Information about:

TOPOLOGICAL METHODS IN GROUP THEORY

by Ross Geoghegan

This book was published by Springer, New York in December 2007.

It is Vol. 243 of their series Graduate Texts in Mathematics

From the Introduction:

"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

PART I: ALGEBRAIC TOPOLOGY FOR GROUP THEORY

CHAPTER 1: CW COMPLEXES AND HOMOTOPY:

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

PART II: FINITENESS PROPERTIES OF GROUPS

CHAPTER 6: THE BOREL CONSTRUCTION AND BASS-SERRE THEORY:

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

PART III: LOCALLY FINITE ALGEBRAIC TOPOLOGY FOR GROUP THEORY

CHAPTER 10: LOCALLY FINITE CW COMPLEXES AND PROPER HOMOTOPY:

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

PART IV: TOPICS IN THE COHOMOLOGY OF INFINITE GROUPS

CHAPTER 13: HOMOLOGICAL GROUP THEORY AND ENDS OF COVERING SPACES:

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

PART V: HOMOTOPICAL GROUP THEORY

CHAPTER 16: THE FUNDAMENTAL GROUP AT INFINITY:

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

PART VI: THREE ESSAYS

CHAPTER 18: ESSAYS

18.1* l_*2 Poincare duality

18.2 Quasi-isometry invariants

18.3 The Bieri-Neumann-Strebel invariant