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Strong Shape and Homology : Springer Monographs in Mathematics - S. Mardesic

Strong Shape and Homology

Springer Monographs in Mathematics

Hardcover Published: 22nd November 1999
ISBN: 9783540661986
Number Of Pages: 489

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Shape theory is an extension of homotopy theory from the realm of CW-complexes to arbitrary spaces. Besides applications in topology, it has interesting applications in various other areas of mathematics, especially in dynamical systems and C*-algebras. Strong shape is a refinement of ordinary shape with distinct advantages over the latter. Strong homology generalizes Steenrod homology and is an invariant of strong shape. The book gives a detailed account based on approximation of spaces by polyhedra (ANR's) using the technique of inverse systems. It is intended for researchers and graduate students. Special care is devoted to motivation and bibliographic notes.

Prefacep. V
Introductionp. 1
Coherent Homotopy
Coherent mappingsp. 9
Mappings of inverse systemsp. 9
Coherent mappings of inverse systemsp. 13
Composition of coherent mappingsp. 23
The coherence operator Cp. 26
Coherent homotopyp. 29
The coherent homotopy category CH(pro-Top)p. 29
Associativity of the compositionp. 34
The identity morphismp. 41
Coherent homotopy of sequencesp. 47
Coherent homotopy of finite heightp. 47
Coherent homotopy of inverse sequencesp. 53
Coherent homotopy and localizationp. 61
An isomorphism theorem in CH(pro-Top)p. 61
Cotelescopes (homotopy limits)p. 72
Localizing pro-Top at level homotopy equivalencesp. 85
Coherent homotopy as a Kleisli categoryp. 93
The Kleisli category of a monadp. 93
CH(pro-Top) is the Kleisli category of a monadp. 95
Strong Shape
Resolutionsp. 103
Resolutions of spaces and mappingsp. 103
Characterization of resolutionsp. 107
Resolutions versus limitsp. 112
Existence of polyhedral and ANR - resolutionsp. 116
Resolutions of direct products and pairsp. 123
Strong expansionsp. 129
Strong expansions of spacesp. 129
Resolutions are strong expansionsp. 134
Invariance under coherent dominationp. 138
Strong shapep. 147
Coherent expansions of spacesp. 147
The strong shape categoryp. 157
Strong shape equivalencesp. 164
Strong shape of metric compactap. 181
The Quigley strong shape categoryp. 181
Complement theoremsp. 192
Selected results on strong shapep. 201
Normal pairs of spacesp. 201
Normal triads of spacesp. 202
Strong shape using the Vietoris systemp. 204
The Bauer - Günther description of strong shapep. 205
Strong shape of compacta via multi-valued mapsp. 208
Strong shape using approximate systemsp. 209
Strong shape and localizationp. 210
Stable strong shapep. 211
Higher Derived Limits
The derived functors of limp. 215
Inverse systems of modulesp. 215
Projective and injective systemsp. 221
lim and its right derived functorsp. 228
Axiomatic characterization of the functors limnp. 240
Explicit formulae for limnp. 244
limn for sequencesp. 249
limn and the extension functors Extnp. 253
The bifunctors Extnp. 253
Expressing limn in terms of Extnp. 262
The vanishing theoremsp. 269
Homological dimensionp. 269
Goblot's vanishing theoremp. 274
Systems with non-vanishing limnp. 277
The cofinality theoremp. 285
Colimits and tensor productsp. 285
The cofinality theorem for limnp. 291
Higher limits on the category pro-Modp. 301
limn as a functor on pro - Modp. 301
Properties of limn on pro - Modp. 305
Homology Groups
Homology pro-groupsp. 319
Homology pro-groups and Čech homologyp. 319
Higher limits of homology pro-groupsp. 321
Strong homology groups of systemsp. 327
Strong homology of pro-chain complexesp. 327
The first Miminoshvili sequencep. 336
The second Miminoshvili sequencep. 342
Isomorphism theorems for strong homologyp. 348
Strong homology on CH(pro-Top)p. 353
Chain mappings induced by coherent mappingsp. 353
Chain mappings induced by congruence classesp. 359
Chain mappings induced by homotopy classesp. 365
Chain mappings induced by compositionp. 368
Induced chain mappings and the coherence functorp. 375
Strong homology of spacesp. 379
Strong homology groups of spacesp. 379
Strong excision propertyp. 383
Strong homology of clustersp. 388
Strong homology and dimensionp. 394
Strong homology of polyhedrap. 396
Strong homology of metric compactap. 399
Spectral sequences. Abelian groupsp. 405
The spectral sequence of a filtered complexp. 405
The spectral sequences of a bicomplexp. 413
The Roos spectral sequencep. 416
Pure extension functors Pextnp. 422
Some theorems on abelian groupsp. 427
Strong homology of compact spacesp. 439
Universal coefficients for compact polyhedrap. 439
Homology of compact spacesp. 443
Universal coefficients for compact spacesp. 446
A filtration of the strong homology groupp. 448
Strong homology with compact supportsp. 453
Generalized strong homologyp. 459
Referencesp. 465
List of Special Symbolsp. 479
Author Indexp. 483
Subject Indexp. 485
Table of Contents provided by Publisher. All Rights Reserved.

ISBN: 9783540661986
ISBN-10: 3540661980
Series: Springer Monographs in Mathematics
Audience: General
Format: Hardcover
Language: English
Number Of Pages: 489
Published: 22nd November 1999
Publisher: Springer-Verlag Berlin and Heidelberg Gmbh & Co. Kg
Country of Publication: DE
Dimensions (cm): 23.39 x 15.6  x 2.87
Weight (kg): 0.89

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