Strong Shape and Homology

486 Pages·2000·14.819 MB·English

by  Sibe Mardešić (auth.)

Springer Monographs in Mathematics
Springer-Verlag Berlin Heidelberg GmbH
Sibe Mardesic
Strong Shape
and Homology
, Springer
Sibe Mardesic
Department of Mathematics
University of Zagreb
Bijenicka cesta 30
10000 Zagreb, Croatia
e-mail: [email protected]
Library of Congress Cataloging-in-Publication Data
Mardesic, S. (Sibe),1927-
Strong shape and homology I Sibe Mardesic.
p.cm. -- (Springer monographs in mathematics)
Includes bibliographical references and index.
ISBN 978-3-642-08546-8 ISBN 978-3-662-13064-3 (eBook)
DOI 10.1007/978-3-662-13064-3
I. Shape theory (Topology) 2. Homology theory. I. Title. II Series
QA612.7 .M353 1999
514'.24--dc21 99-047673
Mathematics Subject Classification (l991): 55NXX,55PXX, 18GXX
ISBN 978-3-642-08546-8
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Originally published by Springer-Verlag Berlin Heidelberg New York in 2000
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Printed on acid-free paper SPIN 10732798 41/3143AT-5 4 3 21 0
Preface
It is well known that standard notions of homotopy theory are not adequate
to study global properties of spaces with bad local behavior. For instance, all
homotopy groups of the dyadic solenoid vanish in spite of the fact that the
solenoid is globally a non-trivial object. Shape theory is designed to correct
these shortcomings of homotopy theory. When restricted to spaces with good
local behavior, like the ANR's, polyhedra or CW-complexes, shape theory
coincides with homotopy theory, therefore, it can be viewed as the appropriate
extension of homotopy theory to general spaces.
Many constructions in topology lead naturally to spaces with bad local
behavior even if one initially considers locally good spaces, e.g., manifolds.
Standard examples include fibers of mappings, sets of fixed points, attractors
of dynamical systems, spectra of operators, boundaries of certain groups. In
all these areas shape theory has proved useful.
It took some time to realize that beside ordinary shape, introduced in
1968 by K. Borsuk, there exists a finer theory, presently called strong shape
theory, which has various advantages over ordinary shape. Its position is in
termediate, between homotopy and ordinary shape. One encounters a similar
situation in homology theory. Beside singular homology, which is a homotopy
invariant, and Cech homology, which is a shape invariant, there exists strong
homology, which is a strong shape invariant. In the special case of metric
compacta, this homology was introduced by N.E. Steenrod in 1940 and is
often referred to as the Steenrod homology.
The main purpose of the present book is to develop in detail strong shape
theory and (ordinary) strong homology groups for arbitrary spaces. To date
there exist five books considering shape theory: (Borsuk 1975), (Edwards,
Hastings 1976), (Dydak, Segal 1978), (Mardesic, Segal 1982) and (Cordier,
Porter 1989). However, only the second one includes considerations on strong
shape (of metric compacta). There exist numerous books on homology theory,
but only a few ((Bredon 1967), (Edwards, Hastings 1976), (Massey 1978) and
(Sklyarenko 1989a, 1989b)) consider strong homology (mostly restricted to
compact or locally compact spaces).
In the present book the approach to strong shape uses the technique
of inverse systems, already successfully applied to ordinary shape, e.g., in
(Mardesic, Segal 1982). One first generalizes the homotopy theory of spaces
VI Preface
to a homotopy theory of inverse systems of spaces. The second step consists
of approximating spaces by polyhedra (or ANR's), i.e., of replacing spaces
by suitable inverse systems of polyhedra (or ANR's), associated with these
spaces. Finally, one applies the developed homotopy theory to the associated
systems and one proves that results depend only on the spaces and not on
the choice of the approximating systems. In contrast to the situation in ordi
nary shape, in strong shape there are considerable difficulties in carrying out
this program. The right modification of homotopy to systems of spaces is the
rather involved coherent homotopy. Moreover, the right approximation pro
cess consists of delicate constructions like resolutions and strong expansions.
Similarly, strong homology groups are first defined for inverse systems of
spaces. An essential feature of the definition is that one defines a complex of
strong chains. Strong homology groups of a system are just homology groups
of this chain complex. This insures exactness of strong homology, a property
lacked by Cech homology. For compact metric spaces, our strong homology
coincides with the Steenrod homology. In general, strong homology does not
have compact supports.
The content of the present book is almost disjoint from the content of
other books on shape. However, acquaintance with the shape theory book
(Mardesic, Segal 1982), to which we often refer, will facilitate the reading.
Results forming the core of the book are fully proved, even when this requires
performing lengthy computations. The reader is advised to skip at first read
ing these computations and concentrate on the nature and structure of the
formulae in question, which are relatively simple and pretty. Additional infor
mation and bibliographic data can be found in Bibliographic notes following
each section, except for Sections 10 and 22, which are surveys of results with
out proofs. In an effort to make the book as selfcontained as possible, I have
included some material from homological algebra. In particular, I discuss
at length derived functors of the functor lim, which play an essential role
in strong homology of non-compact spaces. The only book on this subject
is (Jensen 1972). I also included an introduction to spectral sequences and
some material on abelian groups.
The book consists of four chapters, the first two being devoted to strong
shape and the last two to strong homology. Chapters are divided in sections,
whose numbering runs throughout the whole book. Subsections are num
bered within sections. The same applies to Theorems, Corollaries, Lemmas,
Remarks and Examples, which use the same counter, e.g., Theorem 3.4 refers
to the fourth of these items in Section 3. Formulae are numbered by Sub
sections. When reference is made to a formula from a different Subsection,
its number is added. E.g., (2.3.4) refers to the fourth formula in Subsection
2.3. External referencing is by author and year of publication, e.g., (Borsuk
1968).
In my work on strong shape and homology I benefited very much from
contacts with a number of colleagues in various parts of the world. This
Preface VII
is especially true of Ju.T. Lisica (Moscow), Z. Miminoshvili (Tbilisi), A.V.
Prasolov (Minsk and Tromso) and T. Watanabe (Yamaguchi) with whom
I wrote joint papers on the subject. I am grateful to my colleagues at the
Universities of Zagreb, Split and Ljubljana who attended many of my seminar
talks based on various parts of the manuscript. I would also like to thank
my colleague Bime Ungar (Zagreb) for having patiently guided me through
delicate points of the 1HEX typesetting.
Zagreb, June 1999
Sibe Mardesic
Table of Contents
Preface....................................................... V
Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
I. COHERENT HOMOTOPY
1. Coherent mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.1 Mappings of inverse systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2 Coherent mappings of inverse systems. . . . . . . . . . . . . . . . . . . .. 13
1.3 Composition of coherent mappings. . . . . . . . . . . . . . . . . . . . . . .. 23
1.4 The coherence operator C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 26
2. Coherent homotopy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 29
2.1 The coherent homotopy category CH(pro-Top) . . . . . . . . . . . .. 29
2.2 Associativity of the composition. . . . . . . . . . . . . . . . . . . . . . . . .. 34
2.3 The identity morphism. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 41
3. Coherent homotopy of sequences ......................... 47
3.1 Coherent homotopy of finite height. . . . . . . . . . . . . . . . . . . . . .. 47
3.2 Coherent homotopy of inverse sequences. . . . . . . . . . . . . . . . . .. 53
4. Coherent homotopy and localization. . . . . . . . . . . . . . . . . . . . .. 61
4.1 An isomorphism theorem in CH(pro-Top) ................. 61
4.2 Cotelescopes (homotopy limits) .......................... 72
4.3 Localizing pro -Top at level homotopy equivalences. . . . . . . . .. 85
5. Coherent homotopy as a Kleisli category ................. 93
5.1 The Kleisli category of a monad. . . . . . . . . . . . . . . . . . . . . . . . .. 93
5.2 CH(pro-Top) is the Kleisli category of a monad. . . . . . . . . . .. 95
X Table of Contents
II. STRONG SHAPE
6. Resolutions ............................................... 103
6.1 Resolutions of spaces and mappings ....................... 103
6.2 Characterization ofresolutions ........................... 107
6.3 Resolutions versus limits ................................ 112
6.4 Existence of polyhedral and ANR -resolutions .............. ll6
6.5 Resolutions of direct products and pairs ................... 123
7. Strong expansions ........................................ 129
7.1 Strong expansions of spaces .............................. 129
7.2 Resolutions are strong expansions ........................ 134
7.3 Invariance under coherent domination ..................... 138
8. Strong shape ............................................. 147
8.1 Coherent expansions of spaces ............................ 147
8.2 The strong shape category ............................... 157
8.3 Strong shape equivalences ............................... 164
9. Strong shape of metric compacta ......................... 181
9.1 The Quigley strong shape category ....................... 181
9.2 Complement theorems .................................. 192
10. Selected results on strong shape .......................... 201
10.1 Normal pairs of spaces .................................. 201
10.2 Normal triads of spaces ................................. 202
10.3 Strong shape using the Vietoris system .................... 204
10.4 The Bauer - Gunther description of strong shape ........... 205
10.5 Strong shape of compacta via multi-valued maps ........... 208
10.6 Strong shape using approximate systems .................. 209
10.7 Strong shape and localization ............................ 210
10.8 Stable strong shape ..................................... 2ll
III. HIGHER DERIVED LIMITS
11. The derived functors of lim ............................... 215
11.1 Inverse systems of modules .............................. 215
ll.2 Projective and injective systems .......................... 221
11.3 lim and its right derived functors ......................... 228
11.4 Axiomatic characterization of the functors limn ............ 240
ll.5 Explicit formulae for limn ............................... 244
11.6 limn for sequences ...................................... 249
Table of Contents XI
12. limn and the extension functors Extn ...................... 253
12.1 The bifunctors Extn .................................... 253
12.2 Expressing limn in terms of Extn ......................... 262
13. The vanishing theorems .................................. 269
13.1 Homological dimension .................................. 269
13.2 Goblot's vanishing theorem .............................. 274
13.3 Systems with non-vanishing limn ......................... 277
14. The cofinality theorem .................................... 285
14.1 Colimits and tensor products ............................ 285
14.2 The cofinality theorem for limn .......................... 291
15. Higher limits on the category pro-Mod .................... 301
15.1 limn as a functor on pro-Mod ........................... 301
15.2 Properties of limn on pro-Mod ........................... 305
IV. HOMOLOGY GROUPS
16. Homology pro-groups ..................................... 319
16.1 Homology pro-groups and Cech homology ................. 319
16.2 Higher limits of homology pro-groups ..................... 321
17. Strong homology groups of systems ....................... 327
17.1 Strong homology of pro-chain complexes ................... 327
17.2 The first Miminoshvili sequence .......................... 336
17.3 The second Miminoshvili sequence ........................ 342
17.4 Isomorphism theorems for strong homology ................ 348
18. Strong homology on CH(pro-Top) .......................... 353
18.1 Chain mappings induced by coherent mappings ............ 353
18.2 Chain mappings induced by congruence classes ............. 359
18.3 Chain mappings induced by homotopy classes .............. 365
18.4 Chain mappings induced by composition .................. 368
18.5 Induced chain mappings and the coherence functor ......... 375
19. Strong homology of spaces ................................ 379
19.1 Strong homology groups of spaces ........................ 379
19.2 Strong excision property ................................. 383
19.3 Strong homology of clusters .............................. 388
19.4 Strong homology and dimension .......................... 394
19.5 Strong homology of polyhedra ............................ 396
19.6 Strong homology of metric compacta ...................... 399