ISBN: 3-540-66198-0
TITLE: Strong Shape and Homology
AUTHOR: Mardesic, Sibe
TOC:

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
II. STRONG SHAPE
6. Resolutions 103
6.1 Resolutions of spaces and mappings 103
6.2 Characterization of resolutions 107
6.3 Resolutions versus limits 112
6.4 Existence of polyhedral and ANR-resolutions 116
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 BauerGnther 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 211
III. HIGHER DERIVED LIMITS
11. The derived functors of lim 215
11.1 Inverse systems of modules 215
11.2 Projective and injective systems 221
11.3 lim and its right derived functors 228
11.4 Axiomatic characterization of the functors limn 240
11.5 Explicit formulae for limn 244
11.6 lim^n for sequences 249
12. lim^n and the extension functors Ext^n 253
12.1 The bifunctors Ext^n 253
12.2 Expressing limn in terms of Ext^n 262
13. The vanishing theorems 269
13.1 Homological dimension 269
13.2 Goblot's vanishing theorem 274
13.3 Systems with non-vanishing lim^n 277
14. The cofinality theorem 285
14.1 Colimits and tensor products 285
14.2 The cofinality theorem for lim^n 291
15. Higher limits on the category pro-Mod 301
15.1 lim^n 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 polyhedra396
19.6 Strong homology of metric compacta 399
20. Spectral sequences. Abelian groups 405
20.1 The spectral sequence of a filtered complex 405
20.2 The spectral sequences of a bicomplex. 413
20.3 The Roos spectral sequence 416
20.4 Pure extension functors Pext^n 422
20.5 Some theorems on abelian groups 427
21. Strong homology of compact spaces 439
21.1 Universal coefficients for compact polyhedra 439
21.2 Homology of compact spaces 443
21.3 Universal coefficients for compact spaces 446
21.4 A filtration of the strong homology group 448
21.5 Strong homology with compact supports 453
22. Generalized strong homology 459
References 465
List of Special Symbols 479
Author Index 483
Subject Index 485
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