"This is a book about the interplay between algebraic topology and the theory of infinite discrete groups. I have written it for three kinds of readers. First, it is for graduate students who have had an introductory course in algebraic topology and who need a bridge from common knowledge to the current research literature in geometric and homological group theory. Secondly, I am writing for group theorists who would like to know more about the topological side of their subject but who have been too long away from topology. Thirdly, I hope the book will be useful to manifold topologists, both high- and low-dimensional, as a reference source for basic material on proper homotopy and homology..."
Table of Contents
1.1 Review of general topology
1.2 CW complexes
1.3 Homotopy
1.4 Maps between CW complexes
1.5 Neighborhoods and complements
CHAPTER 2: CELLULAR HOMOLOGY:
2.1 Review of chain complexes
2.2 Review of singular homology
2.3 Cellular homology: the abstract theory
2.4 The degree of a map from a sphere to itself
2.5 Orientation and incidence number
2.6 The geometric cellular chain complex
2.7 Some properties of cellular homology
2.8 Further properties of cellular homology
2.9 Reduced homology
CHAPTER 3: FUNDAMENTAL GROUP AND TIETZE TRANSFORMATIONS:
3.1 Combinatorial fundamental group, Tietze transformations, Van Kampen theorem
3.2 Combinatorial description of covering spaces
3.3 Review of topologically defined fundamental group
3.4 Equivalence of the two definitions of the fundamental group of a CW complex
CHAPTER 4: SOME TECHNIQUES IN HOMOTOPY THEORY:
4.1 Altering a CW complex within its homotopy type
4.2 Cell trading
4.3 Domination, mapping tori and mapping telescopes
4.4 Review of homotopy groups
4.5 Geometric proof of the Hurewicz Theorem
CHAPTER 5: ELEMENTARY GEOMETRIC TOPOLOGY:
5.1 Review of topological manifolds
5.2 Simplicial complexes and combinatorial manifolds
5.3 Regular CW complexes
5.4 Incidence numbers in simplicial complexes
6.1 The Borel construction, stacks and rebuilding
6.2 Decomposing groups which act on trees (Bass-Serre theory)
CHAPTER 7: TOPOLOGICAL FINITENESS PROPERTIES AND DIMENSION OF GROUPS:
7.1 K(G, 1)-complexes
7.2 Finiteness properties and dimension of groups
7.3 Recognizing the finiteness properties and dimension of a group
7.4 Brown's Criterion for finiteness
CHAPTER 8: HOMOLOGICAL FINITENESS PROPERTIES OF GROUPS:
8.1 Homology of groups
8.2 Homological finiteness properties
8.3 Synthetic Morse theory and the Bestvina-Brady Theorem
CHAPTER 9: FINITENESS PROPERTIES OF SOME IMPORTANT GROUPS:
9.1 Finiteness properties of Coxeter groups
9.2 Thompson's Group F and homotopy idempotents
9.3 Finiteness properties of Thompson's Group
9.4 Thompson's simple group T
9.5 The outer automorphism group of a free group
10.1 Proper maps and proper homotopy theory
10.2 CW-proper maps
CHAPTER 11: LOCALLY FINITE HOMOLOGY:
11.1 Infinite cellular homology
11.2 Review of inverse and direct systems
11.3 The derived limit
11.4 Homology of ends
CHAPTER 12 COHOMOLOGY OF CW COMPLEXES
12.1 Cohomology based on infinite and finite (co)chains
12.2 Cohomology of ends
12.3 A special case: Orientation of pseudomanifolds and manifolds
12.4 Review of more homological algebra
12.5 Comparison of the various homology and cohomology theories
12.6 Homology and cohomology of products
13.1 Cohomology of groups
13.2 Homology and cohomology of highly connected covering spaces
13.3 Topological interpretation of H*(G, RG)
13.4 Ends of spaces
13.5 Ends of groups and the structure of H^1(G, RG)
13.6 Proof of Stallings' Theorem
13.7 The structure of H^2(G, RG)
13.8 Asphericalization and an example of H^3(G,ZG)
13.9 Coxeter group examples of H^n(G,ZG)
13.10 The case H*(G, RG) = 0
13.11 An example of H*(G, RG) = 0
CHAPTER 14: FILTERED ENDS OF PAIRS OF GROUPS:
14.1 Filtered homotopy theory
14.2 Filtered chains
14.3 Filtered ends of spaces
14.4 Filtered cohomology of pairs of groups
14.5 Filtered ends of pairs of groups
CHAPTER 15: POINCARÉ DUALITY IN MANIFOLDS AND GROUPS:
15.1 CW manifolds and dual cells
15.2 Poincaré and Lefschetz duality
15.3 Poincaré duality groups and duality groups
16.1 Connectedness at infinity
16.2 Analogs of the fundamental group
16.3 Necessary conditions for a free Z-action
16.4 Example: Whitehead's contractible 3-manifold
16.5 Group invariants: simple connectivity, stability and semistability of groups
16.6 Example: Coxeter groups and Davis manifolds
16.7 Free topological groups
16.8 Products and group extensions
16.9 Sample theorems on simple connectivity and semistability
CHAPTER 17: HIGHER HOMOTOPY THEORY OF GROUPS:
17.1 Higher proper homotopy
17.2 Higher connectivity invariants of groups
17.3 Higher invariants of group extensions
17.4 The space of proper rays
17.5 Z-set compactifications
17.6 Compactifiability at infinity as a group invariant
17.7 Strong Shape theory
18.1 l_2 Poincaré duality
18.2 Quasi-isometry invariants
18.3 The Bieri-Neumann-Strebel invariant