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.
Inhalt
Preface
Introduction
Chapter I. Foundations of Shape Theory
Pro-Categories
1. Inverse Systems
2. Systems with Cofinite Index Sets
3. Level Morphisms of Systems
4. Generalized Inverse Systems
Abstract Shape
1. Inverse System Expansions
2. Dense Subcategories
3. The Shape Category
4. Shape Morphisms as Natural Transformations
Absolute Neighborhood Retracts
1. ANR's for Metric Spaces
2. Homotopy Properties of ANR's
3. Pairs of ANR's
Topological Shape
1. Shape for the Homotopy Category of Spaces
2. Some Particular Expansions
3. Shape of Pairs. Pointed Shape
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
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
Shape Dimension
1. Shape Dimension of Spaces
2. Shape Dimension of Pointed Spaces
Pro-Groups
1. Monomorphisms and Epimorphisms of Pro-Groups
2. Isomorphisms of Pro-Groups
3. Exact Sequences of Pro-Groups
Homotopy and Homology Pro-Groups
1. Homology Pro-Groups and Cech Homology Groups
2. Cech Cohomology Groups
3. Homotopy Pro-Groups and Shape Groups
Hurewicz Theorem in Shape Theory
1. Absolute Hurewicz Theorem
2. Relative Hurewicz Theorem
Whitehead Theorem in Shape Theory
1. n-Equivalences in Pro-Homotopy
2. Whitehead Theorem
3. Homological Versions of the Whitehead Theorem
Movability of Pro-Groups
1. Movability and Uniform Movability in Categories
2. Mittag-Leffler Property and Derived Limits
Movability of Spaces
1. Homotopy Groups of Inverse Limits
2. Movable Spaces
3. Movability of Metric Compacta and Shape Groups
n-Movability of Spaces
1. n-movable Spaces
2. Changing the Base Point in a Continuum
3. Pointed and Unpointed Movability
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
Basic Topological Constructions and Shape
1. Products
2. Sums
3. Quotients
4. Suspensions
5. Space of Components
6. Hyperspaces
Shape Dimension of Metric Compacta
Shape of Compact Connected Abelian Groups
Shape of the Stone-Cech Compactification
LCn-Divisors and Continua with LCn Shape
1. ANR-Divisors and LCn-Divisors
2. Continua with LCn-Shape
Complement Theorems of Shape Theory
1. Infinite-Dimensional Case
2. Finite-Dimensional Case
Embeddings Up to Shape
Shape Fibrations
Strong Shape
. Cell-Like Maps
Appendix 1. Polyhedra
Topology of Simplicial Complexes
1. Simplicial CW-Complexes
2. Attaching Simplicial Complexes
3. Metric Simplicial Complexes
The Homotopy Type of Polyhedra
1. Weak Equivalences with Polyhedral Domains
2. Spaces Homotopy Dominated by Polyhedra
3. Homotopy Domination by Polyhedral Pairs
The Cech Expansion
1. Normal Coverings
2. The Cech System
Appendix 2. Borsuk's Approach to Shape
Shape Category of Metric Compacta
Shape Category of Compact Metric Pairs
Bibliography
List of Special Symbols
Subject Index