reason for this was the fear of censorship.

XXXV
refuge of short or long duration in Zurich. Amongst those he was able to
help were his cousin Ludwig Hopf and Paul Bernays. Issai Schur also spent
a semester in Zurich after he had lost his position in Berlin and before he
emigrated to Palestine.
In his letters, Hopf often writes about his parents in Breslau, whom
Aleksandrov knew personally, and who were not getting on well (10.9.38).
They were lonely and suffering from depression (3.3.40). Hopf had visited
them in Breslau as late as Christmas 1939. He had been in touch with
the immigration authorities and submitted an application to bring them to
Switzerland. The necessary permit had been obtained in the end but his
plans were thwarted by the outbreak of the Second World War.6
The exchange of letters which had become more rare since the start of
the war was totally interrupted for several years after the German invasion
of the Soviet Union.
5· The Period 1941-1970
On 20.3.1946 diplomatic relations between Switzerland and the Soviet Union
were established. Hopf immediately took the opportunity of getting in touch
with Aleksandrov. To be on the safe side, he wrote in French (21.3.46)
and Aleksandrov promptly replied three weeks later, also in French, telling
him part of what he had lived through in 1941-42 with his family and
Kolmogorov in Kazan. The roof of his house had been slightly damaged by
a shell splinter in autumn 1941 but apart from that all was well.
In his memories of Heinz Hopf Aleksandrov tells of the period following
the war (cf. [3]):
We did not meet up again until the last days of April 1950 and
that happened in Rome on the occasion of Severi's 70th birthday,
which was celebrated with splendid festivity. Hopf arrived from
Zurich and I from Moscow to take part. I cannot — nor need
I — describe the agitation and excitement we both felt at this
reunion, coming as it did after so many years. The Severi celebrations
lasted for some time because they were one part of an international
(geometrical-topological) symposium. During the whole stay we
naturally spent a lot of time together and mathematics formed a
6 Hopf s father died in 1942 in Breslau. After the war his mother succeeded, despite her
bewildered state, in making her way from Breslau to Erfurt where her daughter was living.
— In 1943 Hopf was informed by the German administration that his assets in Germany
had been confiscated. Subsequently, he also lost his German nationality. He had, however,
already applied for Swiss citizenship and this was granted in 1943 after a short process.

XXXVI
good part of our discussions because the Severi symposium was
scientifically very interesting and something of a compulsory venue for
the exchange of mathematical thoughts.
The correspondence between the two picked up again after this, but it
was not as regular as it had been and the letters consist mostly of short
messages and information on personal matters. Nevertheless, there was
the occasional reflection on things mathematical, as in Aleksandrov's letter
of 15.11.54 in which he congratulates Hopf on his 60th birthday and goes
on: "I am still feeling the strong influence of the last congress: how difficult
to understand mathematics has become, even in those fields where one, as
a so-called specialist, should be able to understand". A year later in a
letter of 10.10.55 he addresses this theme again by citing first of all Serenus
Zeitblom from Thomas Mann's "Doctor Faustus": "I am an old-fashioned
being, come to a halt at certain romantic notions, which I cherish" and
then continues: "My own passion belongs to a period and a direction in
mathematics which began with Dedekind and Cantor and which probably
comes to an end with Emmy Noether". Again in a letter of 8.12.62 he
reverts to the subject:
[...] how beautiful the old Brouwer-Alexander-Hopf-Lefschetz (in
my thoughts, I modestly add the HausdorfF-Urysohn-Aleksandrov)
topology was. And how beautiful the whole of mathematics in the
days of Poincare-Hilbert-E.Schmidt-Weyl-E. Noether is, or was.
Such mathematics are a thing of the past, and this was of course
always the case. The style of mathematical thought has changed —
perhaps more radically than at any other time in the past hundred
years (which is quite understandable because during this period, the
whole of human life has undergone so many weighty changes).
Hopf was also passing through a similar phase. In a letter dated 19.11.57
he writes that "mathematical production [is] the business of the young. [...]
What is worse is that one has great difficulty (and often no success) when
one tries to understand the new things of the young people".
We will let Pavel Aleksandrov's own words describe the closing years of
this long friendship between himself and Heinz and Anja Hopf (cf. [3]):
After 1950 I saw Hopf the next time in Amsterdam at the
International Congress of Mathematicians in August in 1954, then again
in 1958 at the congress in Edinburgh, in 1962 in Stockholm and
in 1966 in Moscow. At these congresses we were usually together for
8-10 days and they were always very gratifying days, even though

not to be compared to our early journeys before the war. Anja
Hopf accompanied her husband to the congresses in Amsterdam,
Edinburgh and Stockholm, although she was already ill at the time
and could only walk a little. She was not able to come to Moscow
in 1966. Apart from these regular encounters at congresses, Hopf
and I also met at all the meetings of the Executive Committee of
the International Mathematical Union. We were both on the board
of the Executive Committee and this enabled us to meet regularly
and with shorter intervals in between meetings. These encounters
occurred in many different places. I was also with Hopfs in Zurich
a few times for extended visits. Our last unclouded gathering was
in and around Moscow in August 1966. At the time, even though
Anja Hopf had been unable to undertake the journey, her state of
health was at least satisfactory, and Hopf was tranquil and in good
humor. The meeting of the International Mathematical Union took
place before the congress started in a beautiful little place called
Dubna, on the banks of the Volga. We spent a few wonderful days
there. My young pupil Victor Zaicev was with us; some years later
he fell seriously ill, but for now he was young, healthy and full
of zest for life, and he left out no chance to organize rowing and
swimming parties and to help us wherever he could. It was superb,
just like the olden days, and Hopf especially was very happy during
these days, indeed almost youthful again. But this was to be the
last time. After his return from Moscow, Anja's health began to
deteriorate rapidly. After a few slight fluctuations, her strength
declined continuously and she died in February of the following year
(1967). Hopfs own life curve now began to sink. An emptiness
formed in him, a vacuum which, once it exists, can only grow until
finally, it becomes interlaced with the whole life of a person and
destroys it. We were together again for a last time in June 1970.
We were both in Frankfurt a. M. and spent about a week there
together. We stayed at a small hotel near Palmengarten. Hopf
was accompanied by his niece, Elisabeth Ettlinger. His health was
no longer good. He had difficulties with walking, sometimes also
with talking; it was hard to imagine that this was the same man
who, less than four years before, had swam so joyously in the river
Volga. But yet it was the same Heinz Hopf whom I had known
so well for 44 years, since the summer of 1926, and with whom I
had shared a close and affectionate friendship all those years. His
essential being, his character had not changed. In Frankfurt we
shared once again a few fine — even though not especially merry
— days. Despite the difficult outward circumstances, we had many
very good discussions and understood one another as completely as
we always had during the years of our friendship. We made plans

XXXVIII
for future meetings in the coming years in Switzerland or Germany.
In spite of this, the overriding mood of this Frankfurt meeting was
one of farewell. Perhaps we both sensed in our subconscious minds
that this was to be our last. This feeling of parting was especially
strong — even overpowering — as we watched the train for Zurich
pull into the station. In the end, Hopf got into the carriage and I
looked at his face for the last time.
In April 1971 Hopf's health became very bad and I prepared myself
for the journey to Zurich. At the end of April, after I had completed
all the formalities that had to be dealt with and fixed the date for
my departure, I fell ill myself and was taken (still in April) to a
clinic, where I had to stay for more than a month. It was here
that word reached me that Hopf had died on 3rd June. I did not
travel to Zurich until 19th November, Hopf's birthday. On this day
a Hopf-Gedenkfeter was held at ΕΤΗ.
References
[1] Aleksandrov, Paul: In Memory of Emmy Noether. In Emmy Noether, Gesam-
melte Abhandlungen, Springer Verlag, Heidelberg, 1983.
[2] Aleksandrov, Paul: Die Topologie in und um Holland in den Jahren 1920-1930.
Nieuw Archief voor Wiskunde 3, 17 (1969), 109-127.
[3] Aleksandrov, Paul: Einige Erinnerungen an Heinz Hopf. Jahresbericht der
Deutschen Mathematiker Vereinigung 78 (1976), 113-146.
[4] Frei, Gunther und Stammbach, Urs: Hermann Weyl und die Mathematik an der
ΕΤΗ Zurich, 1913-1930. Birkhauser Verlag, Basel-Boston-Berlin, Marz 1992,
XVI + 181 Seiten.
[5] Hopf, Heinz: Nachlass. Wissenschaftshistorische Sammlungen der ETH-Biblio-
thek, Hs. 620-623.
[6] Hopf, Heinz: Einige personliche Erinnerungen aus der Vorgeschichte der heuti-
gen Topologie. CBRM Bruxelles (1966), 9-20.
Acknowledgement
The authors would like to thank Mrs. A. Rast-Margerison for the careful
translation of the manuscript into English.

Plenary Lectures

The enlarged abstracts of the plenary lectures are presented
in the same order as they were delivered at the Conference

TOPOLOGY and APPLICATIONS Moscow: PHASIS, 1996
International Topological Conference Pages 3-9
Dedicated to P.S. Alexandroff's 100th Birthday
Moscow, May 27-31, 1996
Homology manifolds and the topological
characterization of manifolds *
Steven C. Ferry
State University of New York at Binghamton, Binghamton, NY, USA
1. Background
Definition 1.1. A locally compact metric space X is an integral homology
η-manifold if for each xGlwe have
*'(jf,x-w;4-i i fzlil:
{
We will refer to a locally contractible finite-dimensional homology manifold
as an ENR homology manifold.
Notice that a finite polyhedron is a homology manifold if and only if the
link of every vertex is a homology sphere. ENR homology manifolds have
the same duality properties as topological manifolds of the same dimension.
The suspension of an integral homology sphere is an example of a homology
manifold which is not a topological manifold.
Definition 1.2. We will say that a metric space X has the disjoint 2-disk
property if for every ε > 0 and every pair of maps /, g : D2 —l· X, there are
maps /,0 : D2 -+ X so that d(f(z),f(z)) < ε, d(g(z),g(z)) < ε.
In 1978, James Cannon [2] conjectured a characterization of topological
manifolds:
Conjecture 1.3. If X is a compact ENR homology η-manifold, η > 5, and
X has the disjoint 2-disk property, then X is a topological manifold.
* Partially supported by NSF Grant DMS-9305758.

4
This remarkable conjecture was an outgrowth of Cannon's work on
double suspension. To see the relation, let H3 be a homology 3-sphere. Then
the double suspension of Η is 51 * Η. This space is a homology manifold
which is locally Euclidean everywhere except possibly along the suspension
circle. We show that H*SX has the disjoint disk property. If / : D2 —> 5**#
is a map, we can approximate / arbitrarily closely by a map which meets
this 51 in a O-dimensional set: take a fine triangulation of D2 and use
Alexander Duality to approximate by a map such that the image of the
1-skeleton of D2 misses 51. Retriangulate and choose an approximation so
that the new 1-skeleton misses 51, and continue. In the limit, we get a map
of D2 into Sl * Η so that the image meets 51 in a O-dimensional set. If
/, g : D2 —>· 51 * Η are maps we wish to push apart, we can perturb so
that each meets Sl in a O-dimensional set and then slide the O-dimensional
sets off of each other in the 51-direction. After using these moves to make
the intersections of the two D2,s with 51 disjoint, we can use ordinary
general position to make the images disjoint from each other in the rest of
the manifold.
Soon after Cannon made his conjecture, Edwards and Quinn [3, 7, 8]
confirmed the conjecture for any connected homology manifold X which
contains a Euclidean neighborhood.
Theorem 1.4. // Xn, η > 5, is a closed, connected ENR homology
η-manifold which satisfies the disjoint disk property and which contains a
subset homeomorphic to Rn, then X is a topological manifold.
The proof is a concatenation of results of Edwards and Quinn. We begin
the statement of Edwards' theorem with a definition.
Definition 1.5. (i) A compact separable metric space X is cell-like if there
is an embedding X —> Q = ГЩ^О, 1] so that for every neighborhood U of
X there is a neighborhood V of X so that V contracts to a point in U. If
X is cell-like, then an easy argument using the Tietze extension theorem
shows that if X —> Ζ is any embedding of X into an ANR Z, then for every
neighborhood U of X in Ζ there is a neighborhood V of X in U so that V
contracts in U. A similar argument, again using Tietze, shows that compact
separable contractible metric spaces are cell-like. Thus, it is reasonable to
think of cell-like spaces as being "Cech contractible".
(ii) A map / : X -> Υ between metric spaces is said to be cell-like if it
is proper (i.e. if f~x(K) is compact for each compact К С X) and if the
inverse image of every у £ Υ is cell-like.

5
Here is the statement of Edwards' theorem:
Theorem 1.6 (Edwards [3]). If Mn, η > 5, is a topological manifold with-
out boundary and f : Μ -* X is a cell-like map, then X is a topological
manifold if and only if X has the disjoint disk property.
By the Vietoris-Begle Theorem, the cell-like image of a topological
manifold without boundary is a homology manifold. Thus, Edwards' theorem
guarantees that Cannon's conjecture is true for homology manifolds which
are cell-like images of topological manifolds. Quinn's theorem supplies a
necessary and sufficient condition for a homology η-manifold to be the cell-like
image of a topological manifold of the same dimension. Here is the
statement of Quinn's theorem:
Theorem 1.7 (Quinn [7]). If Xn is a connected ENR homology manifold,
η > 4, then there is an index I(X) G 8Z + 1 which is equal to 1 if and only if
there exist a topological manifold Mn and a cell-like map f : Μ -> X. The
invariant I(X) is multiplicative in the sense that I(X xY) = I(X) X I(Y)
and it is local in the sense that if U is an open subset of X, then
I(X) = I(U).
The statement of the Edwards-Quinn theorem given above follows
immediately from these two results: the existence of a euclidean neighborhood
implies that I(X) = 1 and, therefore, that there is a cell-like map from a
topological manifold to X. It then follows from Edwards' theorem and the
disjoint disk hypothesis that X is a topological manifold.
Thus, counterexamples to Cannon's conjecture must be ENR homology
manifolds which are everywhere noneuclidean. Indeed, they must be
stably bad in the sense that if X is a counterexample to Cannon's conjecture
and Υ is any other ENR homology manifold, then Χ χ Υ is also a
counterexample to Cannon's conjecture. By contrast, when the Edwards-Quinn
theorem appeared, every known locally contractible finite-dimensional
homology manifold X had the property that XxR1 was a genuine manifold.
2. Counterexamples to Cannon's Conjecture
There are, in fact, counterexamples to Cannon's Conjecture.
Theorem 2.1 (Bryant, Ferry, Mio, and Weinberger [1]). If Mn, η > 6, is
a closed simply-connected topological manifold, then for every к £ Ζ there

6
is a finite-dimensional locally contractible homology manifold Mk homotopy
equivalent to Μ with I(Mk) = 8k + 1.
In order to give a more precise statement of this result in the simply-
connected case, we recall the definition of the homotopy structure set
Definition 2.2. (i) If Mn is a closed topological manifold, we define the
structure set S(M) to be the set of homotopy equivalences f : Ν —ϊ Μ,
where N is another closed π-manifold, subject to the equivalence relation
(TV, /) ~ (Λ/7, /') if there is a homotopy equivalence φ : N —> Ν' so that
/' ο φ is homotopic to /.
(ii) We define the homology manifold structure set SH(M) to be the
set of homotopy equivalences f : X —l· Μ where X is an n-dimensional
closed homology manifold and (A", /) ~ (X', /') if there is an s-cobordism
(Z, X, X') of homology manifolds and a map F : Ζ —ϊ Μ extending / and /'.
A classical result in surgery theory says that for Μ simply connected,
S(M) = [punc(M), G/Top], where punc(M) is a punctured copy of M. The
corresponding classification theorem for ENR homology manifolds says that
S(M) = [punc(M), G/Top X Z]. These are unpointed homotopy classes and
Quinn's index 8k +1 corresponds to к in the Z-factor.
Rationally, G/Top is just BTop and the map S(M) —>· G/Top measures
the difference between the L-polynomials of punc(M) and рипс(./У). It is
therefore reasonable to think of Quinn's index as measuring the 0th Pontr-
jagin class of a homology manifold X.
In the nonsimply connected case, the classification is by a surgery
exact sequence
...-> £η+ι(Ζττι(Μ)) -+ SH{M) -> [M,G/Top χ Ζ] -> Ьп(Хпг(М)).
For orientable Μ, we can rewrite the next-to-last term dually to obtain
... -+ Ιη+1(Ζτη(Μ)) -+ 5Я(М) -+ ЯП(М; L(e)) -+ Ln(Zm(M)).
This sequence extends to the right, so for any closed orientable homology
manifold Mn, η > 6, for which the assembly map
Ffc(M;L(e))-^Lfc(Z7r1(M))
is an isomorphism for all k) we have SH(M) — 1. This isomorphism is
conjectured to hold for all aspherical manifolds and is known (by work of
Farrell-Jones) to hold for all nonpositively curved manifolds.

7
This shows that for closed aspherical manifolds the general pattern is
"one Quinn index per homotopy type". This contrasts sharply with the
simply-connected world, where every Quinn index appears in every
homotopy type.
Question 2.3. Is there a closed aspherical ENR homology manifold X with
Quinn invariant I(X) not equal to 1?
Such an X would give a counterexample to one of two old conjectures.
Either π\{Χ) would be a Poincare Duality group which is not the
fundamental group of a closed aspherical topological manifold or X would have
the homotopy type of an aspherical polyhedron К for which the assembly
map Hn(K; L(e)) —>- Ln(Zni(K)) is not an isomorphism.
It is natural to wonder whether these new homology manifolds have
geometric properties similar to those of topological manifolds. In particular, one
wonders if they are manifolds modelled on spaces R£, where /(R£) = 8A: + 1.
If this is true, then R£+1 = EJ X R, since the homology manifold on the
right has the disjoint disk property and the correct Quinn index. This would
mean that homology manifolds with nontrivial Quinn index appear first in
some dimension between 3 and 6 and that the model spaces in higher
dimensions are products of their lower-dimensional ancestors with Kl for some
/. Here is a unified conjecture which expresses our current "(conjectural)
world view".
Conjecture 2.4 (Lost Tribes Conjecture (BFMW)). There exist spaces
R|, к £ Ζ, so that every connected ENR homology manifold Xn, η > 5,
with the DDP and I(X) = 8k + 1 is locally homeomorphic to R% χ Εη"4.
These ENR homology manifolds are classified up to homeomorphism by
Ranicki's algebraic surgery theory. In particular·, we conjecture that
high-dimensional ENR homology manifolds with the DDP are homogeneous,
that the s-cobordism theorem holds for ENR homology manifolds with the
DDP and that such homology manifolds are classified up to homeomorphism
by a surgery exact sequence
...^Hn+l(X;L)-+L°n+l(Znl(X))-+SH(X)
This conjecture consists of a guess that ENR homology manifolds with
nontrivial Quinn index begin in dimension 4 together with the closely related
conjectures that such ENR homology manifolds are homogeneous and that

8
the s-cobordism theorem holds in the category of ENR homology manifolds
with the disjoint disk property.
3. Non-ENR homology manifolds
A.N. Dranishnikov [4] has shown that there are cell-like maps f : Μ —ϊ Χ
so that X is an infinite-dimensional homology manifold with finite coho-
mological dimension but infinite covering dimension. These spaces occur
naturally as limits of topological manifolds in certain topological moduli
spaces defined by Gromov. This makes a topological characterization
desirable. We begin by stating a characterization of compact metric spaces
which are cell-like images of finite polyhedra.
Theorem 3.1. Let X be a compact metric space which is LCn and which
has cohomological dimension < n. Then X is the cell-like image of a finite
polyhedron.
Moreover, this finite polyhedron is unique in the sense that if / : Κ —ϊ Χ
and /' : К' —» X are cell-like maps, then for every ε > 0 there is a simple
homotopy equivalence φ : Κ —ϊ Kf such that /' ο φ is ε-close to /. The
situation for manifolds is, however, not quite so simple.
Theorem 3.2 (Dranishnikov-Ferry [5]). There exist high-dimensional
nonhomeomorphic closed topological manifolds Μ and M' and a space X so
that there are cell-like maps f : Μ —> X and ff:M'-+X.
For any given X there are, however, only finitely many homeomorphism
types of possible manifolds M. Since uniqueness fails, one expects a
corresponding failure of existence:
Theorem 3.3. Let X be a closed weakly locally contractible homology
manifold with finite cohomological dimension and formal dimension η > 6. Let
К be a finite polyhedron which admits a cell-like map onto X. There is a
fibration sequence of spectra
(к)
and an obstruction (a controlled version of Ranicki's total surgery obstruc-
tion) Θ(Χ) € 7rn_i<S I -I- I which vanishes if and only if X is the cell-like

9
image of a closed ENR homology η-manifold. Here, L(e) is the periodic
L-theory spectrum of the trivial group. Since К is homotopically n-dimen-
sional, кп-ХЩКМе)) - Vn-гЩК, G/TOP x Z) £ Ή(Χ, G/TOP χ Ζ).
This last uses the fact that the Vietoris-Begle theorem is true for homology
theories which are bounded below. Thus, the obstruction lives in the (n— l)st
homotopy group of the fiber of the map
ЩХ, G/TOP xZ)4 ЩХ, L(e))
and vanishes if and only if X can be resolved to a closed ANR homology
manifold.
At the moment, we can only realize some of the obstructions in this
theory. In particular, we can construct examples of homology manifolds
Xn so that the obstruction Θ(Χ) maps nontrivially to Hn-\{M\ L(e)). The
(K\
construction of examples coming from the image of Hn(X; L(e)) in S I I J
should be similar to the construction of the nonresolvable ENR homology
manifolds of Section 2, provided that there is no nontrivial lim1 term in
Hn(X; L(e)). We are uncertain as to whether nontrivial lim1 terms exist and
as to their geometric effect on our obstruction theory in case they do exist.
References
[1] J. Bryant, S. Ferry, W. Mio and S. Weinberger, The topology of homology
manifolds, Ann. Math., to appear.
[2] J. Cannon, The characterization of topological manifolds of dimension η > 5,
in: Proceedings of the ICM (1978), 449-454.
[3] R. Daverman, Decompositions of manifolds, Academic Press, 1986.
[4] A.N. Dranishnikov, On a problem of P.S. Alexandroff, Mathematics of the
USSR — Sbornik 63 (1989), 539-545.
[5] A.N. Dranishnikov and S. Ferry, Cell-like images of topological manifolds and
limits of manifolds in Gromov-Hausdorff space, Preprint.
[6] S. Ferry, Limits of polyhedra in Gromov-Hausdorff space, Preprint (available
via http://math. binghamton. edu/steve/).
[7] F. Quinn, Resolutions of homology manifolds, and the topological
characterization of manifolds, Invent. Math. 72 (1983), 267-284.
[8] F. Quinn, An obstruction to the resolution of homology manifolds, Michigan
Math. J. 301 (1987), 285-292.

TOPOLOGY and APPLICATIONS Moscow: PHASIS, 1996
International Topological Conference Pages 11—15
Dedicated to P.S. Alexandroff's 100th Birthday
Moscow, May 27-31, 1996
Cohomological dimension theory and applications
Alexander N. Dranishnikov
University of Florida, Gainesville, FL, USA
1. Alexandroff's problem
The definition of the covering dimension dim is due to Lebesgue: dim X < η
if and only if there exist arbitrarily small covers of X of order < η + 1.
We consider only compact spaces here. P.S. Alexandroff made a major
contribution to Dimension Theory. In late twenties he found a remarkable
characterization of dimension in terms of essential mappings onto n-cubes
/n. An equivalent formulation of his characterization of dimension can be
given via the following extension property: dim X < η if and only if for
every continuous map / : A —» Sn of a closed subset А С X to the n-sphere
there is a continuous extension /: X —>- 5n.
Another his contribution to Dimension Theory is the introduction of
the notion of cohomological dimension. The definition of cohomological
dimension dim^ X can also be given in terms of extensions: dimz X < τι if
and only if for every continuous map f : A —l· K(Z,n) of a closed subset
А С X to the Eilenberg-MacLane space there is a continuous extension.
Alexandroff's theorem. The cohomological dimension dim^X agrees
with the covering dimension dim X for compact metric spaces provided that
dim X < oo.
This theorem was proven in early thirties and the formulation was
different since there was no notion of cohomology that time. From then, there
was an open problem:
Alexandroff's problem. Is it true that dim X = dim^ X for all compact
metric spaces ΧΊ

12
2. Bockstein-Boltjanskij's problem
At the beginning RS. AlexandrofF was too optimistic about cohomologi-
cal dimension. He assumed that the cohomological dimension did not
depend on the choice of coefficients. In 1930 L.S. Pontrjagin constructed
his 2-dimensional "surfaces" Up having the property dim Up X Uq = 3 for
(p,q) = 1. The rational dimension dimQUp of his surfaces equals one.
So, coefficients came into the picture and in thirties Alexandroff posed a
problem about the existence of a countable basis of Abelian groups for the
cohomological dimension. This problem was solved by F.M. Bockstein who
proved the following
Bockstein's theorem. Let σ be the family of Abelian groups consisting
of the rationals Q, localizations Z(p), cyclic groups of prime order Zp and
Zpoo = DirLimZp*. Then the dimension dim#X with respect to any
Abelian group Η can be computed via dimensions dim^X, G 6 σ.
For example, for Η = Ζ Bockstein's algorithm gives the following
formula: dim^X = max{dimz(p) X}.
There are relations between cohomological dimensions with respect to
coefficient groups from Bockstein's basis σ which were discovered by
Bockstein in terms of inequalities (Bockstein's inequalities). Here we
formulate the relations in a different way. For every prime number ρ a com-
pactum X can be p-regular or p-singular. In the first case we have
dimz(p) X = a\m%pX = a\mip00 X = dim<Q)A\ In the second case
dimz(p) X = max{dimQ X, dim^poo X + 1} and dimzp00 X equals dim^p X
or dimzp X — 1.
Realization Problem. Let {uq : G 6 σ} be a set of natural numbers
satisfying the above conditions. Does there exist a compactum X with
dimG X — uq for all G £ σ?
3. Cell-like maps
A map between compacta / : X —» Υ is called cell-like if the preimage
f~l of each point has trivial shape. In other words, f~l is cell-like, i.e.
it can be embedded into En or into the Hilbert cube as the intersection
of a nested sequence of topological cells. In dimension greater than 3 the
Siebenmann-Quinn theorem states that a cell-like map / : Μ —)> N between

13
closed manifolds forces the manifolds to be homeomorphic. R. Edwards has
significantly generalized the above theorem. In his assumption he did not
use that the image TV was a manifold. He assumed only that N had the
disjoint disk property and dim N < oo. A natural question arose: is always
dim N < oo? This question is equivalent to the following
Cell-like Mapping Problem. Can cell-like maps between compacta raise
the dimension?
The Cell-like Mapping Problem first time appeared in works of
R.H. Bing.
In seventies R. Edwards proved (see [10]) that the Cell-like Mapping
Problem is equivalent to Alexandroff 's problem.
4. Solutions
The Cell-like Mapping Problem was solved in [3] by a counter-example for all
η-dimensional manifolds, η > 7, and later J. Dydak and J. Walsh extended
that for η > 5. Recently we proved the following
Theorem 1. There is a cell-like map f : S7 —ϊ Χ of the 7-dimensional
sphere such that X does not admit a map of degree one onto 57.
Due to the standard lifting property of cell-like maps it follows that
dim X = oo in Theorem 1. In fact our map / kills some element in homology
A'-theory.
The Realization Problem was solved positively in [2].
5. Applications
There are various applications of Dimension Theory to different areas of
Topology. Here we discuss some of applications related to the Novikov
Conjecture.
Definition 1. An open η-manifold Μ is called hyperspherical if it admits
a Lipschitz map / : Μ —ϊ Rn of degree one onto the Euclidean space.
The Gromov-Lawson-Rosenberg conjecture states that an aspherical
manifold cannot carry a metric of a positive scalar curvature. This
conjecture is a partial case of the Novikov Conjecture about the homotopy

14
invariance of the higher signatures. Gromov and Lawson [8] proved that if
the universal covering space Μ for an aspherical manifold N is hyperspher-
ical (for the metric induced from N) then the Gromov-Lawson-Rosenberg
conjecture holds for N. The universal cover of an aspherical manifold N is
contractible. If N is a closed manifold then the universal cover is uniformly
contractible, i.e. for every R > 0 there is S > 0 such that every ball J5(x, R)
of radius R is contractible to a point in the ball B(x,S) of radius S. In
view of Gromov-Lawson's result it is natural to conjecture [7] that every
uniformly contractible manifold is hyperspherical.
It turns out that it is not the case. Using Theorem 1 we have
constructed [5] a uniformly contractible Riemannian metric on R8 which is not
hyperspherical. To construct that we took a geodesic uniformly 8-connected
metric on the open cone OX over X. Then using the cell-like map cone(/) :
R8 —> OX we constructed a Riemannian metric on R8 which is on a finite
distance from OX in the Gromov-Hausdorff space. It implies that OX and
the metric on R8 have the same Higson corona. The Higson corona vOX of
the cone space OX is closely related to X. It is possible to show that, like
X, the space vOX does not admit a map of degree one onto S7. Then the
following theorem of J. Roe complete the argument.
Roe's theorem. An η-manifold Μ is hyperspherical if and only if the
Higson corona vM admits a map of degree one onto 5n_1.
By definition the Higson corona vM is the remainder of the compactifi-
cation of Μ generated by the algebra consisting of all bounded functions /
on Μ with the property lim diam /(J3(x, R)) = 0 for every R > 0.
x—>oo
According to Gromov, the asymptotic dimension asdim Μ of a metric
space Μ is the smallest number η such that for every L > 0 there exists an
open cover of Μ by uniformly bounded sets with the Lebesgue number > L.
Theorem 2 ([6]). For every proper metric space Μ the inequality
dim vM < asdim Μ holds.
We recall that a metric space is called proper if the closure of every
bounded set is compact.
Conjecture 1. Let Γ be a group with finite complex /f (Γ, 1) supplied with
a word metric, then dim vY < oo.
Conjecture 2. //dim vY < oo for an above Γ then the Novikov Conjecture
holds for Γ.

15
Clearly Conjectures 1, 2 imply the Novikov Conjecture. Conjecture 2 is
more plausible because of the following theorem of Yu.
Theorem 3 ([11]). 7/asdim Γ < oo for an above Γ then the Novikov Con-
jecture holds for Γ.
References
[1] V.I. Kuzminov, Homological dimension theory, Russian Math. Surveys (1968).
[2] A.N. Dranishnikov, Homological dimension theory, Russian Math. Surveys 43
(1988), 11-63.
[3] A.N. Dranishnikov, On the problem of P.S. Alexandroff, Mat. Sbornik 135
(1988).
[4] A.N. Dranishnikov and S. Ferry, Cell-like images of topological manifolds and
limits in Gromov-Hausdorff space. Preprint, 1994.
[5] A.N. Dranishnikov, S. Ferry and S. Weinberger, Large Riemanman manifolds
which are flexible, Preprint, 1994.
[6] A.N. Dranishnikov. J. Keesling and V.V. Uspenskii, On the Higson corona,
Preprint, 1996.
[7] M. Gromov, Large Riemanman manifolds, Lecture Notes in Mathematics
1201 (1985), 108-122.
[8] M. Gromov and H.B. Lawson, Positive scalar curvature and the Dirac
operator, Publ. IHES 58 (1983), 83-196.
[9] J. Roe, Coarse cohomology and index theory for complete Riemanman
manifolds, Memoirs Amer. Math. Soc. 497 (1993).
[10] J.J. Walsh, Dimension, cohomological dimension, and cell-like mappings,
Springer-Verlag, 1981, SLN 870, 105-118.
[11] G. Yu, The Novikov Conjecture and groups with finite asymptotic dimension,
Preprint, 1995.

TOPOLOGY and APPLICATIONS Moscow: PHASIS, 1996
International Topological Conference Pages 17-24
Dedicated to P. S. Alexandroff 's 100th Birthday
Moscow, May 27-31, 1996
Algorithmic classification of sufficiently large
3-manifolds
Sergei V. Matveev
Chelyabinsk State University, Chelyabinsk, Russia
1. Introduction
The aim of the talk is to sketch a modified proof of the following Haken-
Waldhausen-Johannson-Hemion theorem [3, 11, 6, 4].
Theorem 1· There is an algorithm to decide whether two given Haken
3-manifolds are homeomorphic or not
Recall that a compact 3-manifold Μ is called Haken if:
1. Μ is irreducible (every 2-sphere in Μ bounds a ball);
2. Μ is boundary irreducible (every proper 2-disc in Μ cuts off a ball);
3. Μ is sufficiently large (there is a proper incompressible surface
F φ 52, D2 in M; recall that a two-sided surface FcMis
incompressible if the kernel of the induced homomorphism i* : n\(F) —> πχ(Μ)
is trivial).
Let us explain the reasons why Theorem 1 should be considered as a
classification theorem for Haken manifolds. There is an algorithm that
enumerates all compact 3-manifolds (possibly with duplicates). For
example, one can successively enumerate all finite 3-dimensional simplicial
complexes and select manifolds by checking whether the links of all vertices
are spheres and discs or not. The algorithm can be modified for
producing Haken manifolds only. There are two problems here: how to recognize
irreducibility of a manifold, and how to check algorithmically whether the
manifold is sufficiently large. The solution of the first problem is based on
Rubinstein-Thompson algorithm for recognition of S3 [9, 8]. The second
problem was solved in [5].

18
Having constructed the enumeration algorithm for Haken manifolds, we
can apply Theorem 1 for creating a list Mi, M2, ... of all Haken
manifolds without duplicates by inquiring if each next manifold has been listed
before. It is the list that is considered as classifying list of Haken
manifolds. Certainly, this is a classification in a very weak sense; the existence of
the classifying list does not help to answer questions on Haken manifolds.
It is the proof of Theorem 1 that allows one to grasp the intrinsic
structure of Haken manifolds. Remark that the recognition algorithm for Haken
manifolds is highly unefficient like all other algorithms based on Haken's
theory. The only practical recognition algorithm for a class of 3-manifolds
was suggested in [1].
2. Simple polyhedra and simple skeletons
Definition 1. A compact 2-dimensional polyhedron Ρ is called simple if
the link of any point of Ρ is homeomorphic to a circle, a circle with a
diameter, or a circle with three radii.
Simple polyhedra are known also as fake surfaces or generic polyhedra
and, under minor additional assumptions, as standard or special polyhedra.
The set of singular points (i.e. vertices and triple lines) of a simple
polyhedron Ρ is called a singular graph of Ρ and denoted by SP. Connected
components of Ρ — SP are called 2-components of P.
Definition 2. A simple subpolyhedron Ρ of a 3-manifold Μ is called a
simple skeleton of Μ if Μ — Ρ is a collection of open 3-balls and any
2-component of Ρ is a 2-cell.
Particularly, if dM φ 0, then Ρ must contain dM.
Proposition 1. Let Рь Р2 be simple subpolyhedra of 3-manifolds Μχ, M2
respectively. Assume that the boundaries of Μχ, M2 contain no 2-spheres.
Then any homeomorphism h : Ρχ —» P2 can be extended to a homeomor-
phism Η : M\ -* M2.
PROOF. A circle with a diameter as well as a circle with three radii can
be embedded into 52 in a unique way up to homeomorphisms of S2. It
follows that h can be extended to a homeomorphism #' between regular
neighborhoods of the singular graphs. Since the rest of Ρ and the rest of
Μ — Ρ consist of 2- and 3-cells, one can extend Hf to a homeomorphism
between M\ and M2. D

19
Definition 3. A simple subpolyhedron Ρ of a 3-manifold Μ is called ad-
missible if every 2-component α of Ρ is injective and either а С дМ or a
separates two different components of Μ — P. The closures of connected
components of Μ — Ρ are called 3-components of Μ — P.
Note that each 3-component Q of Μ — Ρ is a compact 3-manifold such
thatQnP = dQ.
Let us describe a general way for constructing admissible subpolyhedra
of manifolds. Let Ρ С Μ be an admissible subpolyhedron, and let F С М
be an embedded surface. Suppose that F Π Ρ = #F and dF is in general
position with respect to 5P, that is dF contains no vertices of Ρ and
intersects the edges transversally. If F does not decompose the 3-component of
Μ - Ρ it is contained in, replace F by the boundary of its relative regular
neighborhood. Then PUP is a simple admissible subpolyhedron of M.
3. Proof of Theorem 1 (modulo extension moves)
Let Ρ be an admissible subpolyhedron of M. Below we will describe a
sort of transformations of Ρ called extension moves. Each extension move
transforms Ρ to another admissible subpolyhedron P\ С М. The following
properties should be satisfied:
1. The number of different extensions of Ρ is finite up to homeomor-
phisms of the pair (Μ, Ρ);
2. There is an algorithm to construct all possible extensions of P;
3. Any sequence Ρ С P\ С Рг..., where each Рг+1 is an extension of Рг,
is finite;
4. If Ρ has no extensions, then Ρ is a simple skeleton.
Theorem 2. There is an algorithm that assigns to any Haken Ъ-manijold
Μ a finite set V(M) of simple skeletons of Μ such that Haken manifolds
M\, M2 are homeomorphic if and only if a skeleton Pi £ V(Mi) is homeo-
morphic to a skeleton P2 G P(M2).
Proof. (Under assumption that extension moves have been described.)
Suppose Μ is given. Denote by Po the boundary of Μ if дМ ф 0 and an
injective surface in Μ if Μ is closed. Let us apply to P0 all extension moves
while it is possible. Properties (3) and (2) ensure us that this branched

20
process stops and that it is algorithmic. It follows from conditions (1) and
(4) that we get a finite set V(M) of simple skeletons.
Suppose now that Μχ, M2 are two Haken manifolds. According to
property (1), the result of our branched process depends only on the homeomor-
phism type of M. Therefore V{M\) and V(M2) should consist actually of
the same polyhedra. On the other hand, if V(M\) and V(M2) contain at
least one pair of homeomorphic polyhedra, then M\ and M2 are homeomor-
phic by Proposition 1. D
Note that Theorem 2 reduces the recognition problem for Haken
manifolds to the corresponding problem for 2-dimensional polyhedra and that
the latter admits an evident algorithmic solution. Therefore Theorem 2
implies Theorem 1.
4. Extension moves
Let Ρ be an admissible simple subpolyhedron of Μ and Q a 3-component
of Μ - P.
4.1. Addition of an incompressible torus
Move E\\ Suppose that there is a non-boundary parallel incompressible
torus T2 С IntQ. Then we replace Ρ by PUT2.
4.2. Addition of a longitudinal annulus
A proper annulus A in Q is called clean if A is incompressible and
dAC)SP = 0.
Definition 4. A non-boundary parallel clean annulus А С Q is called
longitudinal if any other non-boundary parallel clean annulus A\ С Q can be
isotoped mod SP so that afterwards дА П дА\ = 0. Otherwise A is called
transverse.
Move E2: Suppose that Q contains an longitudinal annulus A. Then we
replace Ρ by PU A.
4.3. Addition of an annular belt
Definition 5. A clean annulus А С Q is called a non-trivial annular belt if
A is boundary parallel and the annulus bounded by dA in dQ contains at
least one vertex of SP.

21
Move E3: Suppose that Q φ D2 χ Sl, all clean annuli in Q are boundary
parallel, and there exists a non-trivial annular belt А С Q. Then we replace
Ρ by PUA.
4.4. Addition of a minimal non-trivial disc
Move E\\ Suppose that Q is reducible, not homeomorphic to D2 Χ 51,
and all clean annuli in Q are trivial belts. Then we replace Ρ by Ρ U D,
where D is a non-trivial disc in Q such that the number of points in DOS Ρ
is minimal.
4.5. Addition of a minimal incompressible surface
Move E\\ Suppose that Q is irreducible, all clean annuli in Q are trivial
belts, and there is a proper surface F С Q such that F is incompressible and
dF determines a non-trivial element of H\(dQ). Among all such surfaces
choose a surface F0 such that the complexity w{Fq) = #(dFoC\SP) — χ(Ρο)
takes the smallest possible value (here #(dFoC\SP) is the number of points
in dF0DSP and χ(Ρο) is the Euler characteristic. Replace Ρ by PUF0-
4.6. Subdivision of an /-bundle
Move E$\ Let iV = Fx/cMbean /-bundle over a surface such that
the following holds:
1. N C)P = dNU Fx K, where К is a finite set in Int /;
2. NDdM = FxdI.
Let Pjv be a simple skeleton of N having the smallest possible number
of vertices. Then we replace Ρ by (P-Int F X K){JPn.
4.7. Subdivision of a Stallings manifold
Move E5: Let N — Sl X F С М be a Stallings manifold, that is a surface
bundle over the circle. Suppose that Ν Π Ρ = 97V U К X F, where /i is
a finite set in 51. Let Pn be a simple skeleton of N having the smallest
possible number of vertices. Then we replace Ρ by (P-K χ Int F)UPyv.
4.8. Subdivision of a quasi-Stallings manifold
Let α, β : F —l· F be two free involutions on a surface. Denote
by ~ the equivalence relation on F X / generated by equalities (я,1) =

22
(α(χ),1) and (χ,Ο) = (/?(x),0). Then the manifold (F X /)/„ is called a
quasi-Stallings manifold.
Move E6: Let N С Μ be a quasi-Stallings manifold. Suppose that NOP =
ON U (F X K)/~, where /f is a finite set in /. Let P^v be a simple skeleton
of N having the smallest possible number of vertices. Then we replace Ρ
hy(P-{lntFxK)/„)[JPN.
4.9. Addition of meridional discs for genus one handlebodies
Move Εγ: Suppose that all 3-components of Μ - Ρ are 3-balls and genus
one handlebodies. Then we add to Ρ meridional discs of the handlebodies.
Some care is needed to make this process canonical up to a finite number
of possibilities. (We omit details.)
5. Why do extension moves possess the properties (1)—(4)?
The main instrument here is the normal surface theory of W. Haken [2]. Let
ξ be a handle decomposition for a 3-manifold M. Recall that a surface F in
Μ is normal (with respect to ξ) if, roughly speaking, F intersects all
handles in a very nice way. Each incompressible and boundary incompressible
surface in Μ is isotopic to a normal one. A key result of Haken's theory
can be formulated as follows:
There is a finite set of fundamental normal surfaces in Μ such that
any normal surface can be presented as a geometric sum of fundamental
ones. Moreover, the set of fundamental surfaces can be constructed algo-
rithmically.
Proposition 2. For any Haken manifold Μ there exists an algorithmically
computable number n(M) such that any set of disjoint non-parallel
incompressible and boundary incompressible surfaces in Μ consists of no more
than n(M) surfaces.
Proof. Let Τ = {Fi,..., F&} be a set of disjoint non-parallel
incompressible and boundary incompressible surfaces in M. We can assume that all
surfaces are normal. They decompose every index 1 handle onto "paral-
lelity" and "non-parallelity" regions. The number of non-parallelity regions
is less than the valency of the handle. Since every component of Μ - Ц' Fi
must contain at least one non-parallelity region, one can easily find an upper
bound for k. D

23
Proposition 3. For any Haken manifold Μ and for any number g there
exists an algorithmically constructible finite set F\,.. .,Fk of surfaces in Μ
such that every incompressible and boundary incompressible genus g surface
in Μ can be transformed to some F{ by a homeomorphism of Μ isotopic to
a superposition of twists along incompressible annuli and tori.
An idea OF the proof. It follows from Haken's theory that any
incompressible and boundary incompressible surface in Μ can be isotoped
into a regular neighborhood N of the union of some fundamental surfaces
Si,...,5m such that Ц 5« contains no triple points. Since N has a very
simple structure, the proof of the theorem for N instead Μ is easy. The
set of all fundamental surfaces is finite, so we have only a finite number of
different N. This finishes the proof. D
It is easy to see that the properties (1), (2) and (3) for the moves E\
and £?2 follow from Propositions 2 and 3. For the move E$ these properties
are evident. The properties (1) and (2) for the moves £4, E5 also follow
from Propositions 2 and 3, since we apply the moves only under assumption
that there are no non-trivial annuli and tori in Q. Note also that the moves
£4, E5 decrease the Waldhausen complexity (see [10]) of Q. It follows that
the property (3) for them also holds.
There are no problems with the move E5. To get the desired properties
for the move J56, it is sufficient to use Hemion's solution of conjugacy
problem for surface homeomorphisms [4]. The solution provides an algorithm to
decide whether or not two Stallings manifolds are homeomorphic, and an
algorithm to enumerate the (finite up to an isotopy) set of all autohomeo-
morphisms of a Stallings manifold.
The situation with th>e move Εγ is more complicated. One can reduce
the problem to the following theorem:
There is an algorithm that for given autohomeomorphisms w, w : F —l· F
of a compact surface determines whether un is isotopic to w for some
η or not.
One can also reduce the problem to Thurston's hyperbolization theorem
for sufficiently large 3-manifolds or to Thurston's classification theorem for
surface homeomorphisms. I do not know whether proofs of the theorems
have been published in any form.
In conclusion, we explain why after applying all possible extension moves
we get a simple skeleton. Assume that a simple polyhedron Ρ С М does
not admit any extension move. Suppose that a 3-component Q of Ρ is not a
ball or a genus 1 handlebody. Then Q does not contain incompressible tori

24
and longitudinal annuli and contains a transverse annulus, since otherwise
we could apply either the moves Εϊ) E2 or the moves E3) E4. It follows
that Q is homeomorphic to an /-bundle over a surface, and, since we can
not apply the move £5, Q lies in the interior of M. Unions of such bundles
form Stallings or quasi-Stallings manifolds, which is impossible since we can
not apply the moves Ее and Εγ.
It follows that all 3-components of Μ - Ρ should be balls or genus 1
handlebodies, but all genus 1 handlebodies had been killed by the move Εγ.
References
[1] A. Fomenko, V. Kusnezov and I. Volodin, On the algorithmic recognition
problem for the standard 3-dimensional sphere, Uspekhi Mat. Nauk 29(5) (1974),
71-168 (in Russian).
[2] W. Haken, Theorie der Normalflachen. Em Isotopiekriterium fur der
Kreisknoten, Acta Math. 105 (1961), 245-375.
[3] W. Haken, Uber das Homoomorphieproblem der 3-Mannigfaltigkeiten, I. Math.
Z. 80 (1962), 89-120.
[4] G. Hemion, On the classification of homeomorphisms of 2-manifolds and the
classification of 3-manifolds, Acta Math. 142 (1979), 123-155.
[5] W. Jaco and U. Oertel, An algorithm to decide if a Ъ-manifold is a Haken
manifold, Topology 23 (1984), 195-209.
[6] K. Johannson, Topologie und Geometrie von 3-Mannigfaltigkeiten, Jahresber.
Deutsch. Math.-Ver. 86 (1984), 37-68.
[7] K. Johannson, Topology and combinatorics of Ъ-mamfolds, Springer LNM
1599, 1995.
[8] S. Matveev, A recognition algorithm for the Z-dimensional sphere [after
A. Thompson), Mat. Sbornik 186(5) (1995), 69-84 (in Russian).
[9] A. Thompson, Thin position and the recognition problem for S3, Preprint,
1994.
[10] F. Waldhausen, On irreducible Z-manifolds which are sufficiently large, Ann.
Math. 87(2) (1968), 56-88.
[11] F. Waldhausen, Recent results on sufficiently large Ъ-manifolds, Proc. Symp.
in Pure Math., Amer. Math. Soc. 32 (1978), 21-38.

TOPOLOGY and APPLICATIONS Moscow: PHASIS, 1996
International Topological Conference Pages 25-26
Dedicated to P.S. Alexandroff's 100th Birthday
Moscow, May 27-31, 1996
Foundations of algebraic homotopy theory
and dimension theory
Anthony Bak
University of Bielefeld, Bielefeld, Germany
For a century, the notion of homotopy had been confined to the realm of
topological spaces or abstractions thereof such as simplicial complexes. This
talk describes notions of homotopy and homotopy groups for certain
algebraic objects and establishes fundamental results for these objects such as
infinite relative exact sequences of homotopy groups. Even though the
foundational results in the algebraic setting parallel their topological
counterparts, the algebraic constructions differ profoundly from the corresponding
topological ones in several ways. One important difference is that all higher
homotopy groups can be non-abelian. Certain algebraic objects called
standard objects have only abelian higher homotopy groups and the subcategory
they define is adjoint to the category of topological spaces. Thus algebraic
homotopy theory can be viewed as a non-abelian generalization of
topological homotopy theory. An important application of the algebraic theory is
an algebraic construction of all higher Volodin algebraic A'-theory groups.
This construction provides complete justification of the expression algebraic
A'-theory.
The algebraic objects which form the basis of algebraic homotopy theory
are constructed in 2 steps. First we fit together according to a few simple
rules group actions of usually distinct groups acting on usually distinct sets
to form a new object called a global action. A global action is in spirit
similar to that of a sheaf of groups, but the underlying sets are not related
to open sets of topological space. The second step is the crucial one and
defines the notion of a complex of global actions. The homotopy groups of a
complex are constructed using the local group actions of the global actions
making up the complex. Thus homotopy groups in the algebraic setting
are internally defined and do not depend as in the topological setting on
certain external objects, namely spheres, for their definition. This is another
significant difference between the algebraic and topological theories.

As in topological homotopy theory, one wants to prove results
concerning the basic objects of the theory and concerning functors taking values
in the basic objects of the theory. The latter arises for example in
algebraic A'-theory. Here one considers functors from various categories such as
rings or schemes to topological spaces or complexes of global actions and
defines A'-groups of source objects as homotopy groups of target objects.
In either the traditional or algebraic setting of homotopy theory, it would
be useful for tackling problems above to have a general theory of dimension
which provides a uniform conceptualization of notions of dimension found
in diverse mathematical specialties, such as Krull dimension in ring theory
and CW-dimension in topology. The last part of the talk describes such a
dimension theory in terms of concepts of structure and infrastructure for
arbitrary categories. As an application of the theory, we obtain a
sweeping generalization of the Eilenberg-Steenrod axioms for homology functors
to very general systems of (not necessarily abelian) group valued functors
defined on categories with dimension. Examples of such systems of group
valued functors include the A'-groups of various algebraic A'-theories.

TOPOLOGY and APPLICATIONS Moscow: PHASIS, 1996
International Topological Conference Pages 27-33
Dedicated to P.S. Alexandroff's 100th Birthday
Moscow, May 27-31, 1996
Fomenko's invariants in the theory of integrable
Hamiltonian systems
Alexei V. Bolsinov
Lomonosov Moscow State University, Moscow, Russia
The purpose of this article is to present some recent results in topology of
integrable Hamiltonian systems which are based on the approach suggested
by Fomenko about ten years ago [3, 4] and allowing to describe topological
properties of such systems.
We consider an even-dimensional symplectic manifold (Μ2η,ω) and the
Hamiltonian system on it generated by a smooth Hamiltonian Η : Μ —ϊ R
where Хц is uniquely defined by u>(X#, v) = v(H) for any vector field v.
Definition 1. The system (X#, M2n) is called Liouville integrable if it
admits η pairwise commuting integrals /i, /2, ... , fn which are functionally
independent almost everywhere on M2n.
Liouville theorem. If a common level surface L = {f\ = c\, ... , fn = cn}
is regular, compact and connected then L is an η-dimensional torus (called
a Liouville torus) and, moreover, in some neighborhood U(L) = Tn χ Dn
there exists a canonical coordinate system (φ\)..., φη, s\,..., sn) such that
where a{ = a{{s\%..., sn).
As a result on the manifold there appears a very interesting structure,
namely, the structure of the so-called Liouville foliation С whose leaves,
by definition, are connected components of common level surfaces of the
integrals /i, ... , /n.

28
Remark 1. We will assume that the momentum mapping
F=(/b...,/n):M2n-+Rn
is proper (in the sense that the preimage of a compact set is compact).
Thus, almost all leaves of the Liouville foliation are Liouville tori, but
there exist some singular leaves corresponding to the critical values of F.
In terms of this foliation we can say that the Liouville theorem completely
describes its structure near non-singular leaves.
One of the most important problems in topology of integrable Hamilto-
nian systems is to describe the structure of Liouville foliations and classify
them up to the usual equivalence.
Definition 2. Two Liouville foliations C\ and £2 corresponding to
integrable systems (Χ#η M\) and (X#2, M2) respectively are called isomorphic
if there exists a fiberwise homeomorphism (or a diffeomorphism, it depends
on what we want) ξ : M\ —l· M2. In this situation we also say that the
systems (Χ#ηΜι) and (X#2,M2) are topologically equivalent
Remark 2. In this definition instead of the whole symplectic manifolds M\
and M2 we may consider their parts invariant with respect to the Hamilto-
nian flows, for example, isoenergy surfaces Qi = {Нг = const} or saturated
neighborhoods of singular values.
There is another problem (and maybe even more interesting), namely
the problem of topological obstructions to integrability. In other words,
the question is what symplectic manifolds (or isoenergy surfaces) admit
an integrable Hamiltonian system? (To be more precise, we are speaking
about integrable systems with nice integrals in the sense that the structure
of singularities of the momentum mapping F is not too complicated.)
Of course, to begin working with these general questions we should first
of all describe the structure of singularities of Liouville foliations. Questions
of this kind arose in many papers dealing both with pure mathematical
things and applications. In particular, I would like to mention the famous
paper "Topology and Mechanics" by S. Smale [11] and M.P. Kharlamov's
works (see [7]) where he investigated the properties of Liouville foliations
for classical integrable cases in rigid body dynamics.

29
A new approach (and, I would say, a new language) allowing to describe
and classify Liouville foliations was suggested by Fomenko in 1985 [3, 4, 5].
In particular, two problems were solved.
1. Description of non-degenerate singularities of codimension one (semi-
local classification).
2. Global classification of Liouville foliations on isoenergy surfaces for
non-degenerate integrable Hamiltonian systems with two degrees of
freedom.
To explain the main points of this approach, we begin with a very
simple example. Consider a Hamiltonian system with one degree of freedom.
In this case the structure of the Liouville foliation on M2 is given by a
smooth function / : M2 —>· Ε which is the Hamiltonian of the system.
More precisely, the Liouville foliation is just the foliation into level lines of
/. Non-degeneracy in this case means that / is a Morse function.
Consider a regular saturated neighborhood P2 = U(L) of a singular level
L of this foliation. It is clear that in general case a singular level can be of
two types, either an isolated point or a graph with vertices of degree four
(in the simplest case this graph is just an eight-figure).
Definition 3. The neighborhood P2 of a singular leaf L with the structure
of the Liouville foliation on it is called an atom.
There are several methods allowing to classify and, in principal, to obtain
the complete list of atoms (see [2, 10]).
Consider now non-degenerate singularities of codimension one.
"Codimension one" means that such a singularity has points for which
rkciF = η — 1, but has no points for which rkdF < n — 1.
It was shown by A.T. Fomenko that the semi-local structure of a singular
leaf can be of two types. The first type is just a direct product. To describe
it, consider an atom P2 and a trivial Liouville foliation of dimension η — 1,
that is, Tn~l χ Dn~l. Then the direct product M2n = Ρ2 χ Tn~l X Dn~l
has a natural structure of an η-dimensional Liouville foliation. The second
type (called almost direct product) can be obtained in the following way.
Let us assume that an atom P2 admits an involution τ : Ρ2 —ϊ Ρ2 which
preserves the Liouville foliation structure. Then this involution can be
extended to the direct product M2n = Ρ2 χ Τη_1 χ Dn~l by the formula
Ф, ¥>ь · · ·, ¥>η-ι, *i, · · ·, sn-i) = (r(x), φι + тг, φ2,..., <ρΛ-ι, *ι, · · ·, *η-ι),

30
where χ e Ρ2, {<pu · · ·> ¥>η-ι) € Τ71""1, (*i,...,sn) € Ζ?"-1. It is easy to
see that τ is an involution without fixed points and preserves the Liouville
foliation structure. Thus the quotient space M2n = M2n/f obtains a
natural structure of a Liouville foliation (of almost direct product type). As a
result, the classification of non-degenerate singularities of codimension one
is, in essence, the classification of atoms.
Consider now the case of two degrees of freedom. Let Q3 = {H = const}
be a regular isoenergy surface of an integrable Hamiltonian system Хц on
(Μ4,ω), and С the corresponding Liouville foliation on it. The problem of
classification of such foliations was solved in [2, 10]. The complete invariant
allowing to do it is the so-called marked molecule (or Fomenko-Zieschang
invariant). The molecule W* = W*(Q3,C) can be considered as a graph,
namely, the Reeb graph (or the base) of the foliation. The vertices of the
molecule (atoms) correspond to the singular levels. In other words, for
every singular level we indicate its semi-local structure. This invariant also
includes some numerical marks which show how the molecule should be
glued from single atoms.
There appears a natural question whether it is possible to apply the
same invariants (i.e., atoms and molecules) to describe and classify more
complicated singularities (degenerate ones or singularities of arbitrary
codimension). It turns out that this can be done.
To explain the main idea consider the case of two degrees of freedom. Let
Η be the Hamiltonian of an integrable Hamiltonian system, / an additional
integral and F = (#, /) : M4 —»· R2 the corresponding momentum mapping.
Consider the set of critical points of F
K = {x eM4 :rkdF(x) < 2}
and the bifurcation diagram Σ = F(K) С Е2. Usually the bifurcation
diagram Σ is a union of smooth curves (which correspond to one-parameter
families of non-degenerate singularities of codimension one) and some
singular points of the bifurcation diagram (which correspond to more complicated
singularities).
Definition 4. A singular point j/ G Σ С R2 is called isolated if for any
sufficiently small ε > 0 the circle y€ with the center у and radius ε intersects
Σ transversally at non-singular points.
Consider the preimage Q3e = F~~l(ye) and the corresponding marked
molecule W*(y) which describes the Liouville foliation structure on Q3e. It
is easily seen that W*(y) does not depend on ε.

31
Definition 5. W*(y) is called a circle molecule (associated with the
singularity of С corresponding to the isolated singular point у 6 Σ).
Fomenko's conjecture. The circle molecule of a singularity is its
complete topological invariant.
This construction can be generalized to the case of many degrees of
freedom. The molecule in this case should be replaced by the so-called marked
net showing the types of singularities of codimension one in a neighborhood
of a singular point of arbitrary codimension.
First general results describing non-degenerate singularities of
codimension two were obtained by Lerman and Umanskii [8]. Then we tried to look
at these singularities following Fomenko's approach. As a result we have
obtained the classification of non-degenerate singularities for two degrees
of freedom (center-center, center-saddle, and focus type) and a complete
list of saddle-saddle singularities of complexity one and two (the complexity
is the number of singular points in a singular level) [1] and calculated the
corresponding circle molecules for these singularities (V.S. Matveev [9]).
It turns out that Fomenko's conjecture is valid for these cases. Recently
V.S. Matveev has shown that the restriction to the complexity of a
singularity is not important.
Theorem 1. In the case of two degrees of freedom Fomenko's
conjecture is valid for non-degenerate singularities of codimension two (that is,
center-center, center-saddle, saddle-saddle and focus type singularities).
Moreover for systems with two degrees of freedom this conjecture turns
out to be true for degenerate topologically stable singularities. The list of
such singularities has been recently obtained by V. Kalashnikov [6].
Topological stability means the following.
Let С be the Liouville foliation corresponding to a pair of commuting
functions Η and /. Let #e, fe be a smooth family of commuting functions
(in other words, a smooth perturbation of the initial system). For every ε
we have, consequently, an integrable Hamiltonian system with Hamiltonian
H€ and additional integral fe) and can consider the corresponding Liouville
foliation Ce. The singularity of С = Со is called topologically stable if it
does not change its topological type under such perturbations, that is, Ce
is isomorphic to С for small ε.
Theorem 2. In the case of two degrees of freedom Fomenko's conjecture is
valid for topologically stable singularities.

32
Consider finally the case of η degrees of freedom and multidimensional
singularities of Liouville foliations. How can we construct them? The
simplest way to do it is just to consider the direct product of the simplest
singularities, i.e., atoms. We also can, of course, construct
multidimensional singularities of almost direct product type. To do this consider the
direct product of several atoms Μ = P\ Χ Ρ2 X ... X Pk and assume that on
this direct product there is an action of a finite group G which is
1) free,
2) symplectic,
3) component-wise.
In addition we assume that the action preserves the Liouville foliation
structure on each component. If all these conditions are satisfied then the
quotient space Μ = M/G is a symplectic manifold with the natural
structure of a Liouville foliation C. Let us note that in this construction among
the atoms Pi, ... , Pn there can be trivial atoms (without singularities), that
is, 51 X D1, and some new four-dimensional atoms corresponding to focus
type singularities. It is clear that Μ is a regular neighborhood of a singular
level of С We say that such singularities are of almost direct product type.
The following theorem by N.T. Zung [12] can be considered as a topological
singular analog of the classical Liouville theorem.
Theorem (N.T. Zung). Any non-degenerate singularity is isomorphic to a
singularity of almost direct product type.
Remark 3. In this statement "isomorphic" means isomorphic in the sense
of Definition 2. In other words, such a decomposition into atoms is not
necessarily symplectic.
Thus, this theorem shows that, from the viewpoint of Liouville foliations,
the structure of non-degenerate singularities can be completely described in
very simple terms, namely, in terms of atoms.
In [12] N.T. Zung also announced a corollary to his theorem which can
be reformulated as follows.
Corollary· Fomenho's conjecture is valid for non-degenerate
multidimensional singularities of Liouville foliations.
An open question is whether Fomenko's conjecture remains true for topo-
logically stable multidimensional singularities.

33
In conclusion, we would like to explain how Fomenko's conjecture can
be applied to specific integrable systems. In applications sometimes it is not
clear how to describe singularities of codimension greater than one. On the
other hand, usually it is much easier to find out and describe the
codimension of singularities. If we can do it, then Fomenko's conjecture states that
we already know in essence everything about the Liouville foliation and, in
particular, can reconstruct the structure of the other singularities.
References
[1] A.V. Bolsinov, Methods of calculation of the Fomenko-Zieschang invariant,
in: Topological classification of integrable systems, Advances in Soviet Math.
6 (A.T. Fomenko, ed.), Amer. Math. Soc, Providence, RI, 1991.
[2] A.V. Bolsinov, S.V. Matveev and A.T. Fomenko, Topological classification of
integrable Hamiltoman systems with two degrees of freedom. A list of systems
with small complexity, Uspekhi Mat. Nauk 45(2) (1990), 49-77.
[3] A.T. Fomenko, Morse theory of integrable Hamiltonian systems, Doklady AN
SSSR287 (1986), 1071-1075.
[4] A.T. Fomenko, The topology of constant energy in integrable Hamiltonian
systems and obstructions to mtegrabihty, Izv. AN SSSR 50 (1986), 1276-1307.
[5] A.T. Fomenko, Topological classification of all integrable Hamiltonian
differential equations of general type with two degrees of freedom, in: The geometry
of Hamiltonian systems (Proc. Workshop, Berkeley, CA, 1989; T. Ratiu, ed.),
Math. Sci. Res. Inst. Publ. 22, Springer-Verlag, Berlin, 1991, 131-139.
[6] V. Kalashnikov, A class of generic integrable Hamiltonian systems with two
degrees of freedom, Preprint No 907, March 1995, University of Utrecht, Dept.
of Mathematics.
[7] M.P. Kharlamov, Topological analysis of integrable cases in dynamics of rigid
body, Leningrad Univ. Press, Leningrad, 1988.
[8] L.M. Lerman and Ya.L. Umanskii, Structure of the Poisson action ofR2 on a
four-dimensional symplectic manifold, Selecta Math. Sovi-et. 7 (1988), 39-48.
[9] V.S. Matveev, The calculation of the Fomenko invariant for a "saddle-saddle"
point of an integrable Hamiltonian system, in: Trudy Sem. po Vekt. i Tenz.
Analizu 25, Moscow Univ. Press, Moscow, 1993, 75-104.
[10] A.A. Oshemkov, Morse functions on two-dimensional surfaces. Coding
singularities, Trudy MIRAN 205 (1994), 131-140.
[11] S. Smale, Topology and Mechanics, Invent. Math. 10 (1970), 305-331.
[12] Nguen Tien Zung, Symplectic topology of integrable Hamiltonian systems
I: Arnold-Liouville xuith singularities, Compos. Math., to appear.

TOPOLOGY and APPLICATIONS Moscow: PHASIS, 1996
International Topological Conference Pages 35-42
Dedicated to P.S. Alexandroff's 100th Birthday
Moscow, May 27-31, 1996
Geometry and topology of geometrically finite
negatively curved and Carnot-Caratheodory
manifolds
Boris N. Apanasov*
University of Oklahoma, Norman, OK, USA
1. Here we sketch a recent progress in studying noncompact negatively
curved manifolds, their boundaries at infinity having Carnot-Caratheodory
structures, as well as deformations of such manifolds induced by equivariant
quasiconformal homeomorphisms. Here the most interesting are complex
hyperbolic manifolds with Cauchy-Riemannian structure at infinity, which
occupy a distinguished niche. The problems there have a unique appeal
both for the amount of similarity with the model situation of interactions
between real hyperbolic manifolds, their boundaries with natural conformal
structures and their quasiconformal deformations (see [1, 6]), and for the
interesting ways in which the similarity breaks down.
We study the basic class of such manifolds, which consists of
geometrically finite ones. To deform such manifolds, we use maps which are
quasiconformal (with bounded distortion) with respect to negatively curved
geometries and the corresponding sub-Riemannian structures which appear
at their infinity. One of inspiring ideas here came from a well known
theorem of D. Sullivan (see [26, 17]) that homeomorphisms of quasiconformal
(in Euclidean sense) η-manifolds, η φ 4, can be approximated by
quasiconformal ones. In Carnot-Caratheodory spaces, due to M. Gromov [20],
continuous maps can be approximated as well by maps that are Lipschitz
with respect to the Carnot-Caratheodory metric. However, there are
additional constrains for such quasiconformal maps in Carnot-Caratheodory
spaces because, due to P. Pansu [25] and G. Margulis and G. Mostow [22],
these (always a.e. differentiable) maps should preserve their contact
structures (horizontal vector fields). This makes deformations of manifolds with
* Supported in part by the NSF.

36
negative variable curvature more rigid, as well as questions about their Te-
ichmiiller spaces, geometric realization of isomorphisms of their fundamental
groups and their stability more interesting (cf. Sullivan's stability theorem
for Kleinian groups [27, 6]).
2. The main assumption on a negatively curved π-manifold Μ is
its geometrical finiteness, which implies that the fundamental group
7Γι(Μ) = G С Isom X acting in a simply connected space X — Μ is finitely
generated. An important role here is played by parabolic subgroups of G.
Parabolicity in the variable curvature case is not as easy a condition to deal
with as it is in the constant curvature space. However the results below
simplify the situation.
Due to the absence of totally geodesic hypersurfaces in a space X of
variable negative curvature, we cannot use the original Ahlfors definition of
geometrical finiteness which came from an assumption that the
corresponding real hyperbolic manifold Μ = Hn/G may be decomposed into a cell by
cutting along a finite number of its totally geodesic hypersurfaces, that is
the group G should possess a finite-sided fundamental polyhedron. Another
obstacle for such "polyhedral" approach is due to our result on parabolic
groups in the complex hyperbolic space X — Hc where Isom0 X = PU(2,1).
In the punctured sphere at infinity дШс \ {ρ} « Ε , the Kahler geometry
of He induces the nilpotent geometry of the Heisenberg group % = CxE,
where the parabolic stabilizer Gp С G of ρ acts discontinuously by isometries
(the discontinuity set &{GP) — %2 and the limit set A(GP) = {p}). In
contrast to real hyperbolic geometry, geometry of Dirichlet polyhedra of such
parabolic (even unipotent) discrete groups has no finiteness property [11]:
Theorem 1. Let G С PU(2,1) be a discrete parabolic group conjugate to
the subgroup Г = {(га, π) Ε С X Ε : га, η Ε Ζ} of the Heisenberg group
Ή2 = С X R. Then any Dirichlet polyhedron Dy{G) centered at an arbitrary
point у G He has infinitely many sides.
However, in spaces of variable negative curvature, one can give another
definition of geometrically finite groups G С IsomX as those ones whose
limit sets A(G) С дХ consist of only conical limit points and parabolic
cusp points p. The last cusp points ρ have parabolic stabilizers Gv С G
such that the quotients of the limit set (A(G) \ {p})/Gp are compact (see
another equivalent definitions later and in [1, 15]). To study parabolic ends
of negatively curved manifolds, we prove the following theorem [11, 12, 13]
(similar claim for finite index subgroups see in [14]).

37
Theorem 2· Let N be a connected, simply connected nilpotent Lie group,
С a compact group of automorphisms of N, and Г a discrete subgroup of the
semidirect product Ν Ά С. Then there exist a connected Lie subgroup V of
N and a finite index normal subgroup Г* of Г with the following properties:
1. There exists b G N such that 6Г6"1 preserves V.
2. V/bTb~l is compact.
3. ЬГ*Ь-1 acts on V by left translations and this action is free.
3. Now we apply Theorem 2 to complex hyperbolic manifolds (quotients
of the complex ball B^ С Cn with the Bergman metric by discrete groups
G С AutB^ = PU(n, 1)) and Cauchy-Riemannian manifolds at their
infinity. First we have [11]:
Lemma 1. Let ρ £ дШ^ be a parabolic fixed point of a discrete group
G С PU(n, 1). Then ρ is a cusp point if and only if it has a cusp neighbor-
hood UPir.
Assuming for simplicity ρ — oo, we take the subspace 7^oo С %п given by
Theorem 2 for a discrete stabilizer G^ С G and define a cusp r-neighborhood
Ur of oo as the set Ur = {x G H£ U %n : pc{x, Woo) > Vr}· Here pc is the
Cygan metric in H^U^n (identified with Cn~ Χ Ε χ [0, oo)) induced by the
norm: ||(£, v, u)\\c = \ \\ξ\\2 + u- H1/2, (£, v, u) G Cn_1 χ Ε χ [0, oo). The
key condition is that Ur should be precisely invariant under Goo С G, i.e.
G\Goo(Ur)nUr = 0.
This allows us to prove that a complex manifold M(G) = [MqL\Q(G)]/G
is geometrically finite if and only if it has finitely many ends, and each of
them is a cusp end, that is an end whose neighborhood can be taken as
Up,r/Gp « (Sp,r0/Gp) X (0,1], where 5p>ro is the boundary of UPiro in H^.
So we have [11]:
Theorem 3. Let Г С 7ίη Ά U(n — 1) be a torsion-free discrete group acting
on the Heisenberg group %n = С Χ Ε with non-compact quotient. Then
the quotient %n/T has zero Euler characteristic and is a vector bundle over
a compact manifold. Furthermore, this compact manifold is finitely covered
by a nil-manifold which is either a torus or the total space of a circle bundle
over a torus.

38
Corollary 1. The fundamental groups of Heisenberg manifolds and
geometrically finite complex hyperbolic manifolds are finitely presented.
Due to Theorem 2, any Heisenberg manifold Ν = Ήη/Γ is the vector
bundle Wn/r —>· 7/г/'Г where %γ CHn\s г, minimal Γ-invariant subspace.
Although such vector bundles are non-trivial in general, we have [11]:
Theorem 4. Let Г С %п * U(n - 1) be a discrete group and Hr С %п
a connected Γ-invariant Lie subgroup on which Г acts co-compactly. Then
there exists a finite index subgroup Го С Г such that the vector bundle
%η/Γο —> ^г/Го is trivial In particular, any Heisenberg orbifold ?ίη/Γ
is finitely covered by the product of a compact nil-manifold %r/To and an
Euclidean space.
Such finite covering property holds not only for Heisenberg manifolds
alone but for geometrically finite complex hyperbolic manifolds as well [11]:
Theorem 5. Let G С PU(n, 1) be a geometrically finite discrete group.
Then G has a subgroup Go of finite index such that every parabolic subgroup
of Go is isomorphic to a discrete subgroup of the Heisenberg group %n =
С χ Κ. In particular, each parabolic subgroup of Go is free Abelian or
2-step nilpotent.
4. Studying Carnot-Caratheodory manifolds at infinity of negatively
curved non-compact manifolds, we have a sharp contrast to the real
hyperbolic case. In fact, the Kahler structure of a complex manifold
M{G) — (Hc U Q(G))/G is so rigid that existence of a compact component
of its boundary (a closed Cauchy-Riemannian manifold) implies
connectedness of the boundary [18, 24]. In the non-compact case, we have nevertheless
an absolutely different situation [11]:
Theorem 6. For any integers к, ко, к > к0 >0, and η > 2, there exists a
complex hyperbolic η-manifold Μ = Ш^/G, G С PU(n, 1), whose boundary
at infinity splits up into к connected (n— l)-manifolds, dooM = N\U.. .UTVfc.
Moreover, for each boundary component Nj, j < ко, the inclusion ij : Nj С
M(G) induces a homotopy equivalence of Nj to M(G).
However, we show that such a wild situation is impossible for
geometrically finite complex hyperbolic manifold M. Namely, if the manifold M(G)

39
has non-compact boundary dM = Q,(G)/G with a component No С дМ ho-
motopy equivalent to M{G), then there exists a compact homology cobor-
dism Mc С M(G) homotopy equivalent to M(G)) and M(G) can be
easily reconstructed from Mc by gluing up a finite number of standard open
"Heisenberg collars" [11, Theorem 7.7].
5. Here we study deformations of negatively curved manifolds and
geometric realizations of isomorphisms of discrete groups G, Я С IsomX.
This is closely related to Mostow rigidity and Sullivan stability theorems
(see [23, 27]).
Problem 1. Given an isomorphism φ : G —ϊ Η of geometrically finite
groups G, Я С IsomX, find subsets Xq,Xh С X invariant for the
action of groups G and H, respectivelyf and an equivariant homeomorphism
ίφ : Xg —> Xh which induces the isomorphism φ. Determine metric
properties of /φ, in particular, whether it is quasi-symmetric (q-conformal)
with respect to the given negatively curved metric d in X and in the
induced sub-Riemannian structure on the Carnot-Caratheodory space at
infinity Υ = Τ \ {oo}).
As the first result in this direction, we have an isomorphism theorem [7]:
Theorem 7. Let φ : G —>· Η be a type preserving isomorphism of two
поп-elementary geometrically finite discrete subgroups G, Η С Pt/(n, 1).
Then there exists a unique equivariant homeomorphism /φ : A(G) —>· Л(Я)
of their limit sets that induces the isomorphism φ.
However, in contrast to the conformal case, homeomorphic CR-mani-
folds dM{G) and dM(H) may be not quasiconformally equivalent, see [23].
Also, besides the metrical obstructions, some topological obstructions for
extensions of equivariant homeomorphisms /φ : A(G) —>- Л(Я) may exist.
Namely, let G С Pt/(1,1) С Pt/(2,1) and Η С РО(2,1) С Ρί/(2,1) be
two geometrically finite (loxodromic) groups isomorphic to the
fundamental group K\(Sg) of a compact oriented surface Sg of genus g > 1. Then
the equivariant homeomorphism /^ : A(G) —>· Л(Я) cannot be homeomor-
phically extended to the whole sphere дШ^ « 53. The obstruction here is
о о
due to the fact that the quotient manifolds Hc/G and Нс/Я have different
Toledo [28] invariants: r(m2c/G) = 2g-2 and t(Eq/H) = 0.
Very often, manifolds with variable curvature К < 0 are more rigid than
real hyperbolic ones [25, 16]. In particular, a complex hyperbolic manifold

40
Μ homotopy equivalent to its closed totally geodesic complex hypersurface
is rigid [19].
We show however that complex hyperbolic Stein manifolds Μ
homotopy equivalent to their closed totally real geodesic hypersurfaces are
not rigid. Our construction of deformations is somehow influenced by
well know bending deformations of real hyperbolic manifolds along totally
geodesic hypersurfaces, see [2, 1]. In the case of complex hyperbolic (and
Cauchy-Riemannian) structures, it works however in a different way
involving simultaneous bending of the base of the fibration of the complex surface
Μ as well as bendings of each of its totally geodesic fibers. Moreover, our
bendings are induced by equivariant homeomorphisms, which are in addition
quasiconformal with respect to the corresponding metrics [9]:
о
Theorem 8. Let Sp = HR/G be a closed totally real geodesic surface
of genus ρ > 1 in a given complex hyperbolic surface Μ = H^/G,
G С PO(2,1) С PC/(2,1). Then there is an embedding π о В : В2?-2 <-+
Τ (Μ) of a real (2p — 2)-ball into the Teichmuller space of Mf defined by
quasiconformal bending deformations along disjoint closed geodesies in Μ
and the projection π : U(G) -> T(G) = TZ{G)/PU(2) 1).
Applying bendings, we answer a well known question on the Teichmuller
space boundary dT(M) of a complex surface Μ fibered over a surface of
genus ρ [10]:
Theorem 9. LetG С PO(2,1) С Pt/(2,1) be a uniform lattice isomorphic
to the fundamental group of a closed surface Sg of genus ρ > 2. Then, for
any simple closed geodesic а С Sp = H^/G, there is a continuous
deformation pt = /* induced by G-equivariant quasiconformal homeomorphisms
2 2
ft : H<c —>· H<c whose limit representation p^ corresponds to a boundary cusp
point of the Teichmuller space T{G), that is the boundary group poo(G)
has an accidental parabolic element Poo(ga) where ga £ G represents the
geodesic а С Sp. Moreover, there is a continuous quasiconformal
deformation R :RP —> T(G) whose boundary group Gqq = R(oo)(G) has 2p - 2
non-conjugate accidental parabolic subgroups.
References
[1] B.N. Apanasov, Geometry of discrete groups and manifolds, Nauka, Moscow,
1991.

41
B.N. Apanasov, Nontriviality* of Teichmuller space for Kleiman group in space,
Ann. of Math. St. 97, Princeton Univ. Press, 1981, 21-31.
B.N. Apanasov, Geometrically finite hyperbolic structures on manifolds, Ann.
of Glob. Analysis and Geom. 1:3 (1983), 1-22.
B.N. Apanasov, Nonstandard uniformized conformal structures on hyperbolic
manifolds, Invent. Math. 105 (1991), 137-152.
B.N. Apanasov, Deformations of conformal structures on hyperbolic manifolds,
J. Diff. Geom. 35 (1992), 1-20.
B.N. Apanasov, Conformal geometry of discrete groups and manifolds,
W. de Gruyter, Berlin-New York, 1996, to appear.
B.N. Apanasov, Canonical homeomorphisms in Heisenberg group induced by
isomorphisms of discrete subgroups of PU(n,l), Russian Acad. Sci. Dokl.
Math., to appear.
B.N. Apanasov, Quasiconformality and geometrical finiteness in Carnot-
Caratheodory and negatively curved spaces, Preprint Μ SRI at Berkeley,
1996-019.
B. Apanasov and N. Gusevskii, Bending deformations of complex hyperbolic
surfaces, Preprint, Univ. of Oklahoma, 1996.
B. Apanasov and N. Gusevskii, The boundary of Teichmuller space of complex
hyperbolic surfaces, in preparation.
B. Apanasov and X. Xie, Geometrically finite complex hyperbolic manifolds,
Preprint, Univ. of Oklahoma, 1995.
B. Apanasov and X. Xie, Manifolds of negative curvature and nilpotent groups,
Preprint, Univ. of Oklahoma, 1995.
B. Apanasov and X. Xie, Discrete isometry groups of nilpotent Lie groups,
Preprint, 1995.
L. Auslander, Bieberbach's theorem on space groups and discrete uniform
subgroups of Lie groups, II, Amer. J. Math. 83 (1961), 276-280.
B. Bowditch, Geometrical finiteness with variable negative curvature, Duke J.
Math. 77 (1995), 229-274.
K. Corlette, Archimedian superrigidity and hyperbolic geometry, Ann. Math.
135 (1992), 165-182.
S.K. Donaldson and D. Sullivan, Quasiconformal 4-Tnanifolds, Acta Math. 163
(1989), 181-252.
C. Epstein, R. Melrose and G. Mendoza, Resolvent of the Laplacian on strictly
pseudoconvex domains, Acta Math. 167 (1991), 1-106.
W. Goldman and J. Millson, Local rigidity of discrete groups acting on complex
hyperbolic space, Invent. Math. 88 (1987), 495-520.
M. Gromov, Carnot-Caratheodory spaces seen from within, Preprint IHES,
Bures-sur-Yvette, 1994.
A. Koranyi and M. Reimann, Quasiconformal mappings on the Heisenberg
group, Invent. Math. 80 (1985), 309-338.

42
[22] G.A. Margulis and G.D. Mostow, The differential of a quasiconformal mapping
of a Carnot-Caratheodory space, Geom. Funct. Anal. 5 (1995), 402-433.
[23] R. Miner, Qc-equivalence of spherical CR-manifolds, Ann. Acad. Sci. Fenn.
Ser. A I Math. 19 (1994), 83-93.
[24] T. Napier and M. Ramachandran, Structure theorems for complete Kahler
manifolds and applications to Lefschetz type theorems, Geom. Funct. Anal. 5
(1995), 807-851.
[25] P. Pansu, Metriques de Carnot-Caratheodory et quasiisometries des espaces
symmetries de rang un, Ann. Math. 129 (1989), 1-60.
[26] D. Sullivan, Hyperbolic geometry and homeomorphisms, Geometric Topology,
Acad. Press, 1979.
[27] D. Sullivan, Quasiconformal homeomorphisms and dynamics, II: Structural
stability implies hyperbolicity for Kleinian groups, Acta Math. 155 (1985),
243-260.
[28] D. Toledo, Representations of surface groups on complex hyperbolic space, J.
Diff. Geom. 29 (1989), 125-133.
[29] S.K. Vodop'yanov, Quasiconformal mappings on Carnot groups, Doklady
RAN, to appear.
[30] C.T. Wall, Geometric structures on compact complex analytic surfaces, Topol.
25 (1986), 119-153.

Session 1
General and Geometric
Topology

The abstracts are presented in the alphabetical order
of the authors' names

TOPOLOGY and APPLICATIONS Moscow: PHASIS, 1996
International Topological Conference Pages 45-46
Dedicated to P.S. Alexandroff's 100th Birthday
Moscow, May 27-31, 1996
Absolute extensors of spaces
with countable filtration
Sergei M. Ageev
Brest State University, Brest, Belarus
Yu.M. Smirnov posed the problem of describing absolute retracts and
extensors in the category of metric spaces with finite filtration. The answer
was simple and natural: they are exactly those spaces whose all filtration
elements are absolute retracts.
The transition from finite to countable filtration features a qualitative
saltus. Although the notions of absolute retracts and extensors coincide
here, not every countable increasing sequence of closed linear subspaces in
a linear normed space L is an absolute extensor (AE/).
Theorem 1. Any CW-complex (Ot, L) in the Whitehead weak topology with
a countable increasing sequence of subpolyhedra with respect to the triangu-
lation L is AEj.
Every normal functor F in the category of metric compacta generates a
countable filtration
oo
F1(X)CF2(X)CF3(X)C. in Foo(X) = (J Fk(X)
k=l
where Fk(X) is the subset of F(X) consisting of all points of finite degree
not greater than k.
Theorem 2· Let a functor F be the exponent exp or the functor of
probability measures P. // a metric space X is an AE then F\(X) С F2(X) С
Fz(X) С ... С Foo(X) is AEj in the category of spaces with countable
filtration.
The last theorem implies that there exist closed embeddings
h\ :exp00(X) —> L and h2 : Pqo(X) -» L into a linear normed space L

46
and retractions r\ : L —>· exp00(X) and r2 : L —» Poo(X) such that expfc(X)
and Paj(A') are images of retractions r\ and r2 restricted to some closed
linear subspaces Lk С L. From a theorem of N.T. Nhu (Fund. Math. 124(3)
(1984), 243-253) follows only that expk(X) and Pk{X) are images of
retractions r\ and r2 without any relationship between the latter. Without
certain formality, this strengthening of Nhu's theorem may be formulated
as follows: extensor properties of the families of closed subsets {expk(X)}
and {Pk(X)} are as good as those of the family of closed linear subspaces
in a linear normed space.
Question. For what other functors F does the analogous result take place?

TOPOLOGY and APPLICATIONS Moscow: PHASIS, 1996
International Topological Conference Page 47
Dedicated to P.S. Alexandroff's 100th Birthday
Moscow, May 27-31, 1996
Topological and geometric structures
on compact contractible manifolds
Fredric D. Ancel
University of Wisconsin, Milwaukee, WI, USA
Craig R. Guilbault
University of Wisconsin, Milwaukee, WI, USA
For η > 4, every compact contractible π-manifold С has a wild arc spine.
In other words, С is homeomorphic to the mapping cylinder of a map from
its boundary BdryC to [0,1]. Let К < 0. The mapping cylinder
structure on С implies that IntC admits a complete CAT(K) metric. Hence,
every homology (n — l)-sphere is the visual boundary of a contractible open
CAT(K) manifold.
Some compact contractible 4-manifolds С have wild arc pseudo-spines.
In other words, there is a wild arc J in Int С such that С—J is homeomorphic
to BdryC X [0,1). Questions: (1) Do all compact contractible 4-manifolds
have wild arc pseudo-spines? (2) For К < 0, do the interiors of some (all)
compact contractible 4-manifolds admit complete CAT(K) metrics?

TOPOLOGY and APPLICATIONS Moscow: PHASIS, 1996
International Topological Conference Page 49
Dedicated to P.S. Alexandroff's 100th Birthday
Moscow, May 27-31, 1996
Extension properties of the orbit space
Sergey A. Antonian
Yerevan State University, Yerevan, Armenia
Throughout G will denote a compact Hausdorff group. By a G-A(N)E
(resp., G-A(N)E(fc)) where к > 0 is an integer)-space we mean a G-space
(not necessarily metric), which is a G-equivariant absolute (neighborhood)
extensor for the class of all metric G-spaces Μ (resp. with dim M/G < k).
Theorem 1· Let N < G be a closed normal subgroup and suppose that all
the orbits of a G-space are metric. Then:
1. // X is a G-A(N)E-space, then the N-orbit space X/N is
a G-A(N)E-space. In particular, the G-orbit space X/G is an
A(N)E-space.
2. For any k>0, if X is a G-A(N)E(k)-space, then X/N is a
G-A(N)E(k)-space. In particular, X/G is an A(N)E(k)-space.
Corollary· Let Η be a subgroup of the symmetric group Sn, η > 1. Then
the functor SPfi of symmetric n-th power associated with Η preserves the
properties of a G-space to be a G-A(N)E (resp. G-A(N)E(k), к > 0)-space.
In particular, SPfi preserves the properties of a topological space to be an
A(N)E (resp. A(N)E(k), к > 0)-space.
Theorem 2. Let G be a compact Lie group and 2G be its hyperspace of
closed subsets with the Hausdorff metric. Let G act on 2G by left translations
and let X = 2G \ {G}. Then the orbit space X/G is a Q-manifold.
Theorem 3· Let L be a locally convex linear G-space and V be an invariant
convex subset of L. Suppose also that for any point χ £ V with metric orbit
G(x), there is a sequence {xn} С V such that xn —>- x, Gx С GXn and
GXn e G-A(N)E Vn > 1. Then V <E G-A(N)E. If, in addition, V contains
a G-fixed point then V € G-AE.

TOPOLOGY and APPLICATIONS Moscow: PHASIS, 1996
International Topological Conference Pages 51-52
Dedicated to P.S. Alexandroff's 100th Birthday
Moscow, May 27-31, 1996
Embeddings preserving character
and cardinal invariants
Angelo Bella
Catania University, Catania, Italy
Ivan V. Yaschenko
Moscow Center of Continuous Mathematical Education, Moscow, Russia
In 1924 Alexandroff and Urysohn asked if every space can be embedded
into an Я-closed space. In 1930 Tychonoff answered this in affirmative, and
later Katetov proved that every space can be embedded as a dense subset
into an Я-closed space.
We introduce a construction of Я-closed-like extensions of a space
which preserves character. This provides counterexamples to the
following questions [1].
Question A. Does the inequality \X\ < 2χ(χ) hold for every almost
Lindelof Hausdorff space XI
Question B. Does \X\ < 2χ(γ) hold whenever X is an Η-set of a Hausdorff
space У?
These questions were motivated by an attempt to generalize the cardinal
inequality \X\ < 2X^X\ proved for every Я-closed space X by A. Dow and
J.R. Porter [3].
In fact, the notion of an Я-closed space may be weakened either to the
notion of almost Lindelof space — X is almost Lindelof if every open cover
7 has a countable subfamily γ' such that X = \J{c\x(U) : U 6 γ'} —
or to the notion of an Я-set — X is an Я-set in У if every family γ of
open subsets of У satisfying X С U7 has a finite subfamily γ' such that
XC\J{clY{U):UeY}.
The first question has a positive answer if a space has a dense subset
of isolated points (see [3]) and both questions have a positive answer if
restricted to the class of Urysohn spaces (see [2] and [1]).

52
A subset X of the space Υ is relatively Η-closed in Υ if for every open
cover γ of У there is a finite subfamily γ' С γ satisfying X С U{cly({/) :
[/ G 7'}. Given a space X, We say that a filter U С т(^0 has the weak
countable intersection property (W.C.I.P.) if, for every countable W С W,
p|{clx(t/) : U € ZY'} ^ 0. We call a space X weakly realcompact iff every
ultrafilter И С r(X) having the weak countable intersection property has a
nonempty adherence.
Theorem 1. Every first countable weakly realcompact Hausdorff space X
can be embedded as a closed relatively Η-closed subspace into a first
countable Hausdorff space Υ in such a way that there is an extender
φ : τ(Χ) -+ t{Y) satisfying: a) <p{U)HX = U; b) <p(UnV) = <p(U)n<p{V).
Theorem 2. For any non-measurable cardinal a there exists a first count-
able Hausdorff space which contains a discrete Η-set of size at least a.
Theorem 3. Every weakly realcompact first countable space can be
embedded as a closed subset into an almost Lindelof first countable space.
References
[1] A. Bella, A couple of questions concerning cardinal invariants^ Q L· A in General
Topology, to appear.
[2] A. Bella and F. Cammaroto, On the cardinality of Urysohn spaces, Can ad.
Math. Bull. 31 (1988), 153-158.
[3] A. Dow and J.R. Porter, Cardinalities of Η-closed spaces, Topol. Proc. 7 (1982),
27-50.
[4] J.R. Porter and R.G. Woods, Extensions and absolutes of Hausdorff spaces,
Springer-Verlag, Berlin, 1988.

TOPOLOGY and APPLICATIONS Moscow: PHASIS, 1996
International Topological Conference Page 53
Dedicated to P.S. Alexandroff's 100th Birthday
Moscow, May 27-31, 1996
Locally connected subgroups of the Hubert space
and ANR-spaces
Robert Cauty
University Paris 6> Paris, France
We study the following problem of Dobrowolski ([3, Question LS18] and
[1, Question 5.9]: is every closed locally connected subgroup of the
Hubert space an ANR-space? The following result shows that the answer
is negative.
Theorem. There exists a σ-compact subgroup G of the Hilbert space such
that G and its completion G are locally pathwise connected but are not
LC1 -spaces.
Let X = VSi S} be a bundle of 1-spheres with marked point *. We
construct an embedding φ of the compactum X into /2 such that <^_1(0) = {*}
and that φ(Χ \ {*}) is a linearly independent subset. Let G be a subgroup
generated by the set <p(X). We choose φ so that for any г the restriction
(p\S{ : S{ —l· G is essential. To this end, let A(Sl) be the free Abelian group
over the 1-sphere 51 (with marked point *). First we construct an
embedding ψ : S1 —l· I2 such that the natural homomorphism ψ : A(Sl) —>- G
extending ф is a weak homotopic equivalence, this permits us to use a
special case of a result by Dold and Thorn [2] by which the embedding of 51
into A(Sl) is essential.
References
[1] F.D. Ancel, T. Dobrowolski and J. Grabowski, Closed subgroups in Barfach
spaces, Studio Math. 169 (1994), 977-990.
[2] A. Dold and R. Thorn, Quasifaserungen und unendliche symmetrische Pro-
dukte, Ann. Math. 67 (1958), 239-281.
[3] J. West, Open problems in infinite-dimensional topology, in: Open Problems in
Topology (J. van Mill and G.M. Reed, editors), Elsevier, 1990, 524-597.

TOPOLOGY and APPLICATIONS Moscow: PHASIS, 1996
International Topological Conference Page 55
Dedicated to P.S. Alexandroff's 100th Birthday
Moscow, May 27-31, 1996
Isomorphism problems for Banach spaces
of measurable functions on compact spaces
Mitrofan M. Cioban
Tiraspol State University, Tiraspol, Moldova
Let X be a compact space. Denote by B(X) the Banach space of all
bounded functions on X, C(X) = {/ 6 B(X) : f is continuous},
Ba(X) = {/ € B(X) : / is Baire measurable of class a}, where 0 < a < Ω,
Ω is the first uncountable ordinal number.
Consider two functors Φι, Φ2 of the category of compact spaces into the
category of Banach spaces with the properties:
1. C{X) С Фг(Х) С Ф2{Х) С В{Х).
2. If д : X —> Υ is a continuous mapping and / 6 Φ* (У) then
ί·9€Φ*(Χ).
3. Ф%(Х) are subrings of the ring B(X). The functor Ф« is a Baire functor
if Φ{(Χ) = Ba{X) for some a < Ω,
For Baire functors we examine the following questions:
1. Under which conditions are the rings Ф{(Х) and Φί(Υ) isomorphic?
2. Under which conditions are the rings Φί{Χ) and Φj(Y) isomorphic?
3. Under which conditions is the Banach space Φ\(Χ) complemented in
Φ2(Χ)? .
In particular, it is proved that for a non-scattered compact space X and
0 < a < β the space Ba(X) is not complemented in Ββ(Χ).

TOPOLOGY and APPLICATIONS Moscow: PHASIS, 1996
International Topological Conference Page 57
Dedicated to P.S. Alexandroff's 100th Birthday
Moscow, May 27-31, 1996
The Banach-Mazur compactum Q(2)
is an absolute retract
Paul Fabel
Mississippi State University, Mississippi State, MS, USA
The Banach-Mazur compactum Q(n) is a space of equivalence classes of
norms on Rn. Two norms on En are equivalent if they determine isometri-
cally isomorphic Banach spaces. We prove that Q(2) is an ANR by
embedding Q(2) as a retract of a certain quotient space of analytic mappings of
the unit disk.

TOPOLOGY and APPLICATIONS Moscow: PHASIS, 1996
International Topological Conference Page 59
Dedicated to P.S. Alexandroff's 100th Birthday
Moscow, May 27-31, 1996
On completeness of spaces
of τ-additive measures
Vitaly V. Fedorchuk
Lomonosov Moscow State University, Moscow, Russia
Let us recall that a probability measure μ on a Tychonoff space X is said to
be τ-additive if μ(ϋ) = lim^(Ga) for any upward directed family of open
in X sets Ga with G — \JaGa- The set PT(X) of all r-additive probability
measures on X can be identified with the set of all probability measures μ
on βΧ such that μ(Κ) = 0 for any compact set К С βΧ\Χ.
For a uniform space (X,U) let R{U) be the family of all uniformly
continuous bounded pseudometrics on X. Every pseudometric ρ £ R{U)
generates the Kantorovich pseudometric PT(p) on PT(X). The family
{PT(p) : ρ € R{U)} generates a uniformity PT(M) on PT(X) which extends
the uniformity U. T. Banakh asked whether the functor PT preserves
completeness of uniform spaces. In general, the answer is "no".
Theorem 1. The space PT(RC) is not complete.
On the other hand, the next statement holds.
Theorem 2 (MA). If uw(X,U) < c, then the space (PT(X),PT(U)) is
complete.
Here uwX is the uniform weight of the uniform space X.

TOPOLOGY and APPLICATIONS Moscow: PHASIS, 1996
International Topological Conference Page 61
Dedicated to P.S. Alexandroff's 100th Birthday
Moscow, May 27-31, 1996
Permutation groups
Paul M. Gartside *
Oxford University, United Kingdom
The symmetric group on ω, denoted Sym(u>), when considered as a subspace
of the irrationals ωω, is a separable completely metrizable topological group.
We investigate topological and set descriptive properties of subgroups of
Sym(u>). Our results are applied to the theory of infinite groups and to
model theory.
Joint work with R. Knight, Oxford University, United Kingdom, and D. Mclntyre,
Auckland University, New Zealand.

TOPOLOGY and APPLICATIONS Moscow: PHASIS, 1996
International Topological Conference Page 63
Dedicated to P.S. Alexandroff's 100th Birthday
Moscow, May 27-31, 1996
Some points of compactifications
Anatoli A. Gryzlov
Udmurtia State University, Izhevsk, Russia
We regard some types of points of Cech-Stone compactification βτ of a
discrete space r.
The first theorem gives the way of obtaining different points of
compactifications of a discrete space. Other theorems give examples of these points.
Let U(t) be the set of uniform ultrafilters and R(r) — the set of regular
ultrafilters on r.
Theorem 1. For every linked 2T—τ matrix there is a matrix point in U(r).
Theorem 2. For every discrete set D С U(t), \D\ — r, such that
[D] Π R(t) φ 0, there is a point ξ G [D] Π R(r) with χ(ξ) = 2T.
Let MfT = []{Щ : γ < r} be a union of disjoint subsets of r, |(/7| = r,
MT = []{υΊ = [Щ]\т : γ < r}, and let MT = [MT]\MT.
Theorem 3· In r*, ω < τ, there are:
(a) a matrix point ξ € Мт П U(r) of r* such that ξ £ [LK^y : 7 € τ}] if
F7 С t/7, c(Fy) < ω;
(b) a matrix point 'ξ € Мт П U(r) of r* such that ξ G [U{F7 : γ G τ}]
for some \J{F^ : γ G г}, tu/iere F7 С t/7 and c(F7) = ω, bu£
£ £ [U{£>7 : 7 € τ}] ifDy С f/7 and |D7| = ω;
(c) a matrix point ξ G Μτ Π {/(r) o/ r* such that ξ G [LK^y : Ύ € τ}]
/or some U{F7 : γ G г} sucA J/ш* F7 С t/7 and |F7| = ω, but
ζ i [U{^7 : 7 € r}] if D7 С ί/7 and |D7| < ω;
(d) a matrix point ξ G Μτ Π i/(r) sucA Йа£ ξ G [U{^7 : 7 € τ}] where
|D7D[/7| = 1.

TOPOLOGY and APPLICATIONS Moscow: PHASIS, 1996
International Topological Conference Pages 65-66
Dedicated to P.S. Alexandroff's 100th Birthday
Moscow, May 27-31, 1996
Dugundji extension theorems
for ordered spaces and their products
Yasunao Hattori
Shimane University, Shimane, Japan
Haruto Ohta
Shizuoka University, Shizuoka, Japan
For a space X, let C(X) denote the linear space of real-valued
continuous functions on X. Let Л be a subspace of a space X. Then, a map
e : C(A) —> C(X) is called a Dugundji extender if: (i) e is linear, and
(ii) e(/) is an extension of/ and e(f)[X] is included in the closed convex hull
of f[A] for each / £ С (A). We say that A is continuously D-embedded in X
if there is a continuous Dugundji extender e : С (A) —>- C(X) with respect to
both the pointwise convergence topology and the compact open topology.
Moreover, we say that A is continuously π ρ-embedded in X if A X Υ is
continuously D-embedded in Χ Χ Υ for each space Y. A generalized ordered space
(= GO-space) is a subspace of a linearly ordered space. It is known that a
space X with an order < is a GO-space if and only if: (i) the topology of X is
finer than its order topology, and (ii) X has a base consisting of convex sets.
For a subset Л of a GO-space X, let 1(A) = max{x € X : Va 6 Л (х < a)}
and r(A) = min{x GX: Va 6 Л (я > a)} if they exist. We have:
Theorem 1. Let X be a GO-space, A a closed subspace of X, and
X \ A = (Jγ, where γ is a family of convex components of X \A. Assume
that Υ = {U £ γ : t/ /ms neither l(U) nor r(U)} is discrete in X. TAen, Л
is continuously πd-embedded in X.
Corollary 1. Let X be one о/ Йе following spaces: (i) α locally compact
GO-space; (ii) α GO-space such that the underlying order is well-ordered]
(iii) the Sorgenfrey line. Then, every closed subspace A of X is continuously
π D-embedded in X.

ев
Corollary 2. Let X{ be one of the spaces (i)-(iii) in Corollary 1 and A{ a
closed subspace of X{ for each г — 1,..., η. Then, Пг<пЛг is continuously
D-embedded in Пг<пХ;.
For aGO-space X, let E(X) = {x £ X : (f-, x] or [x, —>·) is open in X}.
Theorem 2. Let X be a perfectly normal, GO-space such that E(X)
is σ-discrete in X. Then, every closed subspace of X is continuously
K£)-embedded in X.

TOPOLOGY and APPLICATIONS Moscow: PHASIS, 1996
International Topological Conference Pages 67-68
Dedicated to P.S. Alexandroff's 100th Birthday
Moscow, May 27-31, 1996
On the monotonicity of dimension dim
of continuous mappings
Vasily M. Karaulov
Vyatka State Pedagogical University, Kirov, Russia
Let / : X —> Υ be a fixed continuous mapping of topological spaces and
(7 = /:Z—^ТСУЬеа fixed submapping of /.
Let a set U be open in У, у £ U and Ω be a finite functionally open
cover of f~lU. We put (following Pasynkov) dim / < к, к = 0,1,2,..., if
there exists a neighborhood V С U of у and a finite functionally open cover
ω of f~lV refining Ω of order < к for every open set U and у £ U.
We shall say that a submapping g of f is: a) approximatively С J-embed'
ded over T; b) z-embedded over Τ, if: a) for every у £ Τ, open neighborhood
U of у, continuous function φ : g~~1(UC)T) —l· [0,1] and ε > 0 there exists an
open neighborhood V С U of у and a continuous function ψ : f~lV —)> [0,1]
satisfying the condition \φ(χ) — ψ(χ)\ < ε for every χ £ g~l{V Π Τ); b) for
every у £ Τ, open neighborhood U of у and functionally open set О in
^T1?/ Π Τ there exists an open neighborhood V С U of у and a
functionally open set G in f"lV satisfying the condition f~lV Π Ζ = 5_1(У Π Τ),
respectively.
Lemma 1. Let α mapping f be functionally normal (in the sense of
Pasynkov), a set Τ be open in Υ and a set Ζ be closed in f~lT. Then
the submapping g is approximatively C*j-embedded.
Lemma 2. Let a mapping f be Tychonoff and a submapping g be compact
(= perfect). Then the submapping g is z-embedded over T.
Theorem 1. Let a submapping g be approximatively CJ-embedded over T.
Then dim <7 < dim /.
This theorem and the property of approximative Ci-embedding of
mappings itself give the possibility of establishing the finite sum theorem and a

68
specific case of the countable sum theorem for continuous mappings. The
countable sum theorem and Theorem 1 imply the following properties of
dimension of mappings.
Theorem 2. Let a set Τ be open in Υ and a set Ζ be functionally open in
f~lT. Then dim g < dim/.
Theorem 3. Suppose that a submapping g is z-embedded over T. Then
dim <7 < dim/.
Remark. All the results for continuous mappings generalize dimension
properties of topological spaces, and coincide with that properties if

TOPOLOGY and APPLICATIONS Moscow: PHASIS, 1996
International Topological Conference Page 69
Dedicated to P.S. Alexandroff's 100th Birthday
Moscow, May 27-31, 1996
On Нп-ЪиЪЫе in η-dimensional compacta
Umed H. Karimov
Institute of Mathematics, Academy of Sciences of Tadjikistan, Dushanbe,
Tadjikistan
Dusan Repovs
University of Ljubljana, Ljubljana, Slovenia
A topological space X is called a Hn-bubble (n is a fixed natural number, Hn
is a Cech cohomology group with integer coefficients) if its n-dimensional
cohomology Hn(X) is nontrivial and the η-dimensional cohomology of every
its proper subspace is trivial. The main results are:
1. Any compact metrizable #n-bubble is locally connected;
2. There exists a 2-dimensional 2-cyclic compact metrizable ANR which
does not contain any #2-bubbles;
3. Every η-cyclic finite-dimensional L//n-trivial metrizable compactum
contains an Яп-ЬиЬЫе.

TOPOLOGY and APPLICATIONS Moscow: PHASIS, 1996
International Topological Conference Pages 71-72
Dedicated to P.S. Alexandroff's 100th Birthday
Moscow, May 27-31, 1996
Weak normality, exp (X) and powers
Anatoli P. Kombarov
Lomonosov Moscow State University, Moscow, Russia
A Τι-space Χ is said to be weakly normal [1, 2] if for every two disjoint
closed subsets A and В of X there exists a continuous mapping / of X into
R" such that the images of A and В are disjoint. Clearly, every normal
space is weakly normal. If there exists a one-to-one continuous mapping of
X onto a separable metrizable space, then the space X is weakly normal.
Can one use weak normality instead of normality in Velicko's theorem [5]:
if exp(X) is normal, then X is compact? It is easy to see that this is not the
case. Indeed, there exists a one-to-one continuous mapping of exp(u;) onto
£>", so exp (ω) is hereditary weakly normal, but the space ω is not compact.
Theorem 1. If X is a countably compact space and if exp(X) is weakly
normal, then X is compact
Theorem 2. If X is a countably compact space and if exp(X) is
hereditarily weakly normal, then X is a perfectly normal hereditarily separable
compact space.
Corollary. If X is countably compact and exp(exp(X)) or exp(X χ Χ) is
hereditarily weakly normal, then X is a metrizable compact space.
M.M. Coban [3] proved that if exp(X) is hereditarily normal, then X is
a metrizable compact space. So the next problem seems to be natural.
Problem. Is a compact space X metrizable, ifexp(X) is hereditarily weakly
normal?
The next theorem is a slight generalization of Noble's theorem from [4].
Theorem 3. All powers of a Τχ-space Χ are weakly normal if and only if
X is compact T2.

72
References
[1] A.V. ArhangePskii, Divisibility and cleavability of spaces, in: Recent
Developments of General Topology and its Applications, Math. Research 67, Berlin,
1992, 13-26.
[2] A.V. Arhangel'skii, A survey of cleavability, Topol. Appl. 54 (1993), 141-163.
[3] M.M. Coban, Note sur topologie exponentielle, Fund. Math. 171 (1971), 27-41.
[4] N. Noble, Products with closed projections II, Trans. Amer. Math. Soc. 160
(1971), 169-183.
[5] N.V. Velicko, On the space of closed subsets, Sib. Mat. Zh. 16 (1975), 484-486
(in Russian).

TOPOLOGY and APPLICATIONS Moscow: PHASIS, 1996
International Topological Conference Page 73
Dedicated to P.S. Alexandroff's 100th Birthday
Moscow, May 27-31, 1996
Butterfly points in separable spaces
with π-weight ω\
Sergey A. Logunov
Udmurtia State University, Izhevsk, Russia
B.E. Shapirovskii has introduced the notion of a Ь-point (or a butterfly
point). We shall say that ρ £ X* is a b-point if it is a limit point for some
sets F,GC X* \ {p} which are closed in βΧ \ {p} and disjoint. In spite of
strong efforts, in ω* it has been proved for very special types of points only,
that they are Ь-points. It still remains unknown whether each non-isolated
point in an extremally disconnected compactum is a b-point.
Theorem 1. Let X be a locally compact Lindelof separable space without
isolated points and nw(X) < ωχ. Then each remote point ρ of X* is a
b-point. Therefore the space βΧ \ {p} is not normal.
Theorem 2. Let X be a locally compact Lindelof space with nw(X) < ωχ.
Let TZ be the set of all remote points of X*. Then each remote point ρ is a
b-point in TZ.

TOPOLOGY and APPLICATIONS Moscow: PHASIS, 1996
International Topological Conference Pages 75-76
Dedicated to P.S. Alexandroff's 100th Birthday
Moscow, May 27-31, 1996
Resolvability of А-, С A- and PCA-sets
in compacta
Vyacheslav I. Malykhin
State Academy of Management, Moscow, Russia
The Α-operation was discovered by P.S. Alexandroffin 1915.
In 1943, E. Hewitt [2] called a space resolvable if it has two disjoint
dense subsets, and irresolvable otherwise. In 1994, W.W. Comfort (see [1])
proved the resolvability of every Tychonoff countably compact space. We
prove the resolvability of any Α-set of uncountable dispersion character, CA-
and PCA-sets without isolated points in compacta and construct examples
of Hausdorff countably compact irresolvable space and HausdorfFa-compact
irresolvable space.
Theorem 1. An uncountable Α-set in a countably compact T\-space
contains an infinite subset which is closed in the entire space.
Theorem 2. A regular subspace of uncountable dispersion character which
is an Α-set in a countably compact Τχ-space is resolvable.
Corollary 1. An Α-set of uncountable dispersion character in a regular
countably compact space is resolvable.
Corollary 2. An Α-set of uncountable dispersion character in a compactum
is resolvable.
Corollary 3. A regular σ-compact space of uncountable dispersion
character is resolvable.
Theorem 3. Every non-isolated point of the complement to a Lindelof sub-
space of a regular countably compact space is a limit point of some infinite
subset of this complement which is closed in the entire space.

76
Theorem 4. The complement to a Lindelof subspace in a regular countably
compact space is resolvable.
Recall that every Α-set in a compact space is Lindelof.
Corollary 4. A CA-set in a compactum is resolvable.
Theorem 5. Let Ε be a Lindelof subspace in a regular countably com-
pact product X\ χ X2 of some two spaces, π\ be the projection from
this product onto X\. Then every non-isolated point of the subspace
Τ — 7Γι(ΛΊ χ X2 \ Ε) is a limit point of some infinite subset of Τ which
is closed in the entire space X\.
Theorem 6. Let Ε be a Lindelof subspace in a regular countably compact
product X\ χ Χ2 of some two spaces, πχ be the projection from this product
onto Χχ. Then the subspace π\(Χ\ χ X2 \ E) is resolvable.
Corollary 5. A PCA-set in compacta is resolvable.
Example 1. There exists a Hausdorff countably compact irresolvable
space.1
Example 2. There exists a Hausdorff σ-compact irresolvable space of
uncountable dispersion character.
References
[1] W.W. Comfort and S. Garcia-Ferreira, On the resolvability of countably compact
spaces, to appear.
[2] E. Hewitt, A problem of set-theoretic topology, Duke Math. J. 10 (1943),
309-333.
1A similar example was constructed some earlier by O. Pavlov. He used another idea.

TOPOLOGY and APPLICATIONS Moscow: PHASIS, 1996
International Topological Conference Pages 77-80
Dedicated to P.S. Alexandroff's 100th Birthday
Moscow, May 27-31, 1996
Covers by stars of discrete subspaces
Mikhail V. Matveev
Bauman Moscow State University of Technology, Moscow, Russia
A topological space is absolutely countably compact (ace) [5], [7] provided
that for every open cover U of X and every dense subspace D С X there
exists a finite subset F С D such that St(F,ZY) = X. This definition was
motivated by a characterization of countable compactness (see [3]): a space
X is countably compact iff for every open cover U of X there exists a finite
subset F С X such that St(F, U) = X. After a series of papers on ace spaces
[5]-[7], [14]-[17], [4], [l]-[2] it seems natural to find out, so to say, what is
absolute countable compactness minus countable compactness. Thus we
came to the following definition.
Definition 1. A space X has the property (a) provided that for every open
cover U of X and every dense subspace D С X there exists a subset F С D
such that F is closed and discrete in X and St(F,ZY) = X.
Indeed, for a countably compact X) "closed and discrete in X" means
"finite", and we obtain the definition of ace.
Though it is not clear straightforward from the definition, the property
(a) is rather close to normality. One of the reasons to say so is that it
turns out to be very difficult to distinguish the property (a) and normality
in the class of countably compact spaces. On the one hand, examples of
(a), countably compact (that is, ace), non-normal spaces can be found in
[13] (these examples are countably compact and first countable, hence by [7]
they are (a)). But the construction of these examples needs special efforts,
one can say that "usually" an ace space is normal. On the other hand, the
question whether there exists a normal, countably compact, non-acc space
(asked by A.V. Arhangel'skii, see [7]) remains open.
It is interesting to note that the following property, obtained by removing
"dense subspace D" from the definition of the property (a), is a property of
every T\ -space:

78
(ao) for every open cover U of X there exists a subset F С X such that F
is closed and discrete in X and St(F,ZY) = X.
It is easy to see that every paracompact space is (a). Moreover, we do
not need the whole strength of paracompactness; for example, the following
weaker property (pp) suffices:
(pp) every open cover U of X has an open refinement V such that for every
choice of points py 6 V for all V 6 V the set {py : V 6 V} is closed
and discrete in X.
There are, however, some properties that, though do not imply (a) in
general, do imply it in the class of countably compact spaces where (a)
means ace. These properties are: countable tightness [7], orthocompactness
[15], radialness [12]. It is not clear if the same is true when countable
compactness is replaced by countable paracompactness.
The following lemma is the (a)-analog of the classical Jones' lemma.
Lemma 1. If a separable space X contains a closed discrete subspace of
cardinality с then X is not (a).
By this lemma, such spaces as Niemytzki plane, Kofner plane, the square
of the Sorgenfrey line etc. are not (a).
It is natural to call a space X (ъ)-Dowker provided that X is an (a)-space
while Χ χ (ω + 1) is not an (a)-space. I do not know if (a)-Dowker spaces
exist, but the following theorem implies that a normal (a)-Dowker space
must be a Dowker space.
Theorem 1. If X is a normal, countably paracompact, (a) -space, then the
space Υ — Χ χ (ω + 1) is an (a)-space.
Certain classes of spaces cannot contain (a)-Dowker spaces: perfect
spaces (by [3] they are normal, by [11] they cannot be Dowker); mono
tonically normal spaces (by [12] they are (a), by [11] they are countably
paracompact); paracompact spaces (paracompactness implies (a) and is
preserved by products with compact spaces).
The existence of normal, Moore, non-metrizable spaces was the subject
of a long-term study in General Topology. Since the property (a) behaves
in many aspects like normality, it is natural to ask whether every Moore
(a)-space is metrizable.

79
Theorem 2. (CH) Every separable Moore (a)-space is metrizable.
It is well known that every normal, feebly compact space is countably
compact. So, it is natural to ask whether every (a), feebly compact space is
countably compact. In the class of Hausdorff spaces the answer is negative
[9]. For regular or Tychonoff spaces the question is open (recall that feeble
compactness is equivalent to pseudocompactness for Tychonoff spaces).
Most of the facts mentioned above demonstrate similarity between
normality and the property (a). Yet in one way they are quite different: unlike
normality, the property (a) is not closed-hereditary.
Theorem 3· Every Tychonoff space can be represented as a closed subspace
of a Tychonoff (г)-space.
Theorem 4. Every Tychonoff countably compact space can be represented
as a closed Gs-subspace of a Tychonoff ace space.
Theorem 5· Every Tychonoff countably paracompact space can be rep-
resented as a closed Gs-subspace of a Tychonoff countably paracompact
(a)-space.
This abstract contains a brief exposition of the results from [8]-[10].
References
[1] M. Bonanzinga, Preservation and reflection of ace and hacc spaces, Comment.
Math. Univ. Carol., to appear.
[2] M. Bonanzinga, On the product of a compact space with an hereditarily
absolutely compact space, submitted.
[3] R. Engelking, General Topology, PWN, Warsaw, 1977.
[4] J. van Mill and J.E. Vaughan, /5 ω* \ {u} accl, Annals New York Acad. Sci.
767 (1995), 161-164.
[5] M.V. Matveev, On absolutely countably compact spaces, Abstracts of IX Intern.
Conf. on Topol. and its Appl., Kiev, 1992, 99.
[6] M.V. Matveev, A countably compact topological group which is not absolutely
countably compact,.Q к A in Gen. Topol. 11 (1993), 173-176.
[7] M.V. Matveev, Absolutely countably compact spaces, Topol. Appl. 58 (1994),
81-92.
[8] M.V. Matveev, Some questions on property (a), Preprint.
[9] M.V. Matveev, On feebly compact spaces with property (a), Preprint.
[10] M.V. Matveev, Embeddings into (a)-spaces and ace spaces, Preprint.

80
[11] Μ.Ε. Rudin, Dowker spaces, Handbook of Set-theoretic Topology (K. Kunen
and J.E. Vaughan, eds.), North-Holland, Amsterdam, 1984, 761-780.
[12] M.E. Rudin, I.S. Stares and J.E. Vaughan, From countable compactness to
absolute countable compactness, Preprint, 1995.
[13] J.E. Vaughan, A countably compact, first countable nonnormal space, Proc.
Amer. Math. Soc. 75 (1979), 339-342.
[14] J.E. Vaughan, On X xY where Υ is a compact space with countable tightness
and X is a countably compact GO-space, Abstracts of Tenth Summer Conf.
on Gen. Topol., Amsterdam, 1994, 165.
[15] J.E. Vaughan, On the product of a compact space with an absolutely countably
compact space, to appear in Proc. Vrije Univ. Topol. Conf., Amsterdam, 1994.
[16] J.E. Vaughan, A countably compact, separable space which is not absolutely
countably compact, Comment. Math. Univ. Carol. 36(1) (1995), 197-201.
[17] J.E. Vaughan, On X xY where Υ is a compact space with countable tightness
and X is an absolutely countably compact space, Preprint.

TOPOLOGY and APPLICATIONS Moscow: PHASIS, 1996
International Topological Conference Pages 81-82
Dedicated to P.S. Alexandroff's 100th Birthday
Moscow, May 27-31, 1996
Some contribution to the theory
of partially ordered locally compact groups
Alexander V. Mironov
Amur State University, Blagoveshchensk, Russia
Below there are given some theorems on partially ordered (po) locally
compact groups.
Theorem 1. Let G be a po-locally compact group with closed positive cone.
If every closed interval [0, χ], χ > 0, is connected, then the group G is locally
o-convex.
The cone of a po-topological group G is non-degenerating iff for every
neighborhood U of the unity element there exists a subset A of positive
elements of U such that the union of all closed intervals [—a, a] where a £ A
is the neighborhood of the unity element.
Theorem 2. LetG be a po-topological group with non-degenerating positive
cone and isolated directed order. If, in addition, the positive cone is closed
or the group G is locally o-convex, then every element of the group G is
discrete and therefore no nontrivial compact subgroup is contained in G. In
particular, nontrivial compact elements are not contained in G.
From B.A. Pasynkov's results on almost metrizable locally compact
groups and the preceding theorem one can deduce the following useful
metrizability theorem.
Theorem 3. The po-almost metrizable (in particular, locally compact)
group satisfying the conditions of the preceding theorem is metrizable.
Theorem 4. LetG be a lattice-ordered topological group with
non-degenerating positive cone and with relatively compact generative solid neighborhood
of the unity element. If, in addition, the positive cone is closed or the group
G is locally o-convex, then the group G is finite-dimensional.

82
The notions of the bipolar cone in a po-LCA group and the self-adjoint
cone in the real conjugate group of real characters of the annihilator of a
given cone will be defined.
Theorem 5. The category of po-LCA groups will bipolar cones is dual with
the category of direct products of LCA groups quasi-ordered by closed
subgroups, and associated with it discrete conjugate groups of real characters
ordered by self-adjoint cones.
In particular, this theorem contains the Pontryagin self-adjoint duality
of LCA group with discrete order.

TOPOLOGY and APPLICATIONS Moscow: PHASIS, 1996
International Topological Conference Page 83
Dedicated to P.S. Alexandroff's 100th Birthday
Moscow, May 27-31, 1996
On compact elements
of locally compact groups
Yuri N. Mukhin
Institute of Mathematics and Mechanics, Ural Branch of Russian Academy
of Sciences, Ekaterinburg, Russia
\ subgroup of a locally compact topological group is called compactly
covered if it is a union of compact subgroups. Willis (1995) proved that if G
is zero-dimensional then the union of all compact subgroups of G is closed.
This result implies an affirmative answer to Question 9.38 from "Kourovka
notebook of unsolved problems of the group theory":
Theorem 1. Maximal compactly covered subgroups of a zero-dimensional
group are closed.
The condition of zero-dimensionality is essential here. However, we have
Theorem 2. The closure of a normal compactly covered subgroup in G is
a compactly covered normal subgroup.
Theorem 3. If N is a closed normal subgroup in G and N and G/N are
compactly covered then G is also compactly covered.
Theorem 4· There exists a maximal compactly covered ^rmal subgroup
in G, and it is closed.

TOPOLOGY and APPLICATIONS Moscow: PHASIS, 1996
International Topological Conference Page 85
Dedicated to P.S. Alexandroff's 100th Birthday
Moscow, May 27-31, 1996
Local compactness of C\(X)
Sergei E. Nokhrin
Institute of Mathematics and Mechanics, Ural Branch of Russian Academy
of Sciences, Ekaterinburg, Russia
We consider the λ-open topology on the space of all continuous functions on
a Tychonoff space Χ (λ is a family of subsets of X), i.e. the topology with
a prebase consisting of all sets of the form (F, (/) = {/ 6 C(X) : f(F) С U}
where F £ λ, U is an arbitrary open set in E. The space C(X) endowed
with the λ-open topology is denoted by C\{X).
It is known that the space C(X) in the topology of pointwise convergence
is locally compact iff X is finite. This result proves to be valid for the λ-open
topology. More precisely, the following theorem holds.
Theorem. C\(X) is a locally compact Hausdorff space iffX is finite and
λ contains all points of X.

TOPOLOGY and APPLICATIONS Moscow: PHASIS, 1996
International Topological Conference Pages 87—88
Dedicated to P.S. Alexandroff's 100th Birthday
Moscow, May 27-31, 1996
Note on hereditary normality of product spaces
Akihiro Okuyama
Kobe University, Kobe, Japan
In 1964 K. Morita introduced the concept of P-spaces which characterized
the normality of products with any metric space. We introduce the concept
of weak P(No)-spaces which concern the paracompactness of products with
any /f-analytic space.
Definition. We say that a completely regular Hausdorff space is a weak
P(R0)-space, if for any family
{G(ni,..., щ) : щ,..., щ € TV, г € Ν}
of open subsets of X such that
G(rab...,n;) с G(nb...,n;,n;+i) for nb...,n;,n;+i e N,i e N, (1)
oo
[J G(ni,..., щ) = X for any sequence {пг·} in N, (2)
t=l
there exists a family
{F{nu...,ni) :пь...,пг· e N, ie N}
of closed subsets of X satisfying two conditions below:
F(ni,..., щ) С G(nx,..., щ) for щ,..., щ e Ν, г e Ν, (3)
oo
[J F(ni,..., щ) = X for any sequence {щ} in N. (4)
«=i
It is a slight modification of a P(No)-space and a P(No)-space is always
a weak Ρ (Ко)-space.
As for the difference between two concepts, we use the idea of the Michael
line; that is, for a space X and its subset Л we denote by L(X, A) the space
which re-topologize each point of Л being open in L(X,A). Then we have
the following theorem:

88
Theorem 1. Let X be an uncountable Polish space and A a Bernstein
subset of X. Then L(X,A) is a paracompact weak P(#o)-space but not
a P(No)-space.
This is due to S. Watson's suggestion.
Our main result is:
Theorem 2. For a separable metric space X, if Χ χ Υ is hereditarily
normal for any normal weak P(No)-space У, then X is a countable space.
The proof of Theorem 2 deeply depends on the technique of E. Michael's
proof which showed the non-normality of the product of the Michael line
with the space of all irrationals.

TOPOLOGY and APPLICATIONS Moscow: PHASIS, 1996
International Topological Conference Page 89
Dedicated to P.S. Alexandroff's 100th Birthday
Moscow, May 27-31, 1996
Descriptive theory and discontinuity points
of Baire I functions *
Dmitrii S. Ohezin
Ural State University, Ekaterinburg, Russia
Relations between properties of a topological space X and spaces of
functions defined on X are interesting from the topological point of view. Let
us consider the first Baire class B\(X) — real-valued limits of convergent
sequences of continuous functions. The aim of this work is to describe in
intrinsic terms Tychonoff spaces for which all functions from B\(X) have
small (in topological sense) set of discontinuity points. It is well known that
this set is empty if and only if X is a P-space.
When is the set of discontinuity points finite? Compact? Countably
compact? Discrete? σ-Discrete? Nowhere dense? These problems compose
the contents of the paper.
* This work is a part of the project "Positional Control in Distributed Parameter Systems"
No 95-01-01130a of Russian Foundation for Basic Research.

TOPOLOGY and APPLICATIONS Moscow: PHASIS, 1996
International Topological Conference Page 91
Dedicated to P.S. Alexandroff's 100th Birthday
Moscow, May 27-31, 1996
F-closed spaces
Alexander V. Osipov
Ural State University, Ekaterinburg, Russia
We consider spaces with the functional separation property.
We call a space F-closed if it is closed in any embracing space with
functional separation.
One may assign to any space X its completely regular leader X endowed
with the weak topology with respect to the family C(X).
Proposition 1. X is F-closed iff its leader X is compact
Proposition 2. There exist F-closed but not Η-closed spaces.
Proposition 3· An F-closed regular space is compact.
Proposition 4. If any closed subspace of X is F-closed then X is compact.

TOPOLOGY and APPLICATIONS Moscow: PHASIS, 1996
International Topological Conference Pages 93-94
Dedicated to P.S. Alexandroff's 100th Birthday
Moscow, May 27-31, 1996
On some covering properties
Stanislav A. Peregudov
State Academy of Management, Moscow, Russia
A base (π-base) В for a Τχ-space Χ is a weakly uniform base (π-base) if no
two points of X belong to infinitely many members of β [3].
We assume that all spaces are Hausdorff.
Proposition 1. A space X with a weakly uniform base is hereditarily
s(X)-Lindelof
Using an example from [1] we can prove the following assertion.
Example (MA + u>2 < 2ω). There exists a normal Moore space X with a
weakly uniform base that is not meta-Lindelof and such that for each
separable Μ С X and each open covering % of X there is a refinement Kof%
only countably many members of which intersect M.
This example partially answers a question raised in [2].
It is shown in [3] that the property of having a weakly uniform base is not
preserved by perfect mappings. But it is easy to see that the image of a space
with a uniform base under a perfect mapping has a weakly uniform 7r-base.
A survey of results and questions about spaces with weakly uniform
bases and π-bases will be given.
A family of subsets of a set X is said to be a b-family if any its
centered subfamily is finite. A space X is said to be b-paracompact if every
open cover of X has an open Ь-refinement. These notions were defined by
A.V. ArhangePskii and were studied by the author in a series of papers.
Proposition 2· A regular metacompact, basically compact, locally normal,
locally b-paracompact space is b-paracompact.
Proposition 3· A regular metacompact, basically compact space which has
countable Souslin number is Lindelof

94
A family S of subsets of a set X is an sb-family if V5 £ <S sup{|C| : С
is a centered subfamily of S and 5 € C} < ω. A space is sb-paracompact
space if every its open cover has an open sb-refinement.
Proposition 4· A normal, locally compact, sb-paracompact space is para-
compact
Some other properties and results will be considered.
References
[1] S.W. Davis, G.M. Reed and M.L. Wage, Further results on weakly uniform
bases, Houston J. Math. 2(1) (1976), 57-63.
[2] E.E. Grace and R.W. Heath, Separability and metrizability in pointwise para-
compact Moore spaces, Duke Math. J. 31 (1964), 603-610.
[3] R.W. Heath and W.F. Lindgren, Weakly uniform bases, Houston J. Math. 2(1)
(1976), 85-90.

TOPOLOGY and APPLICATIONS Moscow: PHASIS, 1996
International Topological Conference Page 95
Dedicated to P.S. Alexandroff's 100th Birthday
Moscow, May 27-31, 1996
On σ-additive covers of if-analytic spaces
Evgenii G. Pytkeev
Institute of Mathematics and Mechanics, Ural Branch of Russian Academy
of Sciences, Ekaterinburg, Russia
A family of subsets is σ-additive [σ-multiplicative) if it is closed under
countable unions (intersections). One of N.N. Luzin's results can be formulated
as follows: if γ is a σ-additive family of G$-sets in an analytic space then
(J 7 £ 7· The conclusion become false when analytic spaces are replaced by
/ί-analytic ones (even by compacta). However, we have the following
Theorem· Let у be a σ-additive cover of а К-analytic space X consisting
of Gs-sets. Then X £ γ, i.e. γ contains a countable subcover.
Recall that a map / : X —> Υ is called an Fa-map if the preimage of
any F^-set in Υ is an F^-set in X.
Corollary 1. Let f : X —> Υ be an Fa-map of а К-analytic space X onto
Y. Then the space Υ is Lindelof
Corollary 2· Let Τ be α σ-multiplicative family of Fa-sets in a regular
K-analytic space. Then
fj{i4 : A e T) = f]{A :AeT}^0.

TOPOLOGY and APPLICATIONS Moscow: PHASIS, 1996
International Topological Conference Page 97
Dedicated to P.S. Alexandroff's 100th Birthday
Moscow, May 27-31, 1996
Cell-like mappings and their applications
in Geometric Topology
Dusan Repovs
University of Ljubljana, Ljubljana, Slovenia
Cell-like mappings are a very important class of maps. For example, when
they map between closed manifolds of the same dimension (> 4), they are
limits of homeomorphisms (Armentrout-Quinn-Siebenmann theorem). In
this survey we shall first briefly describe their history and their various
applications and in the sequel, we shall concentrate on the following problem:
The recognition problem for topological manifolds asks for a short list
of topological properties which characterize topological manifolds among
topological spaces and are relatively easy to verify. It splits into two separate
questions each one of which is of interest by itself:
1. Resolution problem. Given a generalized n-mani)old X (i.e. an
Euclidean neighborhood retract which is also a Z-homology n-manifold),
is there a cell-like resolution (i.e. a topological η-manifold Μ and a
cell-like onto mapping f : Μ —> X).
2. General position problem. Given a resolvable generalized
n-manifold X, is there a general position property for X which implies that
the resolution map / : Μ —>- X is approximable by homeomorphisms.
We shall analyze the current status of the recognition problem. In
higher dimensions (> 4) it is pretty much resolved — the general
position problem was resolved by R.D. Edwards in 1977 while the
resolution problem was resolved, modulo a local surgery obstruction, by
F.S. Quinn in 1983 whereas the obstruction was effectively realized by
J. Bryant-S. Ferry-W. Mio-S. Weinberger only recently. On the other hand,
in dimension 3 the existence of resolutions remains for the most part an open
question (closely entangled with the Poincare conjecture). The status of the
3-dimensional general position problem is more satisfactory.

TOPOLOGY and APPLICATIONS Moscow: PHASIS, 1996
International Topological Conference - Page 99
Dedicated to P.S. Alexandroff's 100th Birthday
Moscow, May 27-31, 1996
On stratifiable subspaces
of spaces of continuous functions
with the compact-open topology
Evgenii A. Reznichenko *
Sternberg State Astronomical Institute, Lomonosov Moscow State
University, Moscow, Russia
For a Tychonoff space X, let Ck(X) be the space of real-valued continuous
functions on X with the compact-open topology. We prove the following
theorems.
Theorem 1. If X is a second countable space, then Ck(X) is cometrizable
and any countable subspace of Ck{X) is stratifiable.
Theorem 2· If X is a Polish space, then Ck{X) is stratifiable.
This result implies the following assertions. If X is an No-space, then
Ck(X) is cometrizable and any countable subspace of Ck{X) is
stratifiable. If X is a Lindelof p-space, then any separable subspace of Ck(X) is
cometrizable and any countable subspace of Ck{X) is stratifiable. If X is
a Lindelof Cech-complete space, then any separable subspace of Ck{X) is
stratifiable.
* Joint work with P.M. Gartside, Oxford, United Kingdom.

TOPOLOGY and APPLICATIONS Moscow: PHASIS, 1996
International Topological Conference Page 101
Dedicated to P.S.AlexandrofTs 100th Birthday
Moscow, May 27-31, 1996
On simultaneous extension of continuous
partial functions
Leonid B. Shapiro *
Academy of Labor and Social Relations, Moscow, Russia
For a metric space X let CVC(X) (that is the set of all graphs of real-valued
continuous functions with a compact domain in X) be equipped with the
Hausdorff metric induced by the hyperspace of nonempty closed subsets of
IxR. It is shown that there exists a continuous mapping Φ : CVC(X) —>·
Сь(Х) satisfying the following conditions:
(i) Φ(/)| dom/ = / for all partial functions /.
(ii) For every nonempty compact subset К of X, the restriction
Ф\Сь(К) : Сь(К) -> Сь(Х) is a linear positive operator of norm 1, and
* This is a joint work with H.-P. Kunzi, University of Berne, Switzerland.

TOPOLOGY and APPLICATIONS Moscow: PHASIS, 1996
International Topological Conference Page 103
Dedicated to P.S. Alexandroff's 100th Birthday
Moscow, May 27-31, 1996
On selections of multivalued mappings
Evgenii V. Shchepin
Steklov Mathematical Institute, Russian Academy of Sciences, Moscow,
Russia
A multivalued map F : X —> У is called a submap of a map G : X —>- У if
for every point χ € X one has F(x) С G(z).
A sequence of multivalued maps {Ffc}£=0 is called a filtration of length
η if Ffc is a submap of F^+i for each к < n.
A filtration {Fk} is called open if the graph of Fk is open in the graph
of Fk+ι for all к < η.
A filtration {F&} is called connected if for any к < η and хбХ the
inclusion of Ffc(a;) into Fk+\(x) induces zero homomorphism of ^-dimensional
homotopy group.
A multivalued mapping F : X —>· Υ is called complete if there is a
G$-subset G of the product Χ χ Υ containing the graph of F with the
following property: for each χ e X the product {χ} χ F(z) is closed in G.
At last, a mapping F : X —>Y is called equi-LCn if for every χ £ X and
every open U £ У there exist a neighborhood Ож and open V С U such
that the inclusion F(y)C\V С F(y)C\U induces zero homomorphisms for all
homotopy groups of dimensions < η for у £ Ox.
The main result is the following theorem of the author and N. Brodsky
generalizing the well-known Michael selection theorem [1,2].
Filtered Selection Theorem· If F : X —l· Υ is a lower semicontinuous
equi-LCn mapping of a paracompact space X of dimension dim X < η + 1
into a complete metric space Υ such that F is complete and there exists
an open, connected filtration {Fk} of submaps of F of length η -f 1 with
Fn+i = F, then there exists a continuous function f : X —> Υ such that
f{x) e F(x) for all χ e X.
References
[1] E. Michael, Continuous selections II, Ann. Math. 64 (1956), 562-580.
[2] E. Michael, A generalization of a theorem on continuous selections, Proc. Amer.
Math. Soc. 105 (1989), 236-243.

TOPOLOGY and APPLICATIONS Moscow: PHASIS, 1996
International Topological Conference Page 105
Dedicated to P.S. Alexandroff's 100th Birthday
Moscow, May 27-31, 1996
On the deleted product criterion for embeddability
of manifolds in Rm
Arkadii B. Skopenkov
Lomonosov Moscow State University, Moscow, Russia
We prove the strengthening of Haefliger's theorem:
Theorem. Suppose that N is a closed PL η-manifold. Then N is embed-
dable in Rm if and only if there exists an equivariant map F : N —l· Sm~l,
., , 3n »r . . ι . 3n +1 ^ л
provided m— — + 1 or N ts simply connected and m = > 8.
Here N = {(ж, у) e Ν χ Ν : χ φ у}. The group Z2 acts on N and on
5m_1 by exchanging factors and antipodes respectively. Many corollaries of
Haefliger's theorem are thus also strengthened. Our proof is based on the
extensions of the same theorem for an n-polyhedron N and m > —
(proved by Weber), of the Penrose-Whitehead-Zeeman-Irwin theorem and
of engulfings.

TOPOLOGY and APPLICATIONS Moscow: PHASIS, 1996
International Topological Conference Pages 107-109
Dedicated to P.S. Alexandroff's 100th Birthday
Moscow, May 27-31, 1996
On some compactness-type properties
defined by special open covers
Alexander P. Sostak
University of Latvia, Riga, Latvia
Juris Steprans "
York University, North York,* Canada
Given a topological space (X,T), let 7dp (resp. 7fb> 7cb) denote the
topology on X generated by the family CLP(T) (resp. FB(T), CB(T))
of all clopen sets U € Τ (resp. of all sets U € Τ with finite boundaries,
with compact boundaries). A space (Χ,Τ) is called С LP-compact (resp.
FB-compact, CB-compact) if the space (X, Tc\p) (resp. (X, 7fb), (^>7cb))
is compact. A space (X) T) is called CLP-Hausdorff (resp. FB-Hausdorff,
CB-Hausdorff) if the space {X,TC\P) (resp. (X,7fb), (AT,7cb)) is Haus-
dorff. Obviously,
• compact => СБ-compact => FB-compact => CLP-compact;
• CLP-compactness, FB-compactness and CB-compactness are
preserved by continuous images;
• Hausdorff <= CB-Hausdorff <= FB-Hausdorff <= CLP-Hausdorff.
In the sequel we are mainly interested in the behavior of CLP-, FB- and
CB-compactness under products.
Example. There exists an FB-compact CLP-Hausdorff space whose
square is not even CLP-compact
As an example one can take the space constructed by Stephenson [6].
Theorem 1. Let X\, Xi be CLP-compact (resp. FB-compact,
CB-compact) spaces. Then the following statements are equivalent:

108
(i) the product X\ χ X2 is CLP-cornpact (resp. FB-compact,
С В-compact)]
(ii) the product Χχ χ X2 is С LP-rectangular (i.e. the family {U\ x U2 :
Ui e 7$, i = 1, 2} is a base for the topology TcfplxX2);
(iii) the projection(s) p\ : X\ X X2 ->- ^i or/and p2 : Χχ χ Χ2 —>> X2 ώ/αΓβ
c/open.
(Л mapping is called clopen if the image of each clopen set is clopen.)
Corollary· LetX\, X2 be С LP-compact (resp. FB-compact, CB-compact)
spaces. The product X\ χ X2 is CLP-compact (resp. FB-compact,
CB-compact) in any one of the following cases:
(a) when all quasi-components in X\ are open;
(b) if Χχ is compact;
(c) if the product X\ X X2 is pseudocompact.
Theorem 2. The product of finitely many second countable CLP-compact
spaces is CLP-compact.
The proof ia based on the following auxiliary statement:
Lemma. Let (Хг·, Тг), г = 1, 2, be CLP-compact spaces of countable
weight. If U С X\ X X2 is a clopen set and (a\,a2) € U then there are
sets A{ € 73p, i = 1, 2, such that the interior of the set Αχ χ Α2 \ U with
respect to the topology 7^}p X 72p is empty.
Questions. Is the product of two second countable FB-compact
(CB-compact) spaces again FB-compact (resp. CB-compact)? Does the statement
of Theorem 2 hold for countably many factors? Is the product of two first
countable CLP-compact spaces CLP-compact? Does there exist a pseudo-
compact space X such that its square X2 is CLP-compact but fails to be
pseudocompact?
Remark. CLP-compact spaces were introduced in [3] under the name
clustered spaces. Later such spaces were studied and used in [1, 4, 5] etc. In
particular, in [1] it was proved that a space is compact iff it is
simultaneously CLP-compact and superparacompact. Note also that implicitly the
property of CLP-compactness appears already in [2] (in the definition of a
Λ-mapping).

109
References
[1] D.K. Musaev and B.A. Pasynkov, On compactness and completeness properties
of topological spaces and continuous mappings, FAN Publ. Company, Tashkent,
1994 (in Russian).
[2] V.I. Ponomarev, On star-finite covers and clopen sets, Doklady AN SSSR
186(5) (1969) 1016-1019, (in Russian).
[3] A. Sostak, On a class of spaces containing all compact and all connected spaces,
Proc. 4th Prague Topol. Symp., Part B, Prague, 1976, 445-451.
[4] A. Sondore and A. Sostak, On cip-compact and countably dp-compact spaces,
Acta Univ. Latv. 595 (1994), 129-142.
[5] A. Sondore, On dp-Lindelof spaces, Acta. Univ. Latv. 595 (1994), 143-156.
[6] R.M. Stephenson, Product spaces and Stone-Weierstrass theorem, General
Topol. Appl. 3 (1973), 77-79.

TOPOLOGY and APPLICATIONS Moscow: PHASIS, 1996
International Topological Conference Pages 111-112
Dedicated to P.S. Alexandroff's 100th Birthday
Moscow, May 27-31, 1996
On the dimension of the Higson corona
Vladimir V. Uspenskii
International Institute for Earthquake Prediction Theory and
Mathematical Geophysics, Russian Academy of Sciences, Moscow, Russia
A metric on a space X is proper if every closed bounded set in X is compact.
A topological space X has a compatible proper metric if and only if X
is locally compact and has a countable base. Let X be a metric space
with a proper metric. For χ £ X and r > 0 denote by Br(x) the closed
ball with center χ and radius r. If У is a metric space, we say that a
function / : X —> Υ is slowly oscillating if for any r > 0 the diameter
of the set f(Br(x)) tends to zero when χ £ X tends to infinity. The set
Η of all slowly oscillating bounded complex functions on X is a closed
C*-subalgebra of the C*-algebra C*(X) of all continuous bounded complex
functions on X and thus corresponds to a compactification X of X which is
called the Higson compactification. The compactification X is characterized
by the following property: if К is a compact metric space, then a continuous
function f : X —> К extends over X if and only if / is slowly oscillating.
The Higson corona of X is the compact space vX = X\X.
Novikov's conjecture and some other important open problems can be
reformulated in terms of topological properties of the Higson compactification
and its corona. This justifies our interest in the following question: what is
the dimension of the Higson corona? When is the corona finite-dimensional?
We prove that if X is a Euclidean space with the usual metric, then the
corona i/X is finite-dimensional.
Theorem 1. For the Euclidean space Rn with the usual metric the
dimension of the Higson corona vRn equals n.
The inequality dimi/Rn > η is due to J, Keesling, and the opposite
inequality was proved by the author. A.N. Dranishnikov noted that the proof
can be modified to yield a more general result: the dimension of the Higson
corona vX does not exceed the asymptotic dimension asdim+ X of X.

112
To motivate the definition of the asymptotic dimension, recall that the
dimension of a paracompact space Υ does not exceed η if and only if for
any open cover α of У there exist disjoint families λχ,..., λη+ι of open sets
such that λ = Ur^i1 ^* covers У and refines a. Now let X be a metric space
with a proper metric. We say that a family μ of subsets of X is uniformly
bounded if there is a constant С such that the diameters of all members
of μ are < C. The asymptotic dimension asdim+ X of X does not exceed
η if and only if for any positive number R there exist uniformly bounded
families μ\,..., μη+ι of subsets of X such that for every г = 1, ..., η + 1
all the pairwise distances between members of μ{ are > β and μ = (J^/ M«
covers X. This definition is due to M. Gromov. If X is the n-dimensional
Euclidean space En with the usual metric, then asdim+ X = n.
Theorem 2 (A.N. Dranishnikov-V.V. Uspenskii). Let X be a metric space
with a proper metric. Then the dimension dim vX of the Higson corona does
not exceed the asymptotic dimension asdim+ X of X.

TOPOLOGY and APPLICATIONS Moscow: PHASIS, 1996
International Topological Conference Pages 113-114
Dedicated to P.S.AlexandrofTs 100th Birthday
Moscow, May 27-31, 1996
On semirings of continuous nonnegative functions
on F-spaces
Vera I. Varankina
Vyatka State Pedagogical University, Kirov, Russia
Let X be an arbitrary topological space, R+ be the space of all nonnegative
real numbers with the usual topology and C+ = C(X,R+) (C = C(X,R))
be the semiring of all continuous Revalued (real-valued) functions defined
on X with the ordinary operations of addition and multiplication of
functions. The greatest common divisor (GCD) and the least common
multiple of elements in C+ are defined as usually. A semiring is called a
GCD-semiring (LCM-semiring) if any two its elements have a GCD (an
LCM); a Bezout-semiring if every finitely generated ideal is principal;
distributive if the lattice of all ideals is distributive. We shall say that a
set Л С I is a cozero-set if there exists a function / £ C+ for which
A = {x £ X | f(x) ^ 0}. A space X is called an F-space (Gillman, Henrik-
sen) if for any two disjoint cozero-sets A and В in X there exists / £ C+
such that f(A) = {0} and f{B) = {1}.
Theorem. For every topological space X the following conditions are
equivalent:
(1) C+ is a Bezout-semiring]
(2) C+ is a GCD-semiring;
(3) C+ is an LCM-semiring;
(4) C+ is a distributive semiring;
(5) there exists a canonical isomorphism between the lattice of all ideals
of the semiring C+ and the lattice of all ideals of the ring C;
(6) X is an F-space.

114
Corollary ([1]). For every X the following are equivalent:
(1) С is a GCD-ring;
(2) С is an LCM-ring;
(3) X is an F-space.
Remark. The formulated theorem will be true if the addition is replaced
by the maximum.
References
[1] A.G. Povyshev, On divisibility in rings of continuous functions, Uspekhi Mat.
Nauk49(3) (1994), 185-186.

TOPOLOGY and APPLICATIONS Moscow: PHASIS, 1996
International Topological Conference Pages 115-116
Dedicated to P.S. Alexandroff's 100th Birthday
Moscow, May 27-31, 1996
On the lattice of subalgebras
of a ring of continuous functions
Evgenii M. Vechtomov
Vyatka State Pedagogical University, Kirov, Russia
Let F be a Hausdorff topological field, C(X, F) be the ring of all continuous
F-valued functions on a topological space X, and A(X)F) be the lattice
of all subalgebras of the F-algebra C(X)F). A Τχ-space Χ is called an
F-Туchonoff space if for every point χ £ X and every closed set А С Х\{я}
there exists a function / £ C(X,F) for which f(x) = 1 and f{A) = {0}.
An F-TychonofF space X is called an F-Hewitt space if for every ideal Μ of
the ring C(X) F) with the condition Μ + F = C(X, F) there exists a point
χ € X such that f(x) = 0 for all f e M.
Theorem 1. Any F-Hewitt spaces X and Υ are homeomorphic if and only
if the lattices A(X,F) and A(Y,F) are isomorphic (we can say also that
each F-Hewitt space X is determined by the lattice A(X) F)).
The classes of F-Hewitt spaces and F-compact spaces coincide for many
topological fields F. In this case Theorem 1 implies
Corollary 1. For any topological spaces X and Υ the following are
equivalent:
(1) the lattices A(X) F) and Л(У, F) are isomorphic;
(2) the F-algebras C(X) F) and C(Y) F) are isomorphic;
(3) the rings C(X) F) and C(Y) F) are isomorphic.
Corollary 2· Every R-Hewitt space X (named also real-complete or
functionally closed) is determined by the lattice А(Х,Щ, hence by the ring
C(X,R) too.
In the case of the ring this result was proved by E. Hewitt [1].

116
Theorem 2. Let F be an arbitrary field with discrete topology. Then every
totally disconnected locally compact space X is determined by the lattice of
all F-subalgebras of the ring of all continuous F'-valued functions on X with
a compact support In addition each totally disconnected compact space X is
determined by the lattice of all subalgebras with 1 of the F-algebra C(X, F).
Corollary 3. Every Boolean ring is determined by the lattice of all
its subrings.
Corollary 4 ([2])· Every Boolean algebra is determined by the lattice of all
its subalgebras.
The original technique of minimal and maximal subalgebras was used
for proofs of formulated results.
Theorem 3. For any F - Ту chonojf space X the lattice A(X, F) is modular
(is distributive) if and only if the cardinality \X\ < 2 (\X\ — 1).
References
[1] E. Hewitt, Rings of real-valued continuous functions I, Trans. Amer. Math. Soc.
64(1) (1948), 45-99.
[2] D. Sachs, The lattice of subalgebras of a Boolean algebra, Canad. J. Math, 14(3)
(1962), 451-460.

TOPOLOGY and APPLICATIONS Moscow: PHASIS, 1996
International Topological Conference Page 117
Dedicated to P.S. Alexandroff's 100th Birthday
Moscow, May 27-31, 1996
On sequential completeness of C\(X)
Nikolai V. Velichko
Institute of Mathematics and Mechanics, Ural Branch of Russian Academy
of Sciences, Ekaterinburg, Russia
We consider the space C\(X) of all real-valued continuous functions on
a completely regular space X in the topology of uniform convergence on
elements of λ — a family of bounded subsets of X.
A criterion of sequential completeness of CP(X) was established in [1].
It cannot serve as a sample for a more general case of C\(X). Therefore
another criterion was obtained which says the following.
Theorem· CP(X) is sequentially complete iff for any point-finite sequence
{Уп) of open sets in X and a sequence {Wn} refining {Vn} functionally
(Wn < Vn) we have \JW^ С U Vn.
On the basis of the last criterion, the following condition was established:
(B) Let a sequence σ = {Vn} of open sets in X have the property
Vn = \J{V* : i € N} where V{n are open and V* < VJ+1 < Vn. Let
every A £ λ irftersect only finitely many members of the family
У = {Vn : η e N}. Then (Jy7 С \Ja.
This condition appears to be necessary for sequential completeness of
C\{X) but its sufficiency has not been proved.
A both necessary and sufficient condition was obtained, but its
formulation is too complicated to be placed here.
References
[1] H. Bushwalter and J. Schmets, Sur quelques proprietes de Vespace^ J. Math.
Pures et Appl. 52 (1973), 337-352.

TOPOLOGY and APPLICATIONS Moscow: PHASIS, 1996
International Topological Conference Pages 119-120
Dedicated to P.S. Alexandroff's 100th Birthday
Moscow, May 27-31, 1996
Partial coverings and fibrewise uniformities
Alexei Yu. Zoubov
Lomonosov Moscow State University, Moscow, Russia
For each set Μ and family of sets A we denote A\M — {Α Π Μ : A 6 A}.
Let β be a topological space. For each set X and a mapping / : X —>» В
(also regarded as & fibrewise set over B) let T(/) be a family of all f-partial
coverings, i.e. all pairs (A, V), where V is an open subset of В and A
is a covering of the set f~lV consisting of its subsets. For С С Т(/) and
(A, V) в T(/) we write С < (A, V) if there exists a family {{As, Vs) : s e 5}
of elements of С such that V С \J{VS : s <E 5} and A8 refines A\f"l{V f\V8)
for each sG5.
A family С С Т(/) is said to be a fibrewise uniformity (called F.U.
below) on / if it has the following properties:
(UCO) ({ДВ)6С;
(uci) if с < μ, ν) for μ, ν) e τ(/), then μ, ν) e c,
(UC2) for each (Ль V), (A2, V) e С there exists (A, V) e С such that
A refines both A\ and Аъ\
(UC3) if (Л, V) € С then for each beV there exists (β, W) <E С such
that b € W С V and В is a star-refinement of A\f~lW;
(UC4) for each b € В and different points ж, у 6 /_1Ь there exists
(Л, У) € С such that 6 € V and no element of A contains both
χ and y.
If £ consists of a single point then a F.U. on / is equivalent to a
uniformity on the set X. Each F.U. С С T(/) induces a topology on X: U С X
is open iff for each χ £ U there exists μ, V) € С such that /(ж) G V and
St(x,A) С £/. A F.U. on a continuous mapping / of a topological space X
is said to be uniformizing f if it induces the topology of X. The classical
results of uniform topology (concerning completion, precompactness, Samuel
compactifications etc.) may be expanded to mappings. The following result

120
at that way supplements the generalization of Smirnov's theorem given by
Norin and Pasynkov [2]:
Theorem. For a continuous mapping f there exists a canonical 1-1
correspondence between three sets: all Hausdorff compactifications of f, all
proximities {in the sense of [2]) on f and all precompact F.U.'s on f
that uniformize it The mapping f is uniformizable by a F. U. iff it is
Hausdorff-compactifiable.
The proposed notion of a F.U. generalizes the concepts introduced by
Pasynkov [3] and James [1]. Namely, it is equivalent to Pasynkov's
definition ("completely regular F.U.") if we substitute the (UC3) by a stronger
property, and equivalent to James' one if В is regular. Considering a F.U.
also allows to define a metrizable continuous mapping as a mapping that
can be uniformized by a F.U. having a countable base. The last definition
seems to be a rather appropriate answer to the question about a definition
of such kind stated by Pasynkov and James [1].
References
[1] I.M. James, Fibrewise topology, Cambridge Univ. Press, 1989.
[2] V.P. Norin and B.A. Pasynkov, Proximities on mappings, bicompactifications
of mappings, in: General Topology. Spaces, mappings and functors, Moscow
Univ. Press, 1992, 93-111 (in Russian).
[3] B.A. Pasynkov, Uniformities on mappings, Int. Rep. Prague Topol. Symp. 3,
Prague, 1988, 23-24.

Session 2
Algebraic Topology

The abstracts are presented in the alphabetical order
of the authors' names

TOPOLOGY and APPLICATIONS Moscow: PHASIS, 1996
International Topological Conference Page 123
Dedicated to P.S. Alexandroff's 100th Birthday
Moscow, May 27-31, 1996
Asymptotically flat solutions of Bogomolny
equations with solvable gauge group
Arthur M. Aslanyan
Kazan State University, Kazan, Russia
In a standard domain of the Euclidean space R , consider a pair (φ, Aa)
composed of a scalar function φ and a gauge field Aa assigned to a gauge
group G. The hat φ denotes the matrix representation of the Lie algebra of
the group G. A special type of solutions of Bogomolny equations is studied:
Όαφ = Ba, where Όαφ = даф + [Aa, φ] is the gauge derivative of the scalar
function φ, Ba = \eabcFbc, Fab = даАъ - дьАа + [Λα, Ль] is the curvature
tensor of the field Aa.
In what follows we use as a gauge algebra the Lie algebra of a
harmonic oscillator ho(N, E) which is quasi-nilpotent and admits an invariant
nondegenerate bilinear form.
Let {ei, ea, e^} be a basis of the algebra ho(N, R) where a — 2, N — 1,
and its dimension N = dim ho(N, R) be even.
The structural constants:
[euep] = 0,
[ep,eq] ^u)pqei,
[eN,ex] = 0,
[eN,eq]=uPe^
where upq is a nondegenerate skew-symmetric matrix.
Definition 1. The value of the integral Ε = / з(£а<£, Ба) d3x is called the
enerflfy.
Definition 2. The pair (<£, Aa) is called a monopole if the functions £ and
л О
Аа are regular in the entire space R and the energy Ε < oo.
Theorem. Every asymptotically flat monopole solution of Bogomolny
equations with gauge group HO(N, R) is trivial, i.e. φ = const and A = 0 in an
appropriate gauge.

TOPOLOGY and APPLICATIONS Moscow: PHASIS, 1996
International Topological Conference Pages 125-126
Dedicated to P.S. Alexandroff's 100th Birthday
Moscow, May 27-31, 1996
The orbits of the coadjoint representation
for Lie groups which have some special structure
Dmitrii V. Berzin
Lomonosov Moscow State University, Moscow, Russia
Let us define for a Lie algebra G (dimG = n) the bilinear and skew-
symmetric form in the following way. The structural constants C^ is a
Jordan box such that λ is on the diagonal and units above the diagonal,
C* =0ifi5tn(j=lf 2, ..., n-1).
Under all the above assumptions we can write out the functionally
independent sets of the invariants in the cases λ = 0 and λ φ 0 (see [1]):
F\ = xu
Г~2 r_kz\z\-k-lZk+2 ,_,
* = ^Утфщ^У1) V^w-^1^
*=1
r = 3, 4, ..., η - 2,
G\ = x\ exp(-Ax2/zi),
Gi = —Fh / = 2, 3, ..., n-2.
x[
In the above expressions F{ are the invariants for the case λ = 0, Gi are
the invariants for the case λ φ 0 (г — 1, 2, ..., η-2).
Consider the tensor extension Q(G) which corresponds to the ring
R(x)/(x2) (see [2]). It turns out that the dimension of the general
position orbits is four.
We can write out the invariants of the coadjoint representation for the
Lie algebra Ω(Θ) and consider a general position orbit as a symplectic
manifold. Using the method of argument translation, we can construct involute

126
sets of functions on the orbits. Couples of functions from the involute set
give us a mapping of the moment. Bifurcation diagrams for these cases are
trivial from the topological viewpoint.
References
[1] D.V. Berzin, The invariants of the coadjomt representation for Lie algebras
which have some special structure, Uspekhi Mat. Nauk 51(1) (1996), 141.
[2] V.V. Trofimov and A.T. Fomenko, Algebra and geometry of mtegrable Hamil-
toman differential equations, Moscow, 1995.

TOPOLOGY and APPLICATIONS Moscow: PHASIS, 1996
International Topological Conference Pages 127-129
Dedicated to P.S. Alexandroff's 100th Birthday
Moscow, May 27-31, 1996
Double cobordism, flag manifolds,
and quantum doubles
Victor M. Buchstaber
Lomonosov Moscow State University, Moscow, Russia
Nigel Ray
University of Manchester, Manchester, United Kingdom
In 1978 the first author and Shokurov expressed the Landweber-Novikov
algebra £* as an algebra of differential operators on a certain algebraic
group. Inspired by this work, S.R Novikov more recently described the
algebra Ац of cohomology operations in complex cobordism as the operator
double of 5* and its action on the complex cobordism ring Ω^. This in
turn led the first author to construct the quantum double D(5*) of 5*
(the quantum group of complex cobordism) as a subalgebra of cohomology
operations in double complex cobordism theory β^ί/(')> which had been
introduced (for very different purposes) by the second author in the 1960s.
Our aim here is to develop and extend these results in the contexts of
homotopy theory, algebra, geometry, and combinatorics.
From the point of view of homotopy theory, we construct the double
complex cobordism functor in terms of the doubly indexed spectrum MUAMU.
We employ May's setting of coordinate-free spectra to ensure that the
defining maps, including the product and bimodule structure maps over Μ [/, are
suitably coherent. Geometrically, we describe double [/-structures in terms
of manifolds whose stable normal bundle carries a specific splitting v\ 0 vr
as a sum of left and right [/-bundles; this construction also follows standard
lines, except that we must again take care with the necessary double
indexing. The homotopy theoretic and geometric approaches are linked by an
appropriately refined version of the Pontryagin-Thom construction, which
identifies n*(MUAMU) with the double complex cobordism ring
By definition, double complex cobordism is the universal example of a
doubly complex oriented cohomology theory, and we discuss its properties

128
with this viewpoint in mind. The two orientations are related by a canonical
formal power series, whose coefficients define a polynomial subalgebra G* of
Ωζυ. The orientations correspond to the bimodule structure over β^7, and
lead to left, right, and adjoint actions of 5* on Ωζυ, with respect to which
we may readily show that G* is invariant. Since the work of Adams it has
also been possible to interpret n*(MU Λ MU) as Ajf, the Hopf algebroid of
homology cooperations dual to AJ), and therefore to identify 5*, the integral
dual of 5*, as a sub Hopf algebra. One of our central observations is that the
Pontryagin-Thom construction provides a canonical isomorphism between
5* and G*.
Using this isomorphism, it is important to relate the three actions of 5*
on G* with the actions and coactions which arise from the standard theory
of complex oriented cohomology theories (of which i2jj/(·) is the universal
example) and their Hopf algebroids of cooperations. We carry out this
algebraic part of our programme in terms,of Boardman's eightfold way,
which provides a systematic and unified framework within which to work.
As an immediate consequence, we recover the first author's realization of
D(5*) as a subalgebra of A*DU.
At this juncture we return to the geometry, and investigate the algebraic
structures in the context of Ωζυ. Our starting point is the construction
of double {/-manifolds to represent generators gi of the subring G*, for
which we use certain iterated two-sphere bundles introduced by the second
author to study splittings of normal bundles in complex cobordism. These
manifolds are actually special cases of Bott-Samelson desingularizations of
Schubert varieties, although we interpret them here in terms of restricted
(or bounded) manifolds of flags, and invest them with double {/-structures
which differ from those induced as complex varieties.
We first describe the complex bordism and cobordism of our bounded
flag manifolds Bn) deducing the action of Poincare duality from the
appropriate intersection theory. We are led to subvarieties Xq and Yq of Bn,
indexed by subsets Q С {1, 2,..., η} (with X{\,2,...,n} = Bn), which
underlie a combinatorial calculus based on the Boolean algebra of subsets of an
η-element set. The Xq are the closures of the cells in a natural CW
decomposition of Bn and are always nonsingular; in this sense our calculus is
related to the Schubert calculus of Bressler and Evens for the complex
cobordism of generalized flag manifolds, and some of our computations replicate
theirs. In particular, we may work with any complex oriented spectrum E.
We extend our calculus to cover the double complex bordism and
cobordism of the Bn, taking special care over Poincare duality, which lies at the

129
core of several formulae. The Xq, Yq, and Bn all carry many natural
double [/-structures, whose interplay with the algebra is especially fascinating;
for example, the Xq may be used to represent monomials in the
generators </,·. Once this geometry is fully documented, we have a sufficiently
rich environment to realize our algebra. Our final result therefore records
the geometric description for the structure maps of the Hopf algebroid Л^,
together with the left, right and adjoint actions of 5* on G*, in terms of
the Xq and appropriate double [/-structures. As a corollary, we obtain
a geometrical interpretation of the commutation law which describes the
quantum double.
By way of illustration, we note that the diagonal of A^ may be
evaluated on X[i,2,...,n} a·8 YLq xQ ®Yq·, and the antipode may be obtained by
interchanging left and right summands of the normal bundle (up to double
[/-cobordism class and suitable double [/-structures, in both cases).
We shall explain in future the generalizations available by passing
to η-fold [/-structures for η > 2; they admit corresponding
realizations of operators doubles, and have applications to the Adams-Novikov
spectral sequence.

TOPOLOGY and APPLICATIONS Moscow: PHASIS, 1996
International Topological Conference Pages 131-132
Dedicated to P.S. Alexandroff's 100th Birthday
Moscow, May 27-31, 1996
Strong excision property
for coherent homology
Yuri T. Lisica
Russian Peoples' Friendship University, Moscow, Russia
In papers [1], coherent homologies for inverse systems of topological spaces
and pairs of topological spaces were defined. O-resolutions of pairs of
topological spaces make possible to define strong homology groups Hp(X)A;G)
for any pair (X, A) of topological spaces with coefficients in an Abelian group
G. These homologies have the following strong excision property.
Theorem. Let (X) A) and (У, В) be closed normal pairs such that for any
neighborhoods U and V of the sets A and B, open in X and Υ respectively,
the complements X\U and Υ \ V are normally located in X and Υ
respectively. Further, let f : (X, A) —l· (У, В) be a closed continuous map of pairs
inducing a one-to-one map of the sets X\A and Y\B. Then for any ρ £ Ζ
and any Abelian group G the induced homomorphism
UiHp{X,A\G)^Hp{Y,B\G)
is an isomorphism.
Corollary 1. Let (X) A) be a closed normal pair such that for any open in
X neighborhood U of the set A, X \U is normally located in X. Then the
natural projection
π:(Χ,Α)->(Χ/Α,*)
induces an isomorphism
π* : HP(X, A; G) -> HP(X/A, *; G)
for any ρ £ Ζ and any Abelian group G.

132
Corollary 2. For every collectionwise normal countably paracompact space
X, for each ρ and every Abelian group G there is an isomorphism
HP{X,*;G)*HP+1{SX,*;G),
where SX is the suspension over X.
References
[1] Yu.T. Lisica and S. Mardesic, Strong homology of inverse systems of spaces I;
II; III, Topol. Appl. 19(1) (1985), 29-43; 19(1) (1985), 44-64; 20(1) (1985),
29-37.

TOPOLOGY and APPLICATIONS Moscow: PHASIS, 1996
International Topological Conference Pages 133-135
Dedicated to P.S. Alexandroff's 100th Birthday
Moscow, May 27-31, 1996
Reflexivity of convergence Abelian groups
M. Jesus Chasco
Universidad de Vigo, Vigo, Spain
Elena Martin-Peinador *
Universidad Complutense de Madrid, Madrid, Spain
A number of attempts to extend Pontryagin duality theory to categories of
groups larger than that of locally compact Abelian groups have been made
using different approaches. The extension to the category of topological
Abelian groups created the concept of reflexive group. We deal now with
the extension of Pontryagin duality to the category of convergence Abelian
groups. Reflexivity in this category was defined and studied by E. Binz
and H. Butzmann. A convergence group is reflexive (subsequently called
BB-reflexive by us in our work) if the canonical embedding into the bidual
is a convergence isomorphism.
We denote by TG the set of all continuous homomorphisms (i.e.
continuous characters) from an Abelian topological group G into the circle
group T. If addition in TG is defined pointwise, then TG endowed with
the compact-open topology is a topological Abelian group, which will be
called GA.
The group G is called reflexive if the natural embedding olq from G
into the bidual GAA := (GA)A is a topological isomorphism. The classical
Pontryagin duality theorem states that every locally compact topological
Abelian group (LCA) is reflexive.
Examples of reflexive groups which are not locally compact are known
from the late forties. In [7] it is proved that arbitrary products of
LCA-groups are reflexive, whilst they may not be locally compact, like Κω or
Rc. In [9] it is proved that any infinite dimensional Banach space considered
in its additive structure is a reflexive group.
* Some of the results mentioned have been obtained jointly with W. Banaszczyk and
M. Bruguera.

134
We include here the definition of a convergent <* structure, and of a
convergence group.
Let X be a set and suppose that to each χ in X is associated a collection
Ξ(χ) of filters on X satisfying:
i) the ultrafilter {А С X : x £ ^4} is in Ξ(χ),
ii) if ^ G Ξ(χ) and Q £ Ξ(χ), then the filter /0? = {FUG: FG f,
Gg6} also belongs to Ξ(χ),
iii) if Τ £ Ξ(χ) and £ Э .F then £ £ Ξ(χ).
The totality Ξ of filters Ξ(χ) for χ in X is called a convergence structure
for X, the pair (Χ, Ξ) a convergence space and the filters ^* in Ξ(χ) will be
called convergent to x. We write Τ —ϊ χ instead of Τ £ Ξ(χ). A mapping
f : X —l· Υ between two convergence spaces X and Υ is continuous if
/(.F) —>- /(a?) in У whenever Τ —>- χ in X.
A convergence group (G, Ξ), or briefly G, is a group for which the
convergence structure Ξ is compatible with addition. If G is a convergence group,
we also call TG the set of all continuous homj.norphisms (in the sense of
convergence) from G into Τ and the continuous convergence structure A, in
TG, is defined in the following way: a filter Τ in TG converges in A to an
element ξ £ TG if for every χ £ G and every filter 7^ in G, convergent to x,
w[T X W) converges to £(x) in Τ. (Τ Χ Ή, denotes the filter generated by
the products F X G, F £ ^* and Я £ П, and w(F X Я) := {/(x) : / £ F
and χ £ Я}). Thus A is the coarsest convergence structure in TG for which
the evaluation mapping w is continuous.
E. Binz and H. Butzmann have succeeded to extend Pontryagin duality
theory to the category of convergence Abelian groups and continuous homo-
morphisms, CONABGRP. They first define the "convergence dual" TCG of
a group G £ CONABGRP, as the set of all continuous characters endowed
with the "continuous convergence structure". If G is a LCA group, the
continuous convergence structure in TG is precisely the convergence given
by the compact open topology [3], thus, the "convergence dual" and the
ordinary dual are identical. They call G reflexive if the natural embedding
kg - G —>- rcrcG is an isomorphism in the category CONABGRP. They
have studied many features of this concept of reflexivity. To mention one,
a topological vector space, regarded as an Abelian group, is BB-reflexive if
and only if it is locally convex and complete [4].
Topological Abelian groups are, in an obvious way, convergence groups,
therefore it is natural to compare reflexivity and ВВ-reflexivity for them.

135
We have proved that these two notions are in general independent although
they coincide for some classes of topological groups.
A natural question is to study properties of reflexive groups shared also
by BB-reflexive groups. In previous work [2] we proved the following: 1) If A
is an open subgroup of a topological group G, then G is reflexive if and only
if A is reflexive. 2) If К is a compact subgroup of a group G with sufficiently
many continuous characters, then G is reflexive \f[G/K is reflexive. We have
seen that analogous statements hold for BB-reflexivity.
It is well known the existence of convergences compatible with a group
structure which do not derive from a topology. Thus, the natural forgetful
functor from TOPABGRP into CONABGRP is not surjective. The above
results have been obtained for the image of this functor; we do not know yet
how to extend our proofs to the whole category CONABGRP.
Finally we have used the continuous convergence structure to prove that
every reflexive admissible topological group must be locally compact.
References
[1] W. Banaszczyk, On the existence of Exotic Banach-Lie Groups, Math. Ann.
264 (1983), 485-493.
[2] W. Banaszczyk, M.J. Chasco, and E. Martin-Peinador, Open Subgroups and
Pontryagtn Duality, Mathematische Zeitschrift 215 (1994), 195-204.
[3] E. Binz, Continuous Convergence in C(X), Lecture Notes in Mathematics 469,
Springer-Verlag, Berlin-Heidelberg-New York, 1975.
[4] H.P. Butzmann, Pontrjagin-Dualitat fur topologische Vektorraume, Archiv der
Math. 28 (1977), 632-637.
[5] M.J. Chasco and E. Martin-Peinador, Binz-Butzmann duality versus
Pontryagtn Duality, Archiv der Math. 63(3) (1994), 264-270.
[6] H.R. Fischer, Limesraume, Math. Ann. 137 (1959), 269-303.
[7] S. Kaplan, Extension of the Pontryagtn Duality. I: Infinite Products, Duke
Math. J. 15 (1948), 649-658.
[8] E. Martin-Peinador, A reflexive admissible topological group must be locally
compact, Proc. Amer. Math. Soc. 123 (1995), 3563-3566.
[9] M. Smith Freundlich, The Pontryagtn duality theorem in linear spaces, Ann.
Math. 56(2) (1952), 248-253.

TOPOLOGY and APPLICATIONS Moscow: PHASIS, 1996
International Topological Conference Pages 137-139
Dedicated to P.S. Alexandroff's 100th Birthday
Moscow, May 27-31, 1996
On the orbit topology for co-adjoint representation
of tensor extensions of Lie groups
Tatiana L. Melekhina
Moscow State Pedagogical University, Moscow, Russia
It is known that there exist topological obstacles for the existence of global
canonical coordinates on a symplectic manifold. In the present work we
show a large class of symplectic manifolds such that there is a system of
canonical coordinates for one.
Let Л be a commutative ring and G be a Lie algebra. Then the tensor
product G ® A is a Lie algebra with respect to the product \g ® a, h ® b] =
[<7, h] ® ob, where g,h £ G, a,b € A.
Let us suppose that К = Щж]/(жп) is the quotient ring of the polynomial
ring R[x] by the ideal generated by жп, and G is an arbitrary Lie algebra. If
€i,..., er is a basis of the Lie algebra G, r = dim G, then
€\j . .., 6r,
ei ® ε, ..., er ® ε,'
ex ®εη~χ, ..., er®en~l
is a basis of the Lie algebra G ® K, where ε = π (χ), and π : R[x] ->■ /ί
is the canonical projection. Let us denote the coordinates in the space
(G ® K)* with respect to the dual basis of the space G (8) К by
а?«1(0),ж1-2(1),...,а:»п(п — 1), where жг1(0) is related to the basis e*1 € G*,
and »i2(l) is related to the basis e*2 ®ε € (G®e)*, ..., Xin{n- 1) is related
to the basis etn ® ε71*"1 e (G ® ε71*"1)*, where e1,..., en is the dual basis in
G*,i.e. β·'(β,) = *}·
Now we give a generalization of Trofimov's algorithm (t) [1] to the case
of arbitrary smooth functions. We shall assume that the algorithm (t)
transforms a function F defined on the space G* into the following set of functions

138
F<0\...,Fln-1), defined on the space (G<g>K)*, where
ыт\, ^ 1 dmF(xAm)) h . l4
F(m)<2> = Шаж„ ..^Π-(»-ΐ)
+ 7 ™ 7S я~^ Π ** т " χ χ«—ι т " 2
»т-3
Π xi(m - l)xim_2{m - 3)
1 am-2F(xt(m))
(т-4)!#хг1 ...0s,-TO_2
tm-4
г=г\
1 *m—4
+ (та-2)! В Ж'^т ~ ^ж<—8^го ~ 2^*'—г(т ~ 2^
С/Ж*
Theorem 1. Let G be an indecomposable real Lie algebra, dim G = 3, 4, 5,
G ψ. so(3), or G be an indecomposable real nilpotent Lie algebra, dim G — 6.
Using the algorithm (t) we can construct global canonical coordinate systems
in an explicit form simultaneously on all orbits of general position for the
co-adjoint representation Ad* of the Lie group corresponding to the Lie
algebra ((G <g> Κχ) <g> K2)... ® KS) where Ki = R[x]/(xni).
The proof of this statement is based on the following theorem.
Theorem 2. Let us suppose that functions p\f ... , ps, q\f ... , qS) defined
on the space G*f are giving canonical coordinate system on all generally
positioned orbits of the co-adjoint representation Ad* of the Lie group
corresponding to the Lie algebra G, i.e.
{Pi>Pj} = {ft,ij} = 0, {puqj} = Sij.
Then the functions
Pl __ Jo) ρ _ jo) ρ _ (ι) ρ _ (l)
ρ , ч _ J71-1) ρ _ J71""1)
' *(n-l)+l — Pl > · · ч ίβη — Ps ,
and
^ (n-l) ^ (n-1) ^ (0)

139
on the space (G ® K)* are giving canonical coordinates on all generally
positioned orbits of the co-adjoint representation Ad* of the Lie group
corresponding to the Lie algebra G ® K, i.e.
{P» Pj} = {Qi, Qi) = о. {К, Qj) = *i·
References
[1] V.V. Trofimov and A.T. Fomenko, Algebra and Geometry of integrable Hamil-
tonian systems of differential equations, Factorial, Moscow, 1995 (in Russian).

TOPOLOGY and APPLICATIONS Moscow: PHASIS, 1996
International Topological Conference Pages 141-142
Dedicated to P.S. Alexandroff's 100th Birthday
Moscow, May 27-31, 1996
Metric approach to constructing Fredholm
representations
Alexander S. Mishchenko
Lomonosov Moscow State University, Moscow, Russia
The Fredholm representation theory of discrete groups is useful for
constructing homotopy invariants of non-simply connected manifolds on the
basis of generalized Hirzebruch formula
[σ(Μ)] = (L(M) &A /4, [M]> € K^(pt) <g> Q (1)
where A = C*[n] is the group C*-algebra of the group π, π = 7Γι(Μ),
ξ € Κ^(βπ) is the canonical Α-bundle which is generated by the natural
representation π —> А, / : Μ —> Βπ is the map which is induced
isomorphism of fundamental groups; [σ(Μ)] € K^(pt) is the image of symmetric
signature of the manifold Μ generated by change of rings Ζ j [?r]cA.
Let ρ = (Γι, F, Γ2) be a Fredholm representation of the group π, that is a
pair of unitary representations Γι,Γ2 : π —> B(H) and a Fredholm operator
F : Η -+ Η such that
Fr^-r^FECompitf), g € тт. (2)
Changing the bounded operator algebra £(#) for the Kalkin algebra
/C = J3(#)/Comp(#), one comes to the representation ρ of the group π Χ Ζ
in the Kalkin algebra. The latter induces the homomorphism
p. : KA(X) -^ KAbc(sl)(X χ S1) Λ **:(* x S1). (3)
Here /3 £ Kc(Sl)($l) ls the canonical element generated by the regular
representation of the group Z. Using (3) to (1) one has homotopy invariance
of the corresponding higher signature.
Thus let Γ be a finite sum of copies of the regular representation of the
group π, let F be a blockwise diagonal operator which can be defined as a
matrix function U(g)) g € π. F is Fredholm if
W(g)\\<C, \\U-l(g)\\<C

142
for any g £ π excepting a finite family. The condition (2) means that
ton \\U(g)-U(hg)\\ = Q.
Consider the universal covering Βπ of Βπ with right action of the group
7Γ. Consider a metric on Βπ such that
г{хд,уд)->Ъ, \g\ -xx>.
Let Βπ be the completion of Βπ. Then any continuous mapping
f : (Ш,Ж\В^) ^ (B(V),U(V)) (4)
defines a Fredholm representation p, and (4) defines an element
[/\€К°(Ш,Ш\Вп). (5)
The direct image of [/] coincides with the canonical element p(£a) €
Κ°(Βπ) generated by (3). Hence the problem of constructing a sufficient
family of Fredholm representations is reduced to the study of the homotopy
type of the pair (4).
For example, in the case when Βπ is a compact manifold and Βπ can be
compactified to a disc with an action of the group π, one has a new proof of
the Novikov conjecture in the case [1]. This result can be generalized over
the case of a noncompact manifold Βπ if Βπ is compactified to a disc with
an action of the group π.
References
[1] F.T. Farrell and W.C. Hsiang, On Novikov's conjecture for non-positively curved
manifolds I, Ann. Math. 113 (1981), 199-209.

TOPOLOGY and APPLICATIONS Moscow: PHASIS, 1996
International Topological Conference Page 143
Dedicated to P.S. Alexandroff's 100th Birthday
Moscow, May 27-31, 1996
Parametrized Borsuk-Ulam theorems
for Banach bundles
Neza Mramor-Kosta
University of Ljubljana, Ljubljana, Slovenia
Let G be the group of elements of length 1 in a field F 6 {R,C, H}. If Ε
and E' are F-vector bundles over a paracompact connected base В with an
orthogonal action of G in the fibers and / : S(E) —> E1 is an equivariant
map from the sphere bundle S(E) of Ε then, under the assumption that
the action in S(E) is free (or under various other slightly less restrictive
assumptions), the size of the set
Af = Γ1 (£q), К = 0-section of E'
can be estimated by
dim Aj > dim В + (m — η + 1),
where m and η are the dimensions of the fibers of Ε and E\ and dim
denotes the cohomological dimension. This estimate can be viewed as a
parametrized verion of the Borsuk-Ulam theorem. An analogous estimate
is valid for complex bundles with G equal to Zp or, more generally, Zpa,
where ρ is a prime and α € N. A generalization to maps / : V(E) —>- £',
where V(E) is the associated bundle of Stiefel varieties to the vector bundle
Ε with an action of the orthogonal, unitary or symplectic (depending on F)
group in the fibers, was proved by Jaworowski.
The object of this contribution is to discuss possible generalizations of
these results to the case where the bundles are infinite-dimensional Banach
bundles. In the simpler situation of a map f : S —ϊ Ε from the unit
sphere 5 С Ε of an oo-dimensional Banach space Ε to Ε there exists a
generalization of the Borsuk-Ulam theorem which was originally proved by
Granas and which is valid for maps / that are compact perturbations of the
identity with image in a subspace of Ε of finite nonzero codimension. This
result is extended to Banach bundles over a compact base for all groups
mentioned above.

TOPOLOGY and APPLICATIONS Moscow: PHASIS, 1996
International Topological Conference Page 145
Dedicated to P.S. Alexandroff's 100th Birthday
Moscow, May 27-31, 1996
Splitting along one-sided submanifolds
Yuri V. Muranov
Steklov Mathematical Institute, Russian Academy of Sciences, Moscow,
Russia
Let / : Μ —> Υ be a simple homotopy equivalence of smooth (PL) manifolds
of dimension n, X С Υ be a submanifold of codimension q. In this case
the splitting obstruction groups LSn-q(F) are defined. These groups do not
depend on specific manifolds' pair and depend only on pushout square F of
fundamental groups (with orientation)
πιφιη _> πι(γ\χ)
πι(Χ) —► m(Y)
and η — q mod 4. All maps in the square F are induced by natural
inclusions. In what follows we will consider the case of a one-sided submanifold.
If the horizontal maps in the square F are isomorphisms then the groups
LSn(F) coincide with Browder-Livesay groups LNn(n\{Y \ X) —> π\(Υ)).
We consider L5-groups for the case of a one-sided submanifold when the
horizontal maps in the square F are epimorphisms. We obtain new
connections between groups LS(F) and ^-groups of the manifold У, L-groups of
its submanifold X, and Browder-Livesay groups LNn(K\(Y\X) —> π\(Υ)))
mdLNn(ni(dU)->ni{X)).
In the present talk new results about the relationship between
splitting obstruction groups Novikov-Wall groups for the case of a one-sided
submanifold are presented.

TOPOLOGY and APPLICATIONS Moscow: PHASIS, 1996
International Topological Conference Pages 147-148
Dedicated to P.S. Alexandroff's 100th Birthday
Moscow, May 27-31, 1996
Generalization of the Markov theorem
and Cantorian-like braid groups
Hanna Nencka
Center of Mathematical Sciences, University of Madeira, Funchal,
Portugal
In 1935 A.A. Markov [4] announced a fundamental theorem relating knot
theory with braid group theory.
Theorem 1 ([4])· Let β and β1 be two braids respectively from braid groups
Bn and Bm with η φ m. The links β, β' [1] are ambient isotopically if β' can
be obtained from β by a series of equivalences in a given group, conjugations
in a given group, and Markov moves.
In 1974 Birman [2] gave a proof of the above combinatorial
equivalence theorem. In this paper we present some generalization of the
Markov theorem.
We need to remark that:
Definition. Two braids /?, β' £ Bqq are Markov equivalent if the
corresponding links /?, β' represent the same link.
We consider the Markov theorem in the following form [2, 3] as well:
Theorem 2 ([2, 3]). The Markov equivalence is generated by a conju-
gacy in each Bn and a map fn : Bn —>- Bn+i which takes a word
Η^(σι,...,ση_ι) e Bn to a wordW(ai,...,an-i)a£ 6 Bn+X.
In order to establish a generalization of the Markov theorem, we
introduce some new natural representation of braids in terms of interwoved
strings. This new calculus allows us to generalize the Markov theorem.
The generalization of Markov theorem consists of the following three
theorems.

148
Theorem 3· Let B2%, B2i+i be two braid groups, then for any г £ N there
exists a map fi : B2% —l· B2i+i such that fo is a natural induction map.
Theorem 4· There exists a braid group B2n of rank 2N with N being the
natural number set, having an uncountable set of generators σ, such that
any infinite disjoint union of braid groups Boo С В2ъ.
Theorem 5. The Markov generalized equivalence is generated by a conju-
gacy in each B2n and a natural induction map fn : B2n —> B2n+i, which
takes a word W(a\, σ2,..., σ2η_ι) to a word
W(a\)a2,.. .,σ2η^1)σ2ησ2η+1 .. .^2Λ+1-ι € ®2η+ι
where η £ N.
References
[1] E. Artin, Theorie der Zopfe, Hamburg Abh. 4 (1925), 47-72.
[2] J.S. Birman, The braids, links and mapping class groups, Princeton Univ. Press,
Princeton, 1974.
[3] J.S. Birman, New points of view in Knot Theory, Bull. Amer. Math. Soc. 28
(1993), 253-288.
[4] A.A. Markov, Uber die freie Aquivalenz geschlossener Zopfe, Recueil Mathe-
matique Moscou 1 (1935), 73-78.

TOPOLOGY and APPLICATIONS Moscow: PHASIS, 1996
International Topological Conference Pages 149-151
Dedicated to P.S. Alexandroff's 100th Birthday
Moscow, May 27-31, 1996
General Relativity as a construction of the formal
theory of Lie pseudogroups
Andrew Yu. Neronov
Lomonosov Moscow State University, Moscow, Russia
Alexei M. Boyarskii
Lomonosov Moscow State University, Moscow, Russia
General Relativity can be described in the framework of the formal
theory of Lie pseudogroups. Recently description of continuum mechanics,
electromagnetism and thermodynamics has been obtained in terms of the
construction of non-linear Spenser sequence for the groups of Euclidean
motions of 3-d space (for non-relativistic electrostatics), Poincare (for rel-
ativistic dynamics), and conformal (for electricity). The dynamics of
General Relativity arises as diffeomorphism-invariant dynamics in terms of the
non-linear Spenser sequence in accordance with the general scheme of this
theory. The pairing of matter with gravity is obtained in a unique way,
because the form of the invariant dynamic equations depends only on the group
chosen for the construction (diffeomorphisms in the case of General
Relativity). The dynamics must be interpreted as a dynamics of deformations
of space-time as far as full analogy with continuum mechanics is reached
(the construction of General Relativity is obtained when replacing Poincare
group by diffeomorphisms). So one gets the possibility to use the notions
of "Lagrange variables", "Euler variables", "nondeformed state" and other
notions of continuum mechanics. The topology of the space-time manifold
should not be fixed while solving Einstein's equations, because gluing
functions of the manifold (a special type of transformations or deformations of
space-time in this framework) are determined dynamically. This formalism
is closely related to the tetrad formalism of General Relativity and is in
fact an extension of the latter with the aid of the so-called Jet Theory to all
geometrical objects of Riemannian geometry.
Let Μ be a Riemannian manifold, Jq(M χ Μ) a bundle of 9-jets
of the direct product Μ Χ Μ with local co-ordinates (x^y^). Here

150
μ = (μι,μ2,.. .,μη), where μι = О, 1, 2,..., is a multi-index such that
3μ = (#ι)μι(#2)μ2 ... (9η)μη- The operations of composition and taking the
inverse element could be defined on the invertible jets (dety* φ 0) in
Jq(M X M). The corresponding expressions can be found using the rule
of the derivation of a composite function:
J к Jk\
К/,^)о(у',Л4) = (ЛЛ^)
χ1 = χ; ζ1 = z\
~lk __ ~к „т.
zi — zmiJi J
ι y/fc — yk »/7n7/n ι rA; ».ra
(i)
(2)
The inversion is defined naturally as /g о /~* = jg(id). In this way the
set of invertible sections of Jq(M X M) is equipped with the structure of
groupoid Пд(М, Μ) С Jg(M χ Μ). Surfaces &д С Jg(M χ Μ) defining
different groups of transformations possess the "group" property: they are
closed with respect to composition and inversion. Generally, the equations
which define the surfaces with this property are called Lie equations.
Let us construct the initial part of the non-linear Spenser sequence:
0
г ^ тгд+1 Лг®й?Д л2т* ® #,_!
Γ being the pseudogroup of solutions of the non-linear system 1Zq,
D(fq+i) = f~+x oji(fq) - idg+i and D(jq+i(f)) = 0. The composition law
for 1Zq+i gives rise to gauge transformations for the sections \q of Τ* ® Rq.
If /g+1 = flfg+i о Лд+1 then
A/*+i = hql\ ° £>9q+\ oji(hq) + Dhq+X.
If q = 2:
£>: <
^(uyj-^y<5)
Ι χ*/,·· = *£(ui# ^ vljxb - vlixli)
If 5g+i = Jg+i(fiO ("rigid motion"), £>(/g+i) = D(hq+i) = xg and so
Xg are gauge invariant. Dynamics of continuous media, invariant under the
pseudogroup Г, can be described by the action
5= / L(x)xq(x)) dxl.. .dxn.

151
For the calculus of variations we need "infinitesimal transformations":
y'q = yq° hq or y'q = gq о yg, where hq = (x\ xk + ίξ*, Sf + t#, t£§,...),
gq = («S^ + iC^^^ + 'C^iiC*»···)» £g»Cg € #g. An infinitesimal gauge
transformation takes the form
δχ4(θ) = ί(θ)ϋξ<ι+ι-{χ(θ),ξ<1+ι}
where £>£,+1 = fti* -£j+i, f, = £«+Xe(0 and {&+b fy+i) is the algebraic
bracket on Rq. We can obtain equations of motion over the source (Lagrange
variables x) and over the target (Euler variables y) using
5S = j'Xq.5Xqdx1...dxn = jXq · (y-^DCq+1ji(y,)) dx1...dxn
= Jyq.(DCq+l)dyl...dyn,
where *, = Ц· and Д(Э£.'ф = (**#) fty', Δ = det(fty*).
Theorem 1. Riemannian curvature in Lagrange variables can be expressed
in terms of sections of the first Spenser bundle for the groupoid Π2(Μ, Μ).
So we can describe generally covariant dynamics for continuous media
and gravity in terms of the Spenser sequence for Π2(Μ, Μ) = 9ftg, choosing
the action in the form
" = ^matter ι *bgrav
Theorem 2. With the choice of Einstein action for gravity
Sg™ = const I R,/^dx = const I Anab(XcaCjm(A-l)t - ^(Α-')Γ)ώ
Einstein equations appear as invariant equations of motion at first order jets.
The above construction gives a natural way for describing generally co-
variant dynamics of particles with spin. In this case the equations of
motion are automatically modified. We obtain in a unique way the
necessity of using torsion. The invariance under local translations produces the
energy-momentum tensor of matter, and the invariance under local rotations
produces the spin angular momentum of matter.

TOPOLOGY and APPLICATIONS Moscow: PHASIS, 1996
International Topological Conference Pages 153-154
Dedicated to P.S.AlexandrofTs 100th Birthday
Moscow, May 27-31, 1996
Space of linear differential operators as
a module over the Lie algebra of vector fields
Valentin Yu. Ovsienko
Center of Theoretical Physics — Luminy, Centre National de la Recherche
Scientifique, Marseille, France
One of the basic structures on the space of linear differential operators is
a natural family of module structures over the group of diffeomorphisms
Diff(M) (and of the Lie algebra of vector fields Vect(M)). These Diff(M)-
(and Vect(M))-module structures are defined if one considers the arguments
of differential operators as tensor-densities of degree λ on M. The action of
a vector field X = Xх (x) д/дхх on a differential operator
is given by the commutator with the operator of Lie derivative:
ad X(A) = LXA-ALX,
where Lx = Xi д/дхг - λ дХг/дх\
The problem of isomorphism of Diff(M)- (and Vect(M))-module
structures for different values of λ Was stated in [1] and solved in a series of
papers [1, 3, 2]:
1. dim Μ > 2.
(a) In the case of second order differential operators, different
Diff(M )-module structures are isomorphic to each other for every
λ except 3 critical values: λ = 0, — j, — 1 (corresponding to
differential operators on: functions, ^-densities and volume forms
respectively).
(b) In the case of differential operators of order > 3, Diff (M)-mod-
ules corresponding to λ- and λ'-densities are isomorphic if and
only if λ + λ' = 1 (see [3]). The unique isomorphism in this case
is given by conjugation of differential operators.

154
2. The case dim Μ = 1 (Μ = R or 51) is particular.
(a) The modules of 3-order differential operators are isomorphic to
each other for all values of λ except 5 critical values:
|o,-l,-i-i±^} (see [1,2]).
(b) The Diff(R)-modules on the space of differential operators of
order > 4 corresponding to λ- and λ'-densities are isomorphic if
and only if λ + λ' = -1 (see [2]).
The problem leads to the notion of SLn+1-equivariant symbol of
differential operators and is related to cohomology of Vect(M) with nontrivial
operator coefficients.
References
[1] C. Duval and V. Ovsienko, Space of second order linear differential operators
as a module over the Lie algebra of vector fields, Advances in Math., to appear.
[2] H. Gargoubi and V. Ovsienko, Space of linear differential operators on the real
line as a module over the Lie algebra of vector fields, Int. Math. Res. Notices,
No 5 (1996).
[3] P.B.A. Lecomte, P. Mathonet and E. Tousset, Comparison of some modules of
the Lie algebra of vector fields, Indag. Math., to appear.

TOPOLOGY and APPLICATIONS Moscow: PHASIS, 1996
International Topological Conference Page 155
Dedicated to P.S. Alexandroff's 100th Birthday
Moscow, May 27-31, 1996
Umbral and Schubert calculi
Nigel Ray
University of Manchester, Manchester, United Kingdom
The talk includes joint work with Victor Buchstaber and William Schmitt.
The Roman-Rota umbral calculus is concerned with sequences of
polynomials which behave well with respect to a coproduct map (or generalized
shift). These sequences are especially useful in the study of complex
oriented cohomology theories applied to projective space, and allow symbolic
manipulation of invariants associated with stable homotopy elements in the
image of the J-homomorphism.
We discuss how such a calculus may be formulated in terms of
combinatorial Hopf algebras, freely generated by equivalence classes of intervals in
certain partially ordered sets. Such examples tend to be cocommutative, so
we extend our description to the noncocommutative situation by
considering an appropriate Boolean algebra B. We obtain a structure which may be
used to compose and revert formal power series in one variable. The
associated Hopf algebra is a realization of the dual of the Landweber-Novikov
algebra of operations in complex cobordism.
We also describe certain Schubert varieties, interpreting them as
manifolds of bounded flags and explaining how they admit their own Schubert
calculus, in which the cells are indexed by subsets of a finite set (rather than
permutations or sequences of integers). The closures of these cells are non-
singular subvarieties, which make up a lattice which is isomorphic to В and
therefore closely related to the dual of the Landweber-Novikov algebra.
We enhance this connection by explaining how the bounded flag
manifolds may be invested with a double complex structure, with respect to
which they play an important role in double complex cobordism. In this
context they admit an adjoint action by the Landweber-Novikov algebra,
and therefore provide a geometric realization of the adjoint action of the
Landweber-Novikov algebra on its dual. Applying results of Novikov and
Buchstaber, we interpret this action as an embedding of the Drinfeld (or
quantum) double of the Landweber-Novikov algebra into the ring of
cohomology operations in double complex cobordism.

TOPOLOGY and APPLICATIONS Moscow: PHASIS, 1996
International Topological Conference Pages 157-158
Dedicated to P.S.Alexandroff's 100th Birthday
Moscow, May 27-31, 1996
Invariants of homology 4-cobordisms
from gauge theory
Nikolai N. Saveliev
University of Michigan, Ann Arbor, MI, USA
The homology cobordism group 0| of the oriented integral homology
3-spheres is of central importance in the manifold topology. The result of
D. Galewski and R. Stern [1], for instance, relates the problem of simplicial
triangulability of higher dimensional topological manifolds to the existence
of an element of order two in ©| having non-trivial Rohlin invariant. The
latter problem is well known as one of the famous "Kirby problems".
There has not been much progress in describing the structure of this
group since classical Rohlin's theorems [3] and until introduction of gauge-
theoretical methods in low-dimensional topology in early 80's. We use these
methods, including invariants by S. Donaldson, N. Seiberg and E. Witten,
to detect elements of infinite order in ®% and to show that for homology
spheres from certain classes the infinite order is implied by non-triviality of
the Rohlin invariant. Among others, there are the following results.
• Let a homology sphere Σ be a link of an algebraic surface singularity
— for instance, every Seifert fibered homology sphere is such a link.
We prove that if μ(Σ) < 0, where μ is the Neumann-Siebenmann
invariant, see [2] and [4], then Σ has an infinite order in the group
(5)3
•
•
Let Σ be the homology sphere obtained by 1/m-surgery on a torus
knot with odd m £ Z. Then it is of infinite order in 0|. This result
cannot be improved to cover the case of even m — it is well known
that Seifert fibered homology sphere Σ(2,3,13) which is the result of
( — l/2)-surgery on a (2, 3)-torus knot is homology cobordant to zero.
The μ-invariant is only defined for the so-called plumbed homology
spheres. We use Floer homology to extend the definition over
arbitrary homology spheres and get a new invariant. We investigate its

158
behaviour with respect to homology cobordisms, as well as its relations
with the Д-invariant and the Jones polynomial.
• We compute Floer homology for a variety of homology spheres
including Casson-Harer, Stern, and Mazur homology spheres, some plumbed
and hyperbolic homology spheres.
References
[1] D. Galewski and Ft. Stern, Classification of simplicial triangulations of
topological manifolds, Ann. Math. Ill (1980), 1-34.
[2] W. Neumann, An invariant of plumbed homology spheres, Lecture Notes in
Math. 788 (1980), 125-144.
[3] V. Rohlin, New results in the theory of four-dimensional manifolds, Doklady
Akad. Nauk SSSR 84 (1952), 221-224 (in Russian).
[4] L. Siebenmann, On vanishing of the Rohlin invariant and nonfinitely am-
phicheiral homology ^-spheres, Lecture Notes in Math. 788 (1980), 172-222.

TOPOLOGY and APPLICATIONS Moscow:, PHASIS, 1996
International Topological Conference Page 159
Dedicated to P.S. Alexandroff's 100th Birthday
Moscow, May 27-31, 1996
The homotopy of Z/2-local finite spectra
Katsumi Shimomura
Tottori University, Tottori, Japan
Let Ln denote the Bousfield localization functor from the category of p-local
CW-spectra to itself with respect to K(0) V K(l) V ... V K{n), where K(i)
denotes the г-th Morava /f-theory. Then L$X = ρ"1Χ = ΧΛ SQ for the
Moore spectrum SQ with n*(SQ) = Q, the field of rational numbers, and
L\X = LkX) the /f-theory localization of X. These have much information
on the homotopy theory and so we can expect more by /^-localization. For
example, 7r*(Li5°) gives full information on Im J and n*(L2S°) would give
some information on Coker J. Furthermore, the homotopy groups of LnS°
include the most of information on the category of Ln-local spectra.
Under this situation, we obtain n*(L2S°) for ρ > 3. At the prime 3,
n*(L2V(l)) is computed, where V(l) = (5° U3 e1) Ua (e5 U3 e6) is the
Toda-Smith spectrum for the Adams map a. At the prime 2, there is
no V(l). We instead compute nm(L2D(A\) Λ М2) for the cofiber D(A\) of
the essential map у : Σ5Αίη Λ Μν -> Μη Λ Μ„. Here Mx for χ € Kj(S°)
denotes a cofiber of / : S^ —f 5° representing x. These results give us some
information on Coker J as we expected.

TOPOLOGY and APPLICATIONS Moscow: PHASIS, 1996
International Topological Conference Pages 161-162
Dedicated to P.S. Alexandroff's 100th Birthday
Moscow, May 27-31, 1996
On the topology of the path space
for a symplectic manifold
Valerii V. Trofimov
Lomonosov Moscow State University, Moscow, Russia
Let Xй be a smooth manifold, on which an affine connection Γ^· is defined,
and let Lm be an arbitrary submanifold of Xn. In this situation we can
define unstable characteristic classes of Lm, which are an analog of the
Maslov classes in symplectic geometry, see [1]. We choose and fix a point
x0 e Xй and introduce the notation [Xй, Lm] = {a : [0,1] -* Xn \ a is
a piecewise-smooth map of the interval [0,1] into Xn such that a(0) = xq
and a(l) 6 Lm}. This space and its geometry play an important role in the
calculus of variations. We define a map / : [Xn) Lm] —> Gm(TXoXn) of the
space [Xn) Lm] into the Grassmannian Gm(TXoXn) generated by the tangent
space TXQXn) where m = dim Lm. If a £ [Xn) £m], then the tangent space
Ta(i)Lm is parallel-transported relative to the connection Г* to the point
x0 = a(0) along the path a. Let f(a) denote the image of Ta^Lm under
this parallel transport. The subspace f(a) С TXQXn is by definition the
image of the path a under /. A map /* : h*(Gm(TXoXn)) -* Λ*([Χη, Lm])
is induced in a generalized cohomology theory h*.
Let a € h*(Gm(TXoXn)) be an arbitrary cohomology class of
the Grassmannian Gm(TXoXn). The cohomology classes of the form
/*(a) £ h*([Xn) Lm]) are called generalized Maslov classes of the
submanifold Lm С Хп.
Let {Μ2η,ω) be a symplectic manifold and Ln be its Lagrangian sub-
manifold, where a; is a non-degenerate 2-form. Then, the previous
construction gives us the map / : [M2n, Ln] -* K{TXoM2n), where A(TXQM2n) is a
Lagrangian Grassmannian. In order to do this we must choose a symplectic
connection on the symplectic manifold M2n. It is known (see [1]), that for
the entire symplectic manifold {Μ2η,ω) there exists a symplectic
connection, i.e. the connection Γ*·Λ such that Vk^ij = 0, where Vfc is the covariant
derivative with respect to Тг-к.

162
If α € h*(A(TXoM2n)) is an arbitrary cohomology class of the
Lagrangian Grassmannian Λ(Γΐ0Μ2η), then a cohomology class of the form
/*(a) € h*([M2n) Ln]) is called generalized Maslov class of the Lagrangian
submanifold Ln С M2n. These classes do not depend on the choice of the
symplectic connection and possess the properties of the usual Maslov classes.
References
[1] V.V. Trofimov and A.T. Fomenko, Algebra and Geometry of integrable Hamil-
toman systems of differential equations. Factorial, Moscow, 1995 (in Russian).

TOPOLOGY and APPLICATIONS Moscow: PHASIS, 1996
International Topological Conference Pages 163-164
Dedicated to P.S.Alexandroff's 100th Birthday
Moscow, May 27-31, 1996
Equivariant exterior algebra of finite groups
Alexander V· Zarelua
Lomonosov Moscow State University, Moscow, Russia
The theory of G-invariant multivectors on the augmentation ideal JK[G]
of the group algebra of a finite group G has been developed using methods
of representation theory of finite groups and algebraic number theory. A
special attention is paid to the construction of a biinvariant decomposable
non-degenerated integer-valued 2-vector and to the Lefschetz decomposition
of invariant multivectors connected with such a 2-vector. In constructing
a 2-vector possessing the first three properties essential is the information
about the structure of irreducible representations of finite groups, while the
integer-valuedness involves the action of the Galois group of an appropriate
cyclotomic field on the group G and on its irreducible representations. In
the estimation of the volume of such a 2-vector some divisibility properties
of algebraic numbers are used.
Main results obtained in this direction are the following.
1. The space of biinvariant 2-vectors is isomorphic to the space of
central functions {b(g)} which are skew-symmetric in the sense that
big-1) = -b(g).
2. The space of biinvariant 2-vectors is isomorphic to the subspace of
the space of central functions generated by the complex irreducible
characters. In particular, if all irreducible characters of a group G
are real then in its group algebra there are no non-trivial biinvariant
2-vectors.
3. With any irreducible complex character of a group G, a
decomposable biinvariant 2-vector Qr is connected which is non-degenerated on
the direct sum of the corresponding two-sided ideal and its complex
conjugated ideal. In particular, if the group G has an odd order then
any sum of 2-vectors ΩΓ with coefficients cr different from zero forms
a decomposable non-degenerated 2-vector.

164
4. The set of decomposable non-degenerate biinvariant integer-valued
2-vectors is in a 1-1 correspondence with the skew-symmetric elements
of the center of the group ring Z[G] which are invertible in the
augmentation ideal JC[G\. These results give a possibility to calculate the
volume of non-degenerated decomposable biinvariant integer-valued
2-vectors by means of determinants of the corresponding
multiplication operators and to deduce the formulae for the coefficients of the
Lefschetz decomposition of invariant 2-vectors. Using these formulae
a homomorphism of 2-dimensional exterior homology group in some
residue group is constructed. This homomorphism turns out to be
non-trivial in the case, for instance, when the determinant under
consideration does not exceed the order of the group G. The last result
allows to prove the solvability of some groups and to estimate the
quotient group of a finite group of odd order by its commutator-group.
For the investigation of actions of finite groups on a topological space a
skew-symmetrical analog of homology (cohomology) groups of a group has
been introduced and studied. The formulae mentioned above have been
used to calculate these groups of exterior homology (cohomology) in some
important cases.
The study of connections of the introduced cohomology classes with the
characteristic classes of vector bundles induced by representations of groups
is initiated. At present such connections are established for the first Chern
classes using a diffeomorphism of the space of minimal ideals of the complex
matrix group with the complex projective space.

Session 3
Applications of Topology
and Geometry

The abstracts are presented in the alphabetical order
of the authors' names

TOPOLOGY and APPLICATIONS Moscow: PHASIS, 1996
International Topological Conference Pages 167-169
Dedicated to P.S. Alexandroff's 100th Birthday
Moscow, May 27-31, 1996
The Knaster problem, the Borsuk theorem
and cyclic systems
Semeon A. Bogatyi
Lomonosov Moscow State University, Moscow, Russia
Let f : Sn —l· Rm be a fixed continuous mapping of the standard ra-sphere
Sn into the Euclidean m-space Rm.
K. Borsuk proved that for η > m there exists a point ж £ Sn such
that /(a) = /(—ж). The Borsuk theorem has many generalizations. One
direction of generalizations was initiated in 1944 by H. Hopf. If η > m then
for every angle a, 0 < a < π, there are two points X\,X2 € Sn such that
the angle between these points is equal to a and f{x\) = /(a^)* G. Skordev
estimated the dimension of the set of all collapsing pairs. In 1947, B. Knaster
proposed the following
Problem· Given к = η — m + 2 distinct points αχ,..., Xk 6 Sn, does there
exist a rotation r 6 SO(n+ 1) such that f{rx\) = f(rx2) — ... = /(га*)?
Special cases of the Knaster problem have been studied by many
mathematicians.
In 1986, V.V. Makeev*proposed a method for constructing noncollapsing
configurations and mappings for the Knaster problem. For a finite
configuration αχ,..., Xk of points on 5n, let /C be the space of all configurations
congruent to the given one. And let / be the dimension of the smallest linear
subspace of En+1 containing a copy of the configuration αχ,..., α* and the
point 0. V.V. Makeev proved that the inequality
dim /C = dim Vn+1>/ = In - 1(1 - l)/2 > m(k - 1)
is necessary for the configuration to be a Knaster configuration. He
conjectured that the validity of this inequality is also sufficient. We show that
for all configurations, lying in small enough neighborhood of a
configuration on a circle of large radius (with arbitrary big number I), the stronger
inequality
2n - 1 > m(k - 1)

168
is also necessary. So we show that in general the Makeev problem has
negative solution.
Using the general position method we prove a "noncollapsing" result.
Theorem 1. Let σ\,..., aq : Ρ —ϊ Ρ be continuous piecewise smooth
mappings without points of pairwise coincidence of an η-dimensional polyhedron.
Let the numbers n, m, q and к satisfy the inequality
η + 1 < m(q — k).
Then there exists a map f : Ρ —>- Rm such that for every point χ € Ρ the
set {f{<Ti{x)) : г = 1,... ,ρα} has cardinality > к + 1.
From Theorem 1 and a result by S.I. Bogataya and the author it follows
that for every continuous piecewise smooth free Zp-action on n-dimensional
polyhedron Ρ and every number m satisfying the inequality ra+1 < m(p— 1),
there exists a map f : Ρ —ϊ ]Rm+(Pi"1)/2 which is 1-to-l on every orbit.
Another direction of generalizations is related to works by many authors
(A.S. Schwarz, H.J. Munkholm, H. Steinlein) who considered a free action of
the group Zp on the sphere Sn. Namely, let σ : Sn —l· Sn be a free periodic
homeomorphism of period ρ for some prime integer ρ > 2. If η > m(p — 1)
then there exists a point χ £ Sn such that f(x) = /(σχ) = ... = f(ap~lx).
More precisely, the set B(f) = {x € Sn : f(x) = f(ax) = ... = }{σ*-ιχ)}
has dimension > η — m. E. Lusk obtained a generalization of the Borsuk
theorem with a partial orbit collapsing: If η > (m— l)(p— 1) +1, then there
exists a point χ € Sn such that f(x) = ί(σχ).
We prove analogs of Munkholm's theorem where the assertion about the
coincidence of points from an orbit image is replaced by the assertion that an
orbit image is contained in a certain subspace of (Rm)p. The corresponding
subspace is specified by a homogeneous system of linear equations. We
prove such theorems for the base of some inductive process and first two
steps. Let
ί anyi + auy2 + ... + aiqyq = 0
I a2\y\ + а22У2 + · · · + a2qyq = 0
{ о>к\У\ + ак2у2 + ... + akqyq = 0
be any system with q variables such that the sum of the coefficients in each
equation is zero.

169
Theorem 2. Let the number q be prime, the given system be a cyclically
invariant system of rank r and η > mr. Then there exists a point χ £ Sn
such that the vector у = (/(χ), /(σ(χ)),..., f(aq~l(x))) satisfies the given
system.
If the number q is a power of some odd prime number, Munkholm's
theorem on Zg-actions enables us to obtain some weaker form of a similar
theorem. We show that any cyclically invariant system with q = 1 (mod 2)
variables such that the sum of the coefficients in each equation is zero has
even rank. All systems of this type with rank 2 are described.
Theorem 3. Let the number q be prime, the given system be not
cyclically invariant, the rank of cyclic symmetrization be r and
η > mr — 1. Then there exists a point χ £ Sn such that the vector
у = (/(χ), /(σ(ζ)),..., f(aq~l(x))) satisfies the given system.
As a corollary of this result we obtain that if η > m(p — 1) — 1, then for
every decomposition of the group Ър into two nonempty subsets Ър = AUB,
there exists a point χ £ Sn such that ί{σιχ) = f{a3x) for г and j both lying
either in A or in B.
Theorem 4. Let the number q be prime, the system consisting of the given
equations and one time shifted equations be not cyclically invariant, the rank
of cyclic symmetrization be r and η > mr — 2. Then there exists a point
χ £ Sn such that the vector у = (/(ж), /(σ(χ)),..., f(aq~l(x))) satisfies the
given system.
All three element decompositions of the group Zp which satisfy the
conditions of Theorem 4 are described. As a corollary we obtain a partial
positive solution of the Makeev conjecture — about collapsing given ρ — 2
elements of the group in the image of the orbit of some point. We also give
a partial answer to a question of Cohen and Lusk, which generalizes the
Lusk theorem.
Theorem 5· Let in X > (m - l)(p - 1) + k, where 1 < к < ρ - 1. Then
for every continuous mapping f : X —> Rm there exists a point χ € X
and a decomposition Zp = Αχ U ... U Ap-k into nonempty subsets, such
that Aj consists of consecutive elements (p and 1 are consecutive), and
f(a4x) = f{a%2x) for i\ and i2 both lying in either of Aj.
Actually, in the conditions of Theorems 2-5, we estimate the dimension
of the set of points with a "suitable" orbit image from above and prove
multivalued variants of these theorems.

TOPOLOGY and APPLICATIONS Moscow: PHASIS, 1996
International Topological Conference, Pages 171-172
Dedicated to P.S. Alexandroff's 100th Birthday
Moscow, May 27-31, 1996
Multivalued vector fields with Fredholm
and monotone components
Yuri G. Borisovich
Voronezh State University, Voronezh, Russia
In this lecture we consider proper Fredholm operators A G ФЯСГ(Х, Е),
A : {Х,дХ) -> (£, £\0), where X is a Banach Cr-manifold with the
boundary dX, Ε is a Banach space, г > 1, q is an analytic index and
f(x) = A(x) — g(x) are multivalued vector fields where g : X —» Ε is a
multivalued upper semicontinuous map with convex compact images. The
author's and Yu.G. Sapronov's [1] principle of compact restriction is
generalized. The pair {Л, g} is called completely fundamentally restricted if the
multivalued map F = g о A"1 : (AX) —>- Ε is completely fundamentally
restricted in generalized sense (concerning [2]); the corresponding vector
field / for the condition / : (X,dX) —l· (E,E\ 0) is called a completely
fundamentally restricted field (or CFR-field). Also, all CFR-fields
constitute the class ФдСгМв, r > 1. For q > 0 topological characteristics
d(f) € Fq, d'(f) € F' are defined where F9, Fg' are classes of K. Elworthy's
and A. Tromba's g-bordisms, characterizing the set of coincidences S(A,g)
of inclusions A(x) 6 g(x)· The condition d(f) φ 0 or d'(f) φ 0 is sufficient
for the existence of a solution of the inclusion. For a single-valued map g and
for certain conditions of smoothness and transversality the set S(A,g) = Nq
is a compact ^-dimensional manifold if the topological characteristics are
nontrivial. The definition of characteristics d(/), d'(f) is generalized for the
case q < 0 as an index of coincidence γ(/, к) where к : Lm —> Ε belongs to
the class C°, m + q > 0, where Lm is a compact bounded manifold, and for
the case a priori nonfixed index 9, — 00 < q < 00. Another type of
generalization concerns I.V. Skrypnik's (ao)-°Perators. Connections of mentioned
constructions with a problem of singular in J.-L. Lions' sense problems of
optimization are discussed; applications to controllable dynamical systems
and to nonlinear boundary problems are given.

172
References
[1] Yu.G. Borisovich, Modern approach in the theory of topological characteristics
of nonlinear operators 1; 2, Lect. Notes in Math. 1334 (1988), 199-220; 1453
(1990), 21-50.
[2] Yu.G. Borisovich, B.D. Gelman, A.D. Myshkis and V.V. Obukhovskii,
Topological methods in a fixed point theory of multivalued maps, Uspekhi Mat. Nauk
35(1) (1980), 1-84.
[3] Yu.G. Borisovich, Global analysis of operator equations arising in singular
problems of optimization, Dokl. Akad. Nauk, to appear.

TOPOLOGY and APPLICATIONS Moscow: PHASIS, 1996
International Topological Conference Pages 173-175
Dedicated to P.S. Alexandroff's 100th Birthday
Moscow, May 27-31, 1996
Statistical manifolds, a-geodesics
and λ-Jacobi fields
Guy Burdet
Center of Theoretical Physics — Luminy, Centre National de la Recherche
Scientifique, Marseille, France
Philippe Combe
Center of Theoretical Physics — Luminy, Centre National de la Recherche
Scientifique, Marseille, France
Hanna Nencka
Center of Mathematical Sciences, University of Madeira, Funchal,
Portugal
Let {Ω,.77, m} be a measure space, with Ω a sample space, Τ a σ-algebra
and m a measure. Choose a family of probability measures {Ρζ}ζζΜ labeled
by M) absolutely continuous with respect to the measure m. Let ρζ be
a random variable related to Ρζ. The condition of absolute continuity is
fulfilled if ρζ is a Radon-Nikodym derivative and άΡζ = ρζ dm.
Theorem 1. Let {5, g,i) be a triplet with S = {{Ρζ}ζ^Μ}} 9 the Fisher
metric with g^ = E\d{ In ρ dj lnp], t a completely symmetric tensor of
order 3 such that Ujk = [d{ lnp dj lnp dk lnp]; then {5, g,t} is a statistical
manifold.
Theorem 2· Let a G R, there exists a 1-parameter family of torsionless
affine connections V such that Vg = ott and having the following compo-
nents: Г\ = г\-^.
For any a £ R, α-geodesics и —^ Ca, w 6 / С R, are such that the vector
oe
field Сa = dCa/du is transported in parallel along Ca with respect to V.

174
Let Сa = {χι (и)}, С а = {х*(и) + &{и)} be two α-geodesics, and ξ a
parting vector field.
Let a — a = 2λ £ R, then the vector field ξ satisfies
a a
Proposition 1. The (a, a)-geodesies' deviation ξ = V^V^f, or \-Jacobi
field, fulfills
ik - Ιήά1χψχι = Af*i'V.
Proposition 2· Each component of a \-Jacobi field is the sum of any
particular solution of the above differential equation and any arbitrary linear
combination of the corresponding components of two Jacobi fields.
λ-Jacobi fields are such that:
• for any a £ R they are never tangent to any Ca;
• they are not isocline;
• they do not constitute any infinitesimal 1-parameter variation of Ca
by other α-geodesics, contrary to usual Jacobi fields;
• perpendicularity is not conserved.
Example 1. Let {Ω, T, m} be such that ρ = (ay/2n)~1/2 exp
{-(Щ
is almost continuous with respect to the Lebesgue measure, let 5 = {{i^}},
where dP = ρ dm. The Fisher metric and the totally symmetric tensor of
order 3 are respectively
1 2
gij = —=■ (άμ ® άμ + da ® da), i*· = — (άμ ® άμ ® άμ + 4 da ® da ® da).
az αό
The {5, g,i) triplet is a statistical manifold called the Gaussian manifold
having as a privileged coordinate system (μ/\/2,a). This Gaussian
manifold is the Poincare half-plane, α-geodesics are half-conics generalizing the
Riemannian half-circles. It turns out that the λ-Jacobi fields satisfying
e = 2-(/ia), Γ = -(Α2+4σ2)
a a
are explicitly obtained for a = ±1. Usual Jacobi fields are such that
£ = Au+ B.

175
Example 2. Let us take now {Ω, Τ, m} such that Ω = {x\,..., xn+i} < oo,
the probability Р[{жа;}] being almost continuous with respect to the uniform
measure m on Ω. Then 5 = {Р[{ж*;}]} is an open simplex in Rn+1. The
corresponding statistical manifold is the triple {5, #,£}, where
Sij 1 , l SijSjk
9Ц = -± + ЪГ- and *« = *УМ + ^--
Pl Σ κ Pl
There exists a 1-parameter torsionless affine connection such that the
α-geodesics, satisfying the equations
P· = —^— S — - Pi ι—^— ( >
2 t« i-EjPiJ
Pi
are given by: pi(ti) = p%(0) +p«(0)ti, for α = —1, and
exp гг/ij
Pi(u) =pt(0)
l + EiPiWiexpti/Ci-l)'
for α = +1, where tf,; = £ « + ^«il.
The corresponding λ-Jacobi fields fulfilling the equations £* = Xpi are
as follows: ξ* = Xpi - р»(0) - Pi(0)u.
The problem is motivated by works of Rao [3], Chentsov [2] and
Amari [1].
References
[1] S.I. Amari, Differential geometrical methods in statistics, Lecture Notes in
Statistics 28, Springer-Verlag, 1985.
[2] N.N. Chentsov, Statistical decision and optimal inference, Nauka, Moscow, 1972
(in Russian); English translation in: Amer. Math. Soc. 53, Providence, RI,
1982.
[3] C.R. Rao, Information and accuracy attainable in the estimation of statistical
parameters, Bull. Calcutta Math. Soc. 37 (1945), 81-91.

TOPOLOGY and APPLICATIONS Moscow: PHASIS, 1996
International Topological Conference Pages 177-178
Dedicated to P.S.Alexandroff's 100th Birthday
Moscow, May 27-31, 1996
On a problem of integral geometry
in spaces of constant curvature
Alexander S. Denisiuk
Brest State University, Brest, Belarus
In [4] V.P. Palamodov formulated the following principle.
Theorem· Any inversion formula for Radon transform Rg in the
Euclidean space En can be translated to an inversion formula for Radon
transform Rp in the elliptic space Pn and to an inversion formula for Radon
transform R^ in the hyperbolic space Hn and vice versa.
By Radon transform Rk we mean the transformation mapping function
/ into its integrals over ^-dimensional totally geodesic manifolds. Rl is
called X-ray transform.
In [4] the principle was proved for the elliptic space. We will prove
it for the hyperbolic space and apply it to inversion problem for X-ray
transform in elliptic and hyperbolic spaces. Inversion formulae for complete
manifold of geodesies in Pn and Hn are known [2]. We obtain formulae
for the η-dimensional family of all geodesies which intersect a given curve
7 С Рп(Нп). That is one of contact families. We define contactness as the
following (cf. admissibility in [3]).
Let К be an η-dimensional family of curves in a space X of
dimension η and Kx a set of curves of К which meet a point χ £ X. Let
Cx = U/e/fx l С X. For generic ж, Cx is two-dimensional conoid with vertex
in the point x.
Definition. An η-dimensional family К of curves in space X of dimension
η is called contact if for almost any curve I € K, for almost any pair of points
Х\) Х2 € / conoids CXl and CX2 have the same tangent two-dimensional plane
at any point t G /, t φ Χ\, x2.
This property is invariant under action of a diffeomorphism. So, one
can apply this principle to obtain inversion formulae for any contact family

178
of geodesies in Pn and #n, if there is a formula for the corresponding
family in En.
We translate formulae from [1].
References
[1] A.S. Denisiuk, Inversion of generalized Radon transform, Amer. Math. Soc.
Translations 162(2) (1994), 19-32.
[2] I.M. GePfand and M.I. Graev, Complexes of lines in the space, Funkts. Anal, i
Prilozh. 2 (1968), 219-229 (in Russian).
[3] S. Helgason, Geometric analysis on symmetric spaces, Math. Surveys and
Monographs 39, Amer. Math. Soc, Providence, RI, 1994.
[4] V.P. Palamodov, Selected topics in integral geometry, pure and applied, Lecture
for F. Klein Colloquium, Dusseldorf Universitat, 1994.

TOPOLOGY and APPLICATIONS Moscow: PHASIS, 1996
International Topological Conference Pages 179-180
Dedicated to P.S. Alexandroff's 100th Birthday
Moscow, May 27-31, 1996
Extremality of the Bricard Octahedra *
Nikolai P. Dolbilin
Steklov Mathematical Institute, Russian Academy of Sciences, Moscow,
Russia
Mikhail A. Stan'ко
Steklov Mathematical Institute, Russian Academy of Sciences, Moscow,
Russia
Mikhail I. Stogrin
Steklov Mathematical Institute, Russian Academy of Sciences, Moscow,
Russia
Let S2 be an abstract 2jD-polyhedral sphere S2 whose all faces are Euclidean
convex polygons. Let also / be a continuous mapping / : S2 —>· E3 such that
1) / maps each face of S2 onto a flat polygon of f(S2) С К3;
2) / is an isometric mapping of each face of 52;
3) / is an immersion of the 1-dimensional skeleton of 52.
Let F = /(S*2) admit a continuous non-trivial (i.e. at least one dihedral
angle changes) flexure φ that leaves any face of F flat Then F is called a
flexor (see [1]).
Let Τ denote the set of all flexors that are /-images of polyhedral spheres
(/ being required to fulfil conditions l)-3) above). Denote by V\ (by ei, f\
respectively) the minimal number such that there is a flexor in Τ with щ
vertices (with e\ edges or f\ faces respectively) and denote by ΛΊ, 3^, Z\
subsets of flexors from Τ that have minimal numbers of vertices, edges and
faces respectively. With the help of the flexure graph Γφ introduced and
used in [2, 3, 4] here we prove
The work is supported in part by Russian Foundation for Basic Research,
No 96-01-00166.

180
Theorem· щ = 6, в\ = 12, f\ =8; Х\ = У\ = i?i are families of the
Bricard octahedra.
Thus, given a polyhedron homeomorphic to the sphere and with
immersed edge skeleton, then it has less than 6 vertices, or less than 12 edges,
or less than 8 faces.
References
[1] R. Bricard, Memoires sur la theorie de Voctaedre articular, J. Math. Pures
Appl. 5 (1897), 113-148.
[2] R. Connelly, Conjectures and open questions in rigidity, Preprint supported by
an NSF grant.
[3] N.P. Dolbilin, M.A. Stan'ko and M.I. Stogrin, Rigidity of quadrillages of the
sphere, International Conference on Geometry "in global", Cherkassy, 1995,
22-23.
[4] N.P. Dolbilin, M.A. Stan'ko and M.I. Stogrin, Rigidity of a quadrillage of the
sphere, Russian Doklady, to appear.
[5] N.P. Dolbilin, M.A. Stan'ko and M.I. Stogrin, Rigidity of Zonohedra, Uspekhi
Mat. Nauk, to appear.

TOPOLOGY and APPLICATIONS Moscow: PHASIS, 1996
International Topological Conference Pages 181-182
Dedicated to P.S. Alexandroff's 100th Birthday
Moscow, May 27-31, 1996
Basic topological structures of the theory
of ordinary differential equations
Vladimir V. Filippov
Lomonosov Moscow State University, Moscow, Russia
In [1, 2] we studied a topological structure which is adequate to the question
about the continuity of the dependence of a solution of the Cauchy problem
on parameters of the right-hand side.
The levels of the TODE (the theory of ordinary differential equations)
correspond to the levels of the theory of integration. The TODE with
continuous right-hand side corresponds to the integral of Riemann. The
TODE under the conditions of Caratheodory corresponds to the integral of
Lebesgue. The application of our theory gives us a level corresponding to
the integral of Denjoy.
One of main tools of this approach is a new topological space RC(U),
which can be named "space of solution spaces". Each equation and inclusion
with the right-hand side defined on the set U С Ε χ Εη is represented here
by a point.
To display the position of this notion, let us consider an equation
where the right-hand side depends on a parameter α £ A. Then the
continuity of the mapping A —> RC(U) which associates with a parameter a the
solution space of the equation yf = /(£, y, a), is equivalent to the continuity
of the dependence of solutions of the equation on parameter a (in a form
which does not involve the uniqueness of solutions in general case and in
a strict form under the supposition of the uniqueness), Properties of this
topology are important for investigation of equations.
The existence of the topological space RC(U) allows us to word easily
fundamental properties and relations of solution spaces. Such topological
structures give tools to study equations with singularities in right-hand sides.
Properties of this structures may be taken as axioms for some geometric
chapters of the TODE. This allows us to introduce most general concepts

182
of some notions related to properties of equations and their solutions and
use them in practical investigation of equations. We have new tools to
investigate equations and inclusions and to make this approach to work in
situations which are not covered by the classical theory. The author is able
to show how these tools work is the domain around the Poincare-Bendixson
theorem and in other geometric sections of the TODE, in the investigation
of asymptotically autonomous equations, in the study of stationary points,
in the asymptotic integration etc.
Last results are related with homological properties of a solution set.
Applications of the Leray-Schauder theory in the TODE are based on
estimates of homological properties of the corresponding integral operators.
But if the right-hand side of the equation under consideration is
discontinuous, the integral operator need not exist. Our possibility of description
of topological properties at the level of solution spaces allows us to
construct an equivalent of the Leray-Schauder theory which can be applied to
equations with discontinuous right-hand sides without any supplementary
considerations.
References
[1] V.V. Filippov, Topological structure of solution spaces of ordinary differential
equations, Uspekhi Mat. Nauk 48(1) (1993), 103-154.
[2] V.V. Filippov, Solution spaces of ordinary differential equations, Moscow Univ.
Press, Moscow, 1993.

TOPOLOGY and APPLICATIONS Moscow: PHASIS, 19Э6
International Topological Conference F*age 183
Dedicated to P.S. Alexandroff's 100th Birthday
Moscow, May 27-31, 1996
Costa's minimal surface
Alfred Gray
University of Maryland, College Park, MD, USA
I will give an explicit parametrization of Costa's minimal surface and show
how to create the surface using Mathematica. More generally I will
explain the Mathematica implementation of the differential geometry of curves
and surfaces using my book "Modern Differential Geometry of Curves and
Surfaces".

TOPOLOGY and APPLICATIONS Moscow: PHASIS, 1996
International Topological Conference Page 185
Dedicated to P.S. Alexandroff's 100th Birthday
Moscow, May 27-31, 1996
Convergence in the space of solution spaces
of ordinary differential equations
and its applications
Boris S. Klebanov
Institute of New Education Technologies, Moscow, Russia
A key concept in V.V. Filippov's axiomatic theory of solution spaces of
ordinary differential equations (ODEs) is the notion of convergence in the
space of solution spaces of these equations. On the basis of this concept
appropriate topological structures were introduced in the framework of this
theory for the study of fundamental properties of solutions of ODEs.
Our talk is concerned with applications of these topological structures
to the investigation of limit sets of trajectories of solutions of ODEs (in
particular, to the Poincare-Bendixson theory), as well as stability and dis-
sipativity of the solutions.

TOPOLOGY and APPLICATIONS Moscow: PHASIS, 1996
International Topological Conference Pages 187-190
Dedicated to P.S. Alexandroff's 100th Birthday
Moscow, May 27-31, 1996
Exact smooth classification of divergence-free
vector fields on surfaces of small genus
Boris S. Kruglikov
Lomonosov Moscow State University, Russia
Let us consider a compact two-dimensional manifold P2 and a smooth
measure μ on it, i.e. a symplectic form ω if P2 is oriented and an anti-invariant
symplectic form on the double covering otherwise. Let us call a vector field
υ divergence-free if the corresponding generated flow preserves the measure
μ. We assume all the fields considered complete and Morse nondegenerate.
We present two classifications, each for the category of C^-diffeomorphisms,
к = l,....,oo. Two divergence-free vector fields are called conjugated if
each can be transformed into the other by a diffeomorphism which needs
not preserve the measure. At first we give a classification up to conjugation
of divergence-free vector fields υ with all trajectories but a finite number
closed. Let us stress that the study of periodic divergence-free vector fields
is based on the following reasons: 1) the set of such fields is dense in the set
of all divergence-free fields; 2) the set of periodic fields is easy to study, and
then they may approximate arbitrary dynamical divergence-free systems
like rational numbers approximate irrational ones and polynomials do with
functions, see an example in [4]; 3) they arise from the theory of trajectory
classification of integrable Hamiltonian systems with two degrees of freedom
[2], and that is also the motive we do allow to intersect for separatrix. Next
we consider a classification of arbitrary divergence-free vector fields on the
surfaces S2, EP2, Κ2, Τ2, K2#RP2 with some (possibly none) holes —
deleted disks D2.
The first problem on oriented surfaces goes closely to the classification
of globally Hamiltonian vector fields. The latter is of much importance
for the theory of trajectory classification of Bott integrable Hamiltonian
vector fields with two degrees of freedom and was completely solved for the
category C° in [2], for the smooth categories Ck, к = 1,..., oo — in [1, 3].
In short the results are as follows.

188
Let С be the foliation of P2 by trajectories all but a finite number of
which are closed. The quotient space P2 /C is a graph Г whose vertices
correspond to singularities and are called letter-atoms. Each point of the
edge is a quotient of the periodic trajectory. So we have a period function
on every edge, and we take the conjunction class of this function since there
is no natural parametrization of the edge. To every vertex there is
associated a number of invariants: Invfc = {Ak <E Co(V2;Rfc), Ak <E B2(V2;Rk),
Zk £ #1 [V2; R*1)}, where V2 is a com pact ification of the singular leaf
neighborhood in the surface P2, and chains and cochains are considered with
respect to the accepted cell decomposition for the letter-atom compact-
ifications.
Theorem 1 ([3]). Two Hamiltonian vector fields are equivalent in the
category Ск, к > 1, if their graphs, period functions on edges and invariants
Invfc coincide.
For divergence-free vector fields the picture is similar. There are only
two differences. At first, there can appear Mobius bands instead of annuli as
edges. But we take the period functions anyway. Secondly, neighborhoods
of the singularities may turn out to be nonorientable. We also define the
invariants Л*, Ak) Zk, but this time the homology spaces are considered
with twisted coefficients.
Theorem 2. Two divergence-free vector fields with all trajectories except a
finite number being closed are equivalent in the category Ск, к > 1, if their
graphs Г, period functions on edges and invariants Inv*. coincide.
Let us turn to the second classification problem.
Theorem 3. For any divergence-free vector field on 52, RP2, K2 with holes
only a finite number of trajectories may be nonclosed. So the classification
on these surfaces is given by Theorem 2.
Now consider the case of the torus.
Let us introduce the notion of "mean length" for a divergence-free vector
field ν on the torus T2 (with holes). Let us glue holes by disks and extend
arbitrarily the field υ and the symplectic form ω representing the
measure μ with the only demand that υ is divergence-free. Define v0 — ||5||,
S — — \%νω\ € tf^T^jR), \ω\ = fa, ω. Here || · || is the norm in the space

189
Я1 induced by the norm in H\ with an orthonormal basis of (any) basis
cycles. We call Hamiltonian vector fields with Hamiltonians taking values
in the circle also Hamiltonian.
Proposition 1. For a non-Hamiltonian vector field ν the ((mean length"
v0 φ 0. For almost every winding number λ of the flow ν the "mean length"
vo can be defined as follows. Let u(t) be a regular nonclosed trajectory other
than separatrix. Let и*0 = {u(s) : s € [0, t]}. Let (71,72) be a basis of
transversals and the winding number in it equal λ = tana. Let Ti(t) be the
number of intersections of иг0 with ji. Then
vo = ρ hm —r-^- = Q lim ,
t-юо t sin a f-юо t cos a
\ώ\
where ρ is the relative volume of the "wandering domain", i.e. ρ = -—-,
where ώ = 0 for all points of periodic trajectories and ώ = ω
otherwise. The invariant S ( "winding class") is expressed via v0 by the formula:
S = v0 cos «[72] - v0 sin a[yi] € Hl(T2\ R).
The trajectory portrait of a non-Hamiltonian divergence-free vector field
is easily presented by means of the notions of a 1-marked letter-atom and
equipped transversal. A 1-marked letter-atom is a letter-atom with one edge
being cut. These may be classified in a manner as noncut (Hamiltonian)
letter-atoms. In the interior of 1-marked letter-atoms there can lie (usual)
letter-atoms. We associate all C^-invariants to the first intersection point of
incoming wandering separatrix with a fixed transversal. Such a transversal
with several marked points is called k-equipped.
Theorem 4. For a.e. irrational winding number λ a non-Hamiltonian
divergence-free vector field on the torus T2 (with holes) is classified up to
C°° -conjugation by the invariant S and oo-equipped transversal. If for two
systems with winding numbers λ of the type (Κ,σ) the S-invariants and
(k + r)-equipped transversals coincide, where r = 3 + [σ], then the
systems are Ck-conjugated (k > 1). Under a change of the transversal the
(k + r)-equipment on the marked points is preserved but their order on the
transverse circle might become different.
Let us now consider the last case of the surface P2 = #?RP2 = T2#RP2.

190
Theorem 5. If almost all trajectories of a divergence-free vector field on P2
(with holes) are closed then such a system is classified by Theorem 2.
Otherwise two cases are possible: M\ — the existence of both periodic trajectories
nontrivial in Н\(Р2;Ъ) and nonclosed (wandering) trajectories other than
separatrix, and M2 — the absence of nontrivial periodic trajectories but the
existence of a separatrix connection nontrivial in H\(P2\ Z). In either case
the system is C°°-classified by the invariant S £ Н1(Р2;Ж) and oo-equipped
transversal For the winding numbers λ of the type (Κ, σ) the coincidence
for two systems of their S-invariants and (k + r)-equipment implies their
Ck-conjugation, where r = 3 + [σ].
For the surface P2 there already appear interval exchange
transformations (which are usual for Euler characteristics χ < — 2) but they are nonori-
entable and always periodic:
Proposition 2. Let us consider the Poincare map on the transversal:
ф/ж\ = J (?-aO + 0modl, if χ 6(0,?),
У } \ x + 0mod 1, if x G (9,1).
It is periodic for all q, Θ, and moreover the number of minimal
periods equals 6.
References
[1] A.V. Bolsinov, Smooth trajectory equivalence of integrable Hamiltonian systems
with two degrees of freedom, Mat. Sbornik, No 1 (1995).
[2] A.V. Bolsinov and A.T. Fomenko, Trajectory equivalence of integrable
Hamiltonian systems with two degrees of freedom. Classification theorem. Parts I, II,
Mat. Sbornik 185(4, 5) (1994), 27-80, 27-78.
[3] B.S. Kruglikov, Exact smooth classification of Hamiltonian vector fields on
2-manifolds, Preprint of ICTP, Trieste, November 1994, IC/94/314; Math.
Notices, to appear.
[4] A.B. Katok and A.M. Stepin, On the approximation of ergodic dynamical
systems by periodic transformation, Dokl. AN SSSR 171(6) (1966).

TOPOLOGY and APPLICATIONS Moscow: PHASIS, 1996
International Topological Conference Pages 191-192
Dedicated to P.S. Alexandroff's 100th Birthday
Moscow, May 27-31, 1996
Serret's curves
Aleksandar T. Lipkovski
University of Belgrade, Belgrade, Yugoslavia
In his famous calculus course [1] J.-A. Serret discovered a family of plane
algebraic curves, in response to a problem posed by Legendre, to find all
algebraic curves with their arc length expressed by elliptic integrals of the
first kind. Following the suggestion of Ju.R Solovjev in 1994, the author
studied the family of Serret's curves extensively using the "Mathematica"
computer package, and surprisingly discovered their very nice properties.
Serret's definition of his curves was mechanical (see [2]). The loose end
M(x,y) of two hinged rods of length y/p and y/p + 1, fixed at the point
0(0,0), is being moved according to the rule cosu; = cos(pa — (p + 1)/?).
Here a and β are angles of AOPM. In this way one obtains the curve 5p.
For ρ £ Q it is possible to express cos(pa — (ρ + 1)β) as a polynomial in
sin a, sin/?, cos a, cos β. Clearly, there exist polynomial relations between
χ and r, and у and r respectively:
P(x,r) = 0, Q(y,r) = 0 (I)
Eliminating r one obtains a polynomial relation
F(x,y) = 0 (2)
and the curve Sp (for ρ € Q) is algebraic. Let ρ = k/l be the irreducible
representation (fc, / 6 N). Using the package "Mathematica", the author has
computed several equations. The analysis of computational results shows
that the relations (1) have the form
ao + air2 + ... + anr2n 2 2 2
X = ; , X + У — Г ,
brm
where η = к + I and m = 2k + (I - 1). Also, a0 = (-i)*, an = kk. The
polynomial (2) is
F(X, y) = ЬХ(Х2 + у2)Л+(/-1)/2 _ α() _ βι(χ2 + y2} _ _ аш{х2 + y2)k+l

192
for odd / = 2t +1 (t = 0,1,...). If / is even, the expression contains a radical
y/x2 + y2 and should be squared once more. For odd /, the degree of the
curve equals d — 2n = 2k + 21,
Surprisingly, not only the lemniscate (for ρ — 1), but all Serret's curves
for integer ρ are rational:
Proposition 1. The curve Sv has genus g = 0 for ρ £ N.
The proof is based on the calculation of the genus by resolution of
singularities at infinity and in the finite plane.
Serret showed that the arc length of these curves is expressed by elliptic
differential. The author calculated the explicit equation of the
corresponding elliptic curve:
Proposition 2· The associated elliptic curve for Sp has the equation
y2 — χ(χ - 1) ( χ ) and the j-invariant j = 28 Цт r^-.
V ρ J p2(p+l)2
Therefore, the lemniscate is the only Serret's curve with complex
multiplication.
References
[1] J.-A. Serret, Cours de calcul differentiel et integral^ 2nd French ed., Gauthier-
Villars, Paris, 1879; Lehrbuch der Differential- und Integralrechnung, 2nd
German ed., Teubner, Leipzig, 1899.
[2] V.V. Prasolov and Ju.P. Solovjev, Elliptic functions. A special course, Ed.
Math. College of Independent Moscow University, Moscow, 1993 (in Russian).

TOPOLOGY and APPLICATIONS Moscow: PHASIS, 1996
International Topological Conference Pages 193-194
Dedicated to P.S. Alexandroff's 100th Birthday
Moscow, May 27-31, 1996
On cohomology of a sheaf over foliation
with tangential (X, G)-structure
Mikhail A. Malakhal'tsev
Kazan State University, Kazan, Russia
Let X be a smooth manifold endowed with a quasi-analytic left G-action,
where G is a Lie group. A foliation with tangential (X, G)-structure on a
smooth manifold Μ is a maximal atlas on Μ whose coordinate transforms
have the form (u, x) <E U X V -> {φ{ν), Lg(u){x)) <E Rq X X, where U С R9,
V С X are open sets and φ : U —>- R9, g : U —>· G are smooth mappings. The
class of foliations with tangential (X, G)-structure naturally arises in various
geometrical situations, and, in particular, contains foliations given by locally
free actions of Lie groups, foliations determined by a structure of manifold
over local algebra [2, 5], affine foliations [1,4]. The leaves of a foliation with
tangential (X, G)-structure are Thurston's (X, G)-manifolds [3].
The action of the Lie group G on X induces the fundamental homo-
morphism σ : g —> X(X) of Lie algebras, where X(X) is the Lie algebra of
vector fields on X. Given an (X,G)-manifold L, we denote by X$(L) the
sheaf (over L) of germs of fundamental vector fields σ(α), a £ g. Then for
an (X, G)-foliation (Μ, Τ) we denote by Xq ' ' the sheaf of vector fields
tangent to Τ whose restrictions to any leaf L are sections of XQ(L).
Theorem 1. The cohomology group Ηι(Μ;Χ^ ' ') represents the space
of essential infinitesimal deformations of tangential (X^G)-structure
of{M,T).
References
[1] T. Inaba and K. Masuda, Tangentially affine foliations and ieafwise affine
functions on the torus, Kodai Math. J. 16 (1993), 32-43.
[2] V.V. Shurygin, Jet bundles as manifolds over algebras, Itogi Nauki i Tekhniki
19. Problemy geometrii, VINITI, Moscow, 1987, 3-22 (in Russian).

194
[3] W.P. Thurston, The geometry and topology of 3-manifolds, Mimeographed
Lecture Notes, Princeton Univ., 1978/79, 1980.
[4] I. Vaisman, dj-cohomologies of Lagrangtan foliations, Monatsh. fur Math. 106
(1988), 221-244.
[5] V.V. Vishnevskii, A.P. Shirokov and V.V. Shurygin, Spaces over algebras,
Kazan University, Kazan, 1985 (in Russian).

TOPOLOGY and APPLICATIONS Moscow: PHASIS, 1996
International Topological Conference Pages 195-197
Dedicated to P.S. Alexandroff's 100th Birthday
Moscow, May 27-31, 1996
The standard Cantor set is Lipschitz ambient
homogeneous on the plane
Joze Malesic *
University of Ljubljana, Ljubljana, Slovenia
A map h : Rn —> Rn is said to be bi-Lipschitz if there exists a positive
constant С such that
£ ρ(χ, у) < р(Чх), Чу)) < ср{х, у)
for each pair of points ж, у £ Rn. A Cantor set С С Rn is said to be
Lipschitz ambient homogeneous if for each pair of points c, d £ С there
exists a bi-Lipschitz homeomorphism
/*:(Rn,C,d)-+(Rn,C,c).
Theorem· Let С be the standard ternary Cantor set lying in the interval
[0,1] of the Ox-axis on the plane R2. Then С is Lipschitz ambient
homogeneous in R2.
Proof. The Cantor set С is self-similar:
C = 50(C)U52(C)
where So and £2 are similitudes acting as follows:
So(x, У) = (§, |) , 52(z, V) = (f + |, |) , (x, У) e R2.
Let J3 be a disk centered at the point f-,0j £ R2, with radius
r £ i-,1]. Denote
*bciC2...Cn = *bci ° bC2 О ... О »ЬСп
* Research supported in part by the Ministry for Science and Technology of the Republic
of Slovenia, grant No J1-7039-0101-95. The author would like to thank D. Repovs and
E.V. Shchepin for useful suggestions.

196
and
where cb c2,..., cn € {0, 2}. For example, B0 and B2 are disjoint congruent
disks lying in B, B0o and B02 are disjoint congruent disks lying in B0 etc.
For each point с £ С its first co-ordinate, written in triadic number system,
has the form Qx\c2c^ ... where Ci, c2, С3,... £ {0,2}. Therefore,
{c} = f| BC1C2..
n=l
and in special case,
{Ο} = Π B0»
«=1
where 0n = 00.. .0 (the sequence of η zeroes). Now construct bi-Lipschitz
homeomorphism h : (E2,C, d) —» (E2,C, c) for an arbitrary pair of points
c,rf e С
Without loss of generality we can assume that d — O. It is easy to
construct a diffeomorphism / : E2 —>- E2 such that
f(B) - В, /\я2\в = idR2\S>
f(B0) = B2, f(B2) = B0
and /|в0, /|в2 are translations parallel to the Ox-axis. Because of the above
properties, / is bi-Lipschitz. Now construct a sequence of homeomorphisms
9\, 52,5з? · · · given by the following definition:
9\
_ f id, ci = 0,
\/, c1 = 2f
Γ id, cn+1 = 0,
5n+1 " 1 5clC2..,n ο / ο 5"ΐ2_η, cn+1 = 2,
for every natural number n. Obviously, each homeomorphism </n is
bi-Lipschitz having the same Lipschitz constant С as /. Furthermore,
5n+i|R2\BCiC2 .Cn = id|R2\BciC2 cn,
5п+1(-Ос1С2...спо) — "cic2...cncn+i j

197
and gn+i\Bc c Cn0 is a translation. Now define
K=gno gn-i o...og2ogu η <E N.
The key fact is that there exists a constant D such that
Jjp(x,y) < p{hn(x),hn(y)) < Dp{x,y)
for every natural number η and for each pair of points x, у £ R2. To prove
this fact two cases should be observed:
(1) x, у e B0k \ B0k+2 for some k;
(2) χ £ B0k \ B0k+i for some к and у £ 2?0*+2 ог у^се versa.
In the second case the similarity of disks BClC2.„Cn should be used.
It is easy to prove (directly or by means of the Arzela-Ascoli lemma)
that the sequence of homeomorphisms Λι, Λ2, ^з, · · · converges pointwise to
a map h. Therefore
-рр{х,У) < p(h{x),h{y)) < Dp(x,y).
By construction, h is onto, hence Л is a bi-Lipschitz homeomorphism.
Obviously, h(0) = с and h(C) = C. D
Remark 1. This result is a counterexample to the conjecture
"Lipschitz homogeneous compacta in Mn are Lipschitz submanifolds
ofRn"
raised in the paper
D. Repovs, A.B. Skopenkov and E.V. Shchepin, Cl-homogeneous
compacta in Шп are Cl-submanifolds of Rn, Proc. Amer. Math.
Soc. 124 (1996), 1219-1226.
Remark 2. The homeomorphism h constructed above obviously is smooth
at all points with eventually except of the center О of coordinate system.
But h cannot be smooth at that point since it cannot be smooth everywhere,
by a result of the cited paper.

TOPOLOGY and APPLICATIONS Moscow: PHASIS, 1996
International Topological Conference Pages 199-200
Dedicated to P.S.Alexandroff's 100th Birthday
Moscow, May 27-31, 1996
Orbital classification of integrable Hamiltonian
systems with two degrees of freedom
in a neighborhood of equilibrium
Olga E. Orel
Lomonosov Moscow State University, Moscow, Russia
In this paper, we study the problem on orbital classification of integrable
Hamiltonian systems (IHS) in a neighborhood of nondegenerate elliptic
singular points. The work is based on the new theory of orbital classification of
IHS with two degrees of freedom created by Bolsinov and Fomenko in [3].
Two integrable Hamiltonian systems ν and v' on manifolds Μ and M',
respectively, are called continuously (smoothly) orbitally equivalent at
elliptic singular points x0 6 Μ and x'0 6 M' if there exist neighborhoods
U(xo) С Μ and U'(x'0) С Μ' of these singular points and a homeomor-
phism (diffeomorphism) φ : U —» U' that takes the trajectories of the first
system to the trajectories of the second one preserving the orientation. We
note that the transformation need not preserve the time parameter on the
trajectories.
Let F : M4 —l· R2, F(x) = (h(x),f(x)) be a momentum mapping. We
have rkdF(xo) — 0 at the elliptic singular point xq. We consider the
Hamiltonian system ν = sgrad h and the functions Λ, /, and F to be restricted
to some small four-dimensional neighborhood U of the singular point. The
function h is assumed to be from C2 for the problem on continuous orbital
invariant and from C4 for the problem on smooth invariant. Let Σ = F(K),
where К — {χ £ Μ4 : rkdF(x) < 2}, be a bifurcation diagram. The image
of a neighborhood of x0 under the momentum mapping has the form of a
curvilinear angle. Its two sides are transversal curves that are belong to the
bifurcation diagram and the vertex is the image of the elliptic point F(xo)
(see [2]). The continuous (smooth in the case h £ C4) rotation function ρ
is defined in the domain U \ K. It is equal to the ratio of frequencies of
quasiperiodic motion on Liouville tori. It can be extended by continuity on
the whole domain U.

200
Theorem. Two IHS ν and v' are continuously {smoothly) orbitally
equivalent at nondegenerate elliptic singular points y0 and y'0 if and only if there
exists a homeomorphism (diffeomorphism) φ : (Λ, /) —>· (Λ', /') from some
neighborhood V of the point t/o to a neighborhood V of the point y'0 that
possesses the following properties:
(1) <p(yo) — y'o, ψ{Σ) = Σ'> where Σ, Σ' are the bifurcation diagrams,
(2) φ preserves the rotation function.
To prove Theorem, we use the representation of IHS near elliptic singular
points obtained by Eliasson in [4].
Corollary. Orbital invariant of a nondegenerate elliptic singular point xq
of IHS in general position is a triple (p(xo))sign(p\(xo)))sign(p2{xo))> where
p(xo) is the limit of the rotation function at the point xq, and pi(xo) are the
partial derivatives of the function ρ at the point xo with respect to directions
of two lines tangent to the bifurcation diagram.
The corollary makes it possible to calculate elliptic orbital invariants
for most of classical integrable problems and compare them (from orbital
point of view) at nondegenerate elliptic singular points. The invariants can
be calculated using Birkhoff's theory of normal forms [1].
The author would like to express her gratitude to Prof. A.T. Fomenko
and A.V. Bolsinov for setting the problem and for attention to this work.
The work was performed under the auspices of Soros and INTAS.
References
[1] G.D. BirkhofF, Dynamical systems, Amer. Math. Soc, Providence, RI, 1927.
[2] A.V. Bolsinov, Methods of calculation of the Fomenko-Zieschang invariant,
Advances in Soviet Math. 4 (1991), 147-184.
[3] A.V. Bolsinov and A.T. Fomenko, Orbital equivalence of integrable Hamiltonian
systems with two degrees of freedom. Classification theorem I, II, Mat. Sbornik
185(4, 5) (1994), 27-80, 27-78 (in Russian).
[4] L.H. Eliasson, Normal form for Hamiltonian systems with Poisson commuting
integrals — elliptic case, Comm. Math. Helv. 65 (1990), 4-35.

TOPOLOGY and APPLICATIONS Moscow: PHASIS, 1996
International Topological Conference Page 201
Dedicated to P.S. Alexandroff's 100th Birthday
Moscow, May 27-31, 1996
On topological distinction of Waza continua
in the theory of smooth dynamical systems
Roman V. Ply kin
Obninsk Institute of Atomic Energy, Obninsk, Kaluga region, Russia
Waza continua are the most familiar examples of irreducible continua and
have shapes (and both AlexandrofF-Cech homologies) as a distinguishing
invariant.
However, in the theory of smooth dynamical systems Waza continua
with local structure of the direct product of a closed interval by the
Cantor discontinuum do naturally appear as attractors of codimension 1 of
smooth cascades.
For these continua, appearing in applications, a complete classification
can be given, by adding reals and rotation functionals of natural fibrations
of codimension 1 to groups of AlexandrofF-Cech homologies.

TOPOLOGY and APPLICATIONS Moscow: PHASIS, 1996
International Topological Conference Pages 203-204
Dedicated to P.S. Alexandroff's 100th Birthday
Moscow, May 27-31, 1996
Realization of geodesic flow
within the monopolistic framework
Zinaida G. Psiola
Lomonosov Moscow State University, Moscow, Russia
Consider the complete monopolist who produces and sells к goods. His
optimal strategy is confined in the choice of the prices (or amounts of output
goods) which provides his maximal profit. Let V be a linear space, dim V =
k) V* be the space dual to V. The commodity space is a submanifold X
of the space V, all prices are from a submanifold Μ С V* called the price
manifold. Let the demand function be a smooth map D : T*M —> X on the
cotangent bundle T*M, the cost function be a smooth map С : X —» R [1].
The price dynamics for monopoly is defined by the Lagrangian /(p,p) =
(D(p,p),p) - C(D(p,p)) [4], where (·, ·) is the natural pairing of V and V*.
Note that the corresponding Euler-Lagrange equation
fdl(p,p)\ _ dlfap) _
\ dp. )- dps ' s-1'···'*'
has the energy integral ρ(ρΌφ-Οϋφ) -pD + C.
Theorem. Consider а к-product monopoly on a price manifold
Μ = {(pi,...,p*) | p. > λ5, 5= l,...,fc, 3tpt > At}
with local coordinates (p\,..., рь). Let functions D(p,p) = (xl,..., xk),
С = C(D{Pip)) = Хгх1 + .. . + \кхк, where xs{p) =0(p)pipjf 5=1,...,*,
are amounts of output goods, smooth functions (β%8*(ρ)) define Riemannian
metrics on Μ and λι,..., λ* are arbitrary nonegative constants
corresponding to the cost of the unit output Then:
1. The price dynamics for this model coincides with the motion along
geodesic lines on the Riemannian manifold Μ with the metric
9ij = lib(Pt-*tW(p).

204
2. If M is already a Riemannian manifold with metric gij(p\)..., Pk) then
there exist parameters β^(ρ) such that price dynamics of the
corresponding model of the к-product monopoly coincides with the geodesic
flow on M.
Note that this approach reduces the study of the monopolist's
decisionmaking process to the tracing of geodesic flows on a Riemannian manifold
[2, 3] as well as provides the solution for the problem of optimal synthesis
concerned-with the affect of the current state on the general price strategy.
The author gratefully acknowledges important comments and advice of
Prof. V.V. Trofimov.
References
[1] A.T. Fomenko and T.L. Kunui, Topological Modelling for Visualization, Oxford
Univ. Press, Oxford, 1995, V.V.Trofimov, Appendix F.
[2] Z.G. Psiola, Topological aspects of the monopolist model, Adv. Math. Sc. 4
(1995), 167-168.
[3] Z.G. Psiola, E.R. Rozendorn and V.V. Trofimov, Nonlinear economic dynamics,
Fund, and Appl. Math., to appear.
[4] A. Takayama, Mathematical Economics, Cambridge Univ. Press, Cambridge,
N.Y., 1985.

TOPOLOGY and APPLICATIONS Moscow: PHASIS, 1996
International Topological Conference Pages 205-206
Dedicated to P.S. Alexandroff's 100th Birthday
Moscow, May 27-31, 1996
Local paraconvexity
and local selection theorem
Pavel V. Semenov
Moscow State Pedagogical University, Moscow, Russia
For a Banach space Υ and for its nonempty subsets A and В denote [A]
the convex hull of A, R(A) the Chebyshev radius of A and dist(A : B) the
supremum of all distances dist(a, B), a € A.
Definition. A family С of nonempty closed subsets of a Banach space Υ is
said to be equi-locally paraconvex at a point у G Υ if there exist r = r (y) > 0
and a = ot(y) € [0,1) such that for every L € С which intersects the open
ball D(y)r) and for every А С D(y,r) f]L the following inequality holds:
dist([A]:L) <a-R(A).
The global version of paraconvexity was proposed by E. Michael [2].
In such global notion the quantifier 3a stands on the first place and the
inequality above transforms into the inequality л
di$t{[DnL]:L) < a R{D)
for every open ball D. So, the (global) selection theorem was proved in [2] for
a lower semicontinuous α-paraconvex-valued mappings over paracompacta.
For examples and generalizations, see [3, 4, 5]. We prove the local selection
theorem which is an analog of finite-dimensional selection theorem [1], but
for an arbitrary paracompact domain. Shortly, we replace the equi-LCn
conditions by equiAocal paraconvexity of the family of values.
Theorem. Let F be a lower semicontinuous mapping from a
paracompact space X into a Banach space Y. Let A be a closed subset of X and
h : A-+Y be a continuous single-valued selection of the restriction F\a-
Then h can be continuously extended to a selection of F\u for some open
neighborhood U of A, whenever the family of values of F is equi-locally
paraconvex at every point у € h{A).

206
References
[1] Ε. Michael, Continuous selections II, Ann. Math. 64 (1956), 375-390.
[2] E. Michael, Paraconvex sets, Scand. Math. 7 (1959), 372-376.
[3] D. Repovs and P.V. Semenov, On functions of nonconvexity for graphs of
continuous functions, J. Math. Anal, and Appl. 196 (1995), 1021-1029.
[4] P.V. Semenov, Functionally paraconvex sets, Mat. Zametki 50 (1993), 75-80
(in Russian).
[5] P.V. Semenov, On paraconvexity of star-like sets, Sib. Mat. Zh. 37 (1996),
399-405 (in Russian).

TOPOLOGY and APPLICATIONS Moscow: PHASIS, 1996
International Topological Conference Pages 207-208
Dedicated to P.S. Alexandroff's 100th Birthday
Moscow, May 27-31, 1996
Relative structural stability and relative structural
instability of different degrees
in Topological Dynamics
Maxim V. Shamolin
Lomonosov Moscow State University, Moscow, Russia
By a structurally stable (coarse) system of differential equations we mean
a system such that under small deformations that belong not to the entire
class of functions, but only to some subclass, it is equivalent to the original
system [1].
A system of differential equations specifying a sufficiently smooth vector
field V will be referred to as structurally stable (coarse) with respect to a
class of functions К (relatively structurally stable) if any vector field W
defined with the aid of the class of functions К and obtained by deforming
the field V in the standard topology relative to the class of functions К is
topologically equivalent to the field V [2, 3].
It is known that coarse systems are not dense in the standard topology
[4]. But if one considers coarse systems with respect to some subclass,
then it may turn out that in the standard topology the given systems can
generate an everywhere dense set.
Relatively structurally instable vector fields which are the vector fields
of different degrees of structural instability can be defined analogously.
References
[1] M.V. Shamolin, Relative structural stability in the problem of a body
motion in a resisting medium, ICM'94. Abstracts of Short Communications,
Zurich, 1994, 207.

208
[2] M.V. Shamolin, New two-parameter families of the phase patterns in the
problem of a body motion in a resisting medium, ICIAM'95. Book of Abstracts,
Hamburg, 1995, 436.
[3] M.V. Shamolin, Qualitative methods to the dynamic model of an interaction of
a rigid body with a resisting medium and new two-parametric families of the
phase portraits, DynDays'95. Program and Abstracts, Lyon, 1995, 185.
[4] S. Smale, Coarse systems are not dense, Period. Sbornik Perev. Ino'str. Statei
11(4) (1967), 107-112.

TOPOLOGY and APPLICATIONS Moscow: PHASIS, 1996
International Topological Conference Pages 209-210
Dedicated to P.S.Alexandroff's 100th Birthday
Moscow; May 27-31, 1996
Hausdorff dimension
and dynamics of diffeomorphisms
Evgenii V. Shchepin
Steklov Mathematical Institute, Russian Academy of Sciences, Moscow,
Russia
There are two purposes of the lecture. The first is to point out an elementary
proof of the Hilbert-Smith conjecture for Lipschitz actions [1]. The second
is to point out consequences of this result for diffeomorphisms.
We start with the second part. A diffeomorphism Λ of a manifold Mn
is called a Lie diffeomorphism if it belongs to a compact Lie subgroup of
Diff Mn — the group of all diffeomorphisms of Mn endowed by C^-topology.
The following characterization theorem is implied by the Hilbert-Smith
conjecture for Lipschitz actions.
Theorem 1. A diffeomorphism h of a manifold Mn without boundary
is a Lie diffeomorphism iff all orbits of points are bounded and, at any
point χ £ Mn, the partial derivatives in any direction are bounded for all
iterations of h.
The proof of the above theorem relied on a non-elementary theorem
of Yang [2] which stated that the cohomological dimension of the orbit
space for a p-adic action have to be greater than the dimension of the
original manifold.
Yang's theorem may be replaced by the following elementary theorem
for actions with infinite orbits.
Theorem 2· Let a compact group G act effectively via Lipschitz homeo-
morphism on a metric space with finite k-dimensional Hausdorff measure
and all orbits of the action be infinite. Then the topological dimension of
M/G is less than к.
The last theorem implies the nonexistence of p-adic Lipschitz actions on
every Hausdorff regular metric space. Here we call a metric on a space

210
Μ of topological dimension к Hausdorff regular if Μ has locally finite
fc-dimensional Hausdorff measure. An important example of such metric is
given by Riemannian metric on manifolds.
References
[1] D. Repovs and E.V. Scepin, A proof of the Hilbert-Smith conjecture for actions
of Lipschitz maps, Institute of Mathematics at University of Ljubljana, Preprint
series 34 (1996), 502.
[2] C.T. Yang, p-Adic transformation groups, Michigan Math. J. 7 (1960), 201-218.

TOPOLOGY and APPLICATIONS
International Topological Conference
Dedicated to P.S.Alexandroff's 100th Birthday
.Moscow, May 27-31, 1996
On the bigraduated cohomology of manifolds
over local algebras and its applications *
Vadim V, Shurygin
Kazan State University, Kazan, Russia
Let M„ be a smooth η-dimensional manifold over a local
algebra A [6, 5], AVM^ the bundle of exterior p-forms over M^, and
A.A-lin<Mn^ the subbundle of Α-linear p-forms. Using the
decomposition A<g>AlM£ = ΑιΑ_ιϊηΜ£ φ A1 M£ one can construct the bundles
дг,5д^А с А®ЛГ+5М^ whose fibers ArjfM£ are spanned by exterior
products ξι Λ ... Λ fr Λ ηχ Λ ... Λ η8, where fi,...,fr € ΑιχΜ* and
are Α-linear 1-forms. Let us denote by ΩΓ'5({7) the A-ipodule of smooth
sections of ЛГ,5М£ over an open set U С M^, and by Ωτ*8Μ£ the sheaf of
germs of smooth sections of ΛΓ·5Μ^. For ω € ΩΓ'5(£7) we define άω to be
the component of the exterior differential άω which belongs to ΩΓ+1»*(ί/).
There holds dod = 0, and let Нг>а(М£) denote the corresponding
cohomology groups:
Let I bis an ideal of A, A = A/I the quotient algebra, and
ЛХ-1шМ$ С A ® ЛрМ^ the subbundle of Α-linear forms. There arises the
decomposition
A ® Л*МА = A^_linM^ φ Λ^Μ^
and the corresponding bundles Λ^ΜηΑ. Let f&'(U), Jfc'M* and № .._MA
A A A A,—Qllt
denote, respectively, Α-module of smooth sections of the bundle ΑψΜ^ over
J7, the sheaf of germs of smooth sections of this bundle, and the sheaf of
Α-smooth A-valued s-forms on M„.
The following statements take place:
The operator d can be extended to the case of forms of Ω~5(ί7), and
Hr/(M£) S Я" (мА, П|_аш(МА)).
The research described in this publication was made possible in part by Grant
No JGX100 from the International Science Foundation and Russian Government.
Moscow: PHASIS, 1996
Pages 211-212

212
The canonical epimorphism ρ : A —> A induces the morphism of com-
plexes ρ : Ω*'5(Μ^) —> Ω~5(Μ^) and the corresponding cohomology mor-
Q
phisms Η ρ : НГ>3(М£) -> #£5(М£). In the case when I = A, the
maximal ideal of A, p : A -» R induces Hp : Hr>s(M£) -> Hr/(M%), where
HpS(M£) are dp-cohomology groups [4] of M^ with respect to the canonical
о
Α-foliation (see [5]).
Forl-valuedforms, there holdsЩ>8(М%) й Нг(м£,Of A_diff(M*))·
For JAe bundle J^Wn of Α-jets in the sense of A. Weil (see [6, 5, 3])
tfiere Ao/d tf °'5(JAWn) S A®Q*(Wn), tf r'5(JAWn) =0i/r>0.
The isomorphisms indicated above allow to represent in terms of ci-coho-
mology the space Я1 (MA,T(MA)J, which contains infinitesimal
deformations of Α-smooth structure on MA in the sense of Kodaira and Spenser
[2], and the Atiyah classes [1] which are obstructions to the existence of
Α-smooth connections in Α-smooth principal bundles. In addition, in terms
of d-cohomology the obstruction to the prolongation of transversal con-
o
nection on Μ£ with respect to the canonical A-foliation [4] to A-smooth
Α-linear connection is represented.
References
[1] M.F. Atiyah, Complex analytic connections in fibre bundles, Trans. Amer.
Math. Soc. 85 (1957), 181-207.
[2] K. Kodaira, Complex Manifolds and Deformations of Complex Structures,
Springer-Verlag, 1986.
[3] I. Kolar, P.W. Michor and J. Slovak, Natural Operations in Differential
Geometry, Springer-Verlag, 1993.
[4] P. Molino, Riemannian Foliations, Birkhauser, 1988.
[5] V. V. Shurygin, Manifolds over algebras and their applications to the geometry of
jet bundles, Uspekhi Mat. Nauk (Russian Math. Surveys) 43(2) (1993), 75-106.
[6] V.V. Vishnevsky, A.P. Shirokov and V.V. Shurygin, Spaces over algebras, Kazan
University, 1985.

TOPOLOGY and APPLICATIONS Moscow: PHASIS, 1996
International Topological Conference Page 213
Dedicated to P. S. Alexandroff 's 100th Birthday
Moscow, May 27-31, 1996
Minimal topologies on acting groups
Yuri M. Smirnov
Lomonosov Moscow State University, Moscow, Russia
R. Arens has proved that if a group of homeomorphisms acts on a locally
compact space then among all its topologies under which both group
operations and the action are continuous, there is a minimal one. He defined it
as a slight modification of the compact-open topology.
It turns out that a criterion for the existence of a minimal topology can
be given with the help of equivariant compactifications of a TychonofF space
with a group of homeomorphisms acting on it. In general, a minimal
topology need not exist. Arens' constructive definition of a topology, minimal
in the case of a locally compact space, does not provide minimality in the
general case.

TOPOLOGY and APPLICATIONS Moscow: PHASIS, 1996
International Topological Conference Pages 215-216
Dedicated to P.S. Alexandroff's 100th Birthday
Moscow, May 27-31, 1996
Summands of the Turaev-Viro invariants
Maxim V. Sokolov
Chelyabinsk State University, Chelyabinsk, Russia
In 1990 V.G. Turaev and O.Ya. Viro obtained an infinite set of 3-manifold
numerical invariants TV(M)q [6]. The invariants are parametrized by roots
of unity q of degree 2r such that q2 is a primitive root of degree r > 3.
Let X be a simple spine of a 3-manifold M3. Fix r > 3. Let
Gi,...,Gb be the 2-components of X. By a coloring of X we mean an
arbitrary mapping
<p:{Gu...,Gb}-> Zr_! = {0,1,..., r - 2}.
Let us denote the set of colorings of X by Col(X).
The Turaev-Viro invariant for a 3-manifold Μ is computed by the
following formula
TV(M)q= Σ I*. И,
φ£θο\(Χ)
where \X, <p\q € Q(q) is the weight of the colored spine X. A definition of
\X, φ\4 is contained in [6].
Set Adm(X) = {ψ e Col(X) : \X,ip\q ^0}, and .
S(<p) = {J{Cl(Gi): <p(Gi) = l (mod 2)},
where Cl(G{) denotes the closure of G{. It is known, that if φ 6 Adm(X)
then S(cp) is a closed surface.
Let us represent the set Adm(X) as the union of three sets
Adm0(X)UAdmi(X)UAdm2(X) where
0) φ € Adm0(X) «=* ψ € Adm(X), S{<p) = 0;
1) ψ € Admi(X) <i=> y? € Adm(X), x{S{<p)) = 1 (mod 2);
* Supported in part by ISSEP, Grant No a96-1639, INTAS, Grant No 94201.

216
2) φ € Adm2(X) <=»> j χ(5(ν)) ^ Q (mod 2)
Theorem 1 ([2, 3]). The numbers
tvn{M)4= Σ \χΜ9,
where N £ {0,1,2}, are non-trivial (except the case r = 3, Af = 0)
invariants of M, and
TV(M)q = TV0{M)q + TVi(M)q + TV2(M)q.
Lemma ([4]). Let X be a simple spine of a 3-manifold M. Then for any
φ € Adm(A') and for any parameter q we have
\X,<p\q = (-l)xW«»\X,<p\-r
The following theorem is an easy corollary of the previous lemma.
Theorem 2 ([4]). For any 3-manifold Μ and any q we have
TVN(M)q = (-l)NTVN(M).q, where N G {0,1,2},
TV0(M)q + TV2(M)q = ±(TV(M)q + TV(M).q),
3ViWf = \{TV{M)q-TV{M).q).
Remark. It follows from the Turaev-Walker theorem (see [5, 1]) that if r
is odd then, up to normalization, the invariant TVo(M)q coincides with the
square of the modulus of the so-called SO(3)-invariant re(M) defined in [5].
References
[1] J.D. Roberts, Skein theory and Turaev-Viro invariants, Topology 34 (1995),
771-787.
[2] M. Sokolov, Calculation of Turaev-Viro invariants for 3-manifolds and
solution of Kauffman-Lins conjecture, Abstracts of XXX International Scientific
Student Conference, Novosibirsk, 1992, 82-88 (in Russian).
[3] M. Sokolov, The Turaev-Viro invariant for Ъ-manifolds is a sum of three
invariants, Canad. Math. Bull., to appear.
[4] M. Sokolov, On the absolute value of the SO ($)-invariant and other summands
of the Turaev-Viro invariants, Submitted for publication to Banach Center
Publications, available via q-algCeprints. math. duke. edu, 9601013.
[5] V.G. Turaev, Quantum invariants of knots and 3-manifolds, Walter de Gruyter,
Berlin-New York, 1994.
[6] V.G. Turaev and O.Ya. Viro, State sum invariants of 3-manifolds and quantum
6j-symbols, Topology 31 (1992), 865-902.

TOPOLOGY and APPLICATIONS Moscow: PHASIS, 1996
International Topological Conference Pages 217-218
Dedicated to P.S. Alexandroff's 100th Birthday
Moscow, May 27-31, 1996
Entire rational approximation of G-maps
Dong Youp Suh
Korea Advanced Institute of Science and Technology, Taejon, Korea
Let G be a compact Lie group. A real algebraic G-variety in an orthogonal
representation Ω is the common zeros of polynomials pi,.. .,pm : Ω -» R,
which are invariant under the action of G on Ω. Let X, respectively Υ, be a
real algebraic G-variety in an orthogonal representation Ω, respectively Ξ.
A G-map / : X —> Υ is entire rational if there are polynomials Ρ : Ω —> Ξ
and Q : Ω -> R with Q(x) φ 0 for all χ <E X such that f(x) = P(x)/Q(x)
for all χ € X.
The problem we are interested in is as follows:
Equivariant Entire Rational Approximation Problem. When can
a smooth G-map f : Χ —ϊ Υ between two nonsingular real algebraic
G-varieties X and Υ be approximated by an entire rational G-map?
The problem is motivated by non-equivariant works by Bochnak and
Kucharz in [1], [2] and [3]. Here we consider the problem in the case when
the target space Υ is the unit sphere of an orthogonal representation of
G. Especially when Υ is either 1- or 2-dimensional sphere, the problem is
closely related with equivariant vector bundle theory. Namely, sometimes Υ
can be viewed as Grassmann G-variety, and the smooth G-map / : X —> Υ
is a classifying G-map for some G-vector bundle over X.
There are several results on these cases, and we state two of them.
Theorem A ([5])· Let G be either an odd order group or any compact
Abelian group. Let W be any-dimensional and V a 2-dimensional
representation of G. Then any smooth G-map f : S(W) —>· S(V) can be
approximated by an entire rational G-map.
Theorem В ([5]). Let G be a finite cyclic group or any compact Abelian
group. Let Ε and U%} г = 1,..., η be any-dimensional and V a 1-dimensional
unitary representation of G. Let X = S(E) χ CP(U\) X ... X CP(Un) and

218
Υ = 5(R® V). Then any smooth G-map f : Χ χ Υ can be approximated by
an entire rational G-map.
Equivariant entire rational approximation is not always possible. In
fact, under certain conditions any entire rational G-map to even-dimensional
G-spheres are G-homotopically trivial [5]. On the other hand, if we allow the
algebraic variety structure of X in a given differentiable manifold structure
in the approximation of /, the problem becomes more topological and we
r^fer the reader to [4] for such a case.
References
[1] J. Bochnak and W. Kucharz, Algebraic approximation of mappings into spheres,
Michigan Math. J. 43 (1979), 119-125.
[2] J. Bochnak and W. Kucharz, Realization of homotopy class by algebraic
mappings, J. Reiner Angewandte Math. 377 (1987), 159-169.
[3] J. Bochnak and W. Kucharz, On real algebraic morphisms into even-dimension-
al spheres, Ann. Math. 128 (1988), 415-433.
[4] K.H. Dovermann, M. Masuda and D.Y. Suh, Algebraic realization of equivariant
vector bundles, J. Reiner Angewandte Math. 448 (1994), 31-64.
[5] D.Y. Suh, Entire rational approximation of G-maps and strongly algebraic
G-vector bundles, Preprint, 1996.

TOPOLOGY and APPLICATIONS Moscow: PHASIS, 1996
International Topological Conference Pages 219-221
Dedicated to P.S. Alexandroff's 100th Birthday
Moscow, May 27-31, 1996
Cohomological sphere bundles
and parametrized Borsuk-Ulam theorems
Alexei Yu. Volovikov
Moscow Institute of Radiotechnics, Electronics and Automatics, Moscow,
Russia
We present a generalization of parametrized Borsuk-Ulam theorems proved
by Dold [2] and Nakaoka [5]. Using the results of [3] we generalize
Dold-Nakaoka's theorems to the case of cohomological sphere bundles.
1. All spaces are assumed to be paracompact and maps continuous. We
use Cech cohomology groups with coefficients in a field k.
Let G be a compact Lie group. For a G-space X define its equivari-
ant cohomology using the Borel construction, i.e. Hq{X) = H*(Eq Xg X)
[4, Chapter 3] where Eq —* Bq is the universal principle G-bundle.
Assume that Τ С Υ and С С Ζ are closed invariant subspaces of a
G-space Ζ and g : Ζ -* Υ is a G-map. A G-map (С,СП g~xT) -* (У,Т)
arising from g induces a homomorphism of equivariant cohomology groups.
Following [3] we denote its kernel by IndyTG.
Let Ρ С Ε be a closed invariant subset of a G-space Ε and assume that
the following diagram of G-spaces and G-maps is commutative:
Χ Α Ε D E\P
ψ
Υ > В
i.e. μ/χ = φι/χ for any x 6 f~l(E\P).
We put A = f~lP and assume that Τ С Υ is a closed and G-invariant
subspace.

220
Proposition, a) (φ%Ιηά%(Ε\Ρ)) · (Ш$т А) С Ind£TX.
b) Let L be a linear subspace in Hq(Y,T) and e G lnd%(E \ P).
If acp^e £ ΙηάγΤΧ for any 0 ф a 6 L then L Π lndyTA = 0, i.e.
L —» Hq(A, Α Π г/_1Т) г*5 α monomorphism.
Here <p£ : #£(£) -> Я£(У) is induced by ψ.
2. Let π : Ζ —> Υ be a surjective map. Suppose that the fibers of π are
cohomology π-spheres over к and that the Leray sheaf of π is constant with
fibers H*(Sn). In this case we say that π is an orientable cohomological
sphere bundle. Then for connected Υ there is a Gysin sequence of π, we can
define the Euler class e G Hn+l(Y) (depending on orientation) and Indy Ζ
is the ideal (e) generated by e.
Now assume that in the diagram (*) Υ and В are connected and that
the maps Eq^gX -> EqXgY and EqXg{E\P) -> EqXgB arising from
ν and μ are orientable cohomological sphere bundles with Euler classes e'G)
eo respectively.
Theorem. If a 6 Indy A then онр^ео G (^g)·
Assume further that G acts trivially on Υ and B. Then Hq{Y) =
Л* ® #*(У), ff£(fl) = Л* ® tf*(B) where A* = H*{BG) = H%(pt). Let
ν and μ be orientable cohomological sphere bundles and assume that G
preserves the orientations. Denote the equivariant Euler classes of fibers
v~x у and μ~ιΒ by e'G and ео,ь respectively.
Corollary. // the classes e'Gy and еа%ь are n°t zero divisors in A*
then Σ A* ® H*(Y,T) -¥ Hq(A,A Π ι/"1^ is a monomorphism for
j < dege^-degea.
In particular, this assertion can be applied in the situation of torus or
p-torus action without fixed points.
We can define characteristic classes of an orientable cohomological
π-sphere bundle (with an orientation preserving G-action) via the
equivariant Euler class (cf. [1, Chapter 4, § 10]). For example, if G = Z2, к = Z2,
г n+1
the fixed point set XG is empty, then ec = Σ tn+l~* X ti7j, Λ* = Ζ2Μ,
i=o
degi = 1, Wj are Stiefel-Whitney classes (by definition). If G = S1, к = Zp
and XG = 0 then the equivariant Euler class coincides (for locally trivial
bundle) with the characteristic polynomial of Nakaoka [5].

221
The above results can be generalized to the relative case [6] and to the
case of multivalued mappings.
References
[1] G. Bredon, Sheaf theory, McGraw-Hill, New York, 1988.
[2] A. Dold, Comment. Math. Helv. 63 (1988), 275-285.
[3] E. Fadell and S. Husseini, Ergodic theory and dynamic systems 8 Special Issue
(1988), 259-268.
[4] W.C. Hsiang, Cohomology theory topological transformation groups, Springer-
Verlag, Berlin, 1979.
[5] M. Nakaoka, Lecture Notes in Math. 1411 (1989), 155-170.
[6] A. Volovikov, Uspekhi Mat. Nauk 51(3) (1996), 189-190.

TOPOLOGY and APPLICATIONS Moscow: PHASIS, 1996
International Topological Conference Pages 223-224
Dedicated to P.S. Alexandroff's 100th Birthday
Moscow, May 27-31, 1996
On some variant of the degree theory and its
applications to problems of Hydrodynamics
Victor G. Zvyagin
Voronezh State University, Voronezh, Russia
Let D be a boundary domain of a Banach space E. We introduce the idea of
the degree of a map deg(A—g—k€, D, j/o) for maps of type A—g—ke : D —l· F
where F is a Banach space, Л is a homomorphism, k€ is a completely
continuous map and g is an Α-condensing map. This idea is used for the
investigation of the problem of weak solutions of an initial-boundary problem
for a vector function ν : Qt —* Kn, ν = (t>i,,.., vn) and a scalar function
dv n dv fl
— - μ0Αν + J2 Ъ -сг^- + Bi (v) " ] Div[a(t, s, v(s), Dv(s)] ds
t=l
- / L(t,s)B2{v)ds + gr<idp = f{t,s), {x,t)eQT, (1)
Jo
divv(a:,i) = 0, (x,t) G Qt, (2)
v(3,t) = o, te[o,T], xedn, (3)
υ(χ,0) = υ°(χ), же Ω, (4)
where Ω С Rn is a bounded domain with C°°-smooth boundary,
QT = Ω χ [Ο, Τ], μ0 > 0 is a constant, / : QT -¥ Rn, v° : Ω -* En are given
functions, η = 2, 3. Here B{{v) = - Όίν[2μί(/2(υ))(ε(υ))], г = 1, 2, where
ε(υ) is the matrix-function with components Sij(v) = - I —— + —— J,
' 2 у oxj oxi J
/ η \i/2
h(v) — ( Σ [ε«^(υ)]2 I j M«(5) is a continuous differential function and
α(ί, 5, ν, w) is a matrix-function with components at-j(i, 5, v, it;) which satisfy
the Lipschitz condition on variable w.
Different models describing the motion of non-Newton fluids lead to
investigation of the problem (l)-(4). Particular cases of this problem were
investigated earlier by Litvinov, Oldroit, Sobolevskii, Agranovich and others.

224
Under some conditions on functions /zt-, г = 1, 2, and atJ(t, 5, v,tt;) the
following theorem holds:
Theorem. For any f e £2((0, Г), V*) and v° e Η the problem (l)-(4) has
at least one solution υ € Χ Π L°°((0,T), Я), and tAi* solution satisfies the
following estimate:
n II ftv
£«N011»+ Σ"
- - t=i
dxi
L2((0,T),H)
+ ΙΙνΊΐ^((0,Τ),ν·)
< С (l + ||/|b((o,T),v·) + \\А\н)
with a constant С which does not depend on v°, f and v.
Here Η is the closure of C°°-smooth finite solenoidal functions in L2(ii)
о
and V is the closure of the same functions in W^i^).

PHASIS Publishing House
(Licence LR No 064705 / 09.08.1996)
42-44 Presnenski val, 123557 Moscow, Russia
Phone/Fax: (7 095) 253-0820
URL: http://www.aha.ru/~phasis
E-mail: phasis@aha.ru
Printed in Russia
by the RAS Printing House No 2
6 Shubinski per., 121099 Moscow

ISBN 5-7036-0017-0
The International Topological Conference, which was held
in Moscow in the summer of 1996, had gathered a brilliant
assembly of mathematicians.
The Conference was dedicated to the 100th anniversary
of the birth of Pavel Sergeevich Alexandroff, the founder
of the world-famous Moscow topological school.
The traditions of this school were sharply featured in the
lectures delivered by mathematicians from Moscow State
University, Steklov Mathematical Institute and other Russian
scientific centers. The lectures presented by foreign guests
of the Conference also comprised a great deal of interest.
The papers collected in this book are written by the
participants of this Conference. They give a clear impression
of the current state of Topology, as one of the most actively
developing mathematical sciences. Nowadays topological
methods often find applications not only in other branches
of Mathematics, but also in various problems of Physics
and other sciences. Investigations in such fields as the behavior
of dynamical systems and their stability, wave phenomena
in a continuous medium, quantum mechanics and field theory
are especially fruitful when they are carried out from the
topological viewpoint.
The present book will be of use both for professional
mathematicians and for all investigators who apply
topological methods in their research.
PHASIS Publishing House
Moscow