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Shape Theory

The Inverse System Approach

1st Edition - January 1, 1982

Authors: S. Mardešic, J. Segal

eBook ISBN:

9 7 8 - 0 - 0 8 - 0 9 6 0 1 4 - 2

North-Holland Mathematical Library, Volume 26: Shape Theory: The Inverse System Approach presents a systematic introduction to shape theory by providing background materials,… Read more

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North-Holland Mathematical Library, Volume 26: Shape Theory: The Inverse System Approach presents a systematic introduction to shape theory by providing background materials, motivation, and examples, including shape theory and invariants, pro-groups, shape fibrations, and metric compacta. The publication first ponders on the foundations of shape theory and shape invariants. Discussions focus on the stability and movability of spaces, homotopy and homology pro-groups, shape dimension, inverse limits and shape of compacta, topological shape, and absolute neighborhood retracts. The text then takes a look at a survey of selected topics, including basic topological constructions and shape, shape dimension of metric compacta, complement theorems of shape theory, shape fibrations, and cell-like maps. The text ponders on polyhedra and Borsuk's approach to shape. Topics include shape category of metric compacta and metric pairs, homotopy type of polyhedra, and topology of simplicial complexes. The publication is a valuable source of data for researchers interested in the inverse system approach.

Preface

Introduction

Chapter I. Foundations of Shape Theory

§1. Pro-Categories

1. Inverse Systems

2. Systems with Cofinite Index Sets

3. Level Morphisms of Systems

4. Generalized Inverse Systems

§2. Abstract Shape

1. Inverse System Expansions

2. Dense Subcategories

3. The Shape Category

4. Shape Morphisms as Natural Transformations

§3. Absolute Neighborhood Retracts

1. ANR's for Metric Spaces

2. Homotopy Properties of ANR's

3. Pairs of ANR's

§4. Topological Shape

1. Shape for the Homotopy Category of Spaces

2. Some Particular Expansions

3. Shape of Pairs. Pointed Shape

§5. Inverse Limits and Shape of Compacta

1. Inverse Limits in Arbitrary Categories

2. Inverse Limits of Compact Hausdorff Spaces

3. Shape of Compact Hausdorff Spaces

4. Compact Pairs

§6. Resolutions of Spaces and Shape

1. Resolutions of Spaces

2. Characterization of Resolutions

3. Resolutions and Inverse Limits

4. Existence of Polyhedral Resolutions

5. Resolutions of Pairs

Chapter II. Shape Invariants

§1. Shape Dimension

1. Shape Dimension of Spaces

2. Shape Dimension of Pointed Spaces

§2. Pro-Groups

1. Monomorphisms and Epimorphisms of Pro-Groups

2. Isomorphisms of Pro-Groups

3. Exact Sequences of Pro-Groups

§3. Homotopy and Homology Pro-Groups

1. Homology Pro-Groups and Čech Homology Groups

2. Čech Cohomology Groups

3. Homotopy Pro-Groups and Shape Groups

§4. Hurewicz Theorem in Shape Theory

1. Absolute Hurewicz Theorem

2. Relative Hurewicz Theorem

§5. Whitehead Theorem in Shape Theory

1. n-Equivalences in Pro-Homotopy

2. Whitehead Theorem

3. Homological Versions of the Whitehead Theorem

§6. Movability of Pro-Groups

1. Movability and Uniform Movability in Categories

2. Mittag-Leffler Property and Derived Limits

§7. Movability of Spaces

1. Homotopy Groups of Inverse Limits

2. Movable Spaces

3. Movability of Metric Compacta and Shape Groups

§8. n-Movability of Spaces

1. n-movable Spaces

2. Changing the Base Point in a Continuum

3. Pointed and Unpointed Movability

§9. Stability of Spaces

1. Stability and Pointed Stability

2. Stability and Shape Domination

3. Strong Movability

4. Algebraic Characterization of Stability

5. Shape Retracts

Chapter III. A Survey of Selected Topics

§1. Basic Topological Constructions and Shape

1. Products

2. Sums

3. Quotients

4. Suspensions

5. Space of Components

6. Hyperspaces

§2. Shape Dimension of Metric Compacta

§3. Shape of Compact Connected Abelian Groups

§4. Shape of the Stone-Čech Compactification

§5. LCn-Divisors and Continua with LCn Shape

1. ANR-Divisors and LCn-Divisors

2. Continua with LCn-Shape

§6. Complement Theorems of Shape Theory

1. Infinite-Dimensional Case

2. Finite-Dimensional Case

§7. Embeddings Up to Shape

§8. Shape Fibrations

§9. Strong Shape

§10. Cell-Like Maps

Appendix 1. Polyhedra

§1. Topology of Simplicial Complexes

1. Simplicial CW-Complexes

2. Attaching Simplicial Complexes

3. Metric Simplicial Complexes

§2. The Homotopy Type of Polyhedra

1. Weak Equivalences with Polyhedral Domains

2. Spaces Homotopy Dominated by Polyhedra

3. Homotopy Domination by Polyhedral Pairs

§3. The Čech Expansion

1. Normal Coverings

2. The Čech System

Appendix 2. Borsuk's Approach to Shape

§1. Shape Category of Metric Compacta

§2. Shape Category of Compact Metric Pairs

Bibliography

List of Special Symbols

Subject Index