Topological Methods in Group Theory

Ross Geoghegan (auth.)0387746110, 9780387746111

Topological Methods in Group Theory is about the interplay between algebraic topology and the theory of infinite discrete groups. The author has kept three kinds of readers in mind: 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, combinatorial and homological group theory; group theorists who would like to know more about the topological side of their subject but who have been too long away from topology; and manifold topologists, both high- and low-dimensional, since the book contains much basic material on proper homotopy and locally finite homology not easily found elsewhere.

The book focuses on two main themes:

1. Topological Finiteness Properties of groups (generalizing the classical notions of “finitely generated” and “finitely presented”);

2. Asymptotic Aspects of Infinite Groups (generalizing the classical notion of “the number of ends of a group”).

Illustrative examples treated in some detail include: Bass-Serre theory, Coxeter groups, Thompson groups, Whitehead’s contractible 3-manifold, Davis’s exotic contractible manifolds in dimensions greater than three, the Bestvina-Brady Theorem, and the Bieri-Neumann-Strebel invariant. The book also includes a highly geometrical treatment of Poincaré duality (via cells and dual cells) to bring out the topological meaning of Poincaré duality groups.

To keep the length reasonable and the focus clear, it is assumed that the reader knows or can easily learn the necessary algebra (which is clearly summarized) but wants to see the topology done in detail. Apart from the introductory material, most of the mathematics presented here has not appeared in book form before.

Table of contents :
Front Matter….Pages I-XIV
Front Matter….Pages 1-1
CW Complexes and Homotopy….Pages 3-34
Cellular Homology….Pages 35-72
Fundamental Group and Tietze Transformation….Pages 73-100
Some Techniques in Homotopy Theory….Pages 101-123
Elementary Geometric Topology….Pages 125-140
Front Matter….Pages 141-141
The Borel Construction and Bass-Serre Theory….Pages 143-159
Topological Finiteness Properties and Dimension of Groups….Pages 161-179
Homological Finiteness Properties of Groups….Pages 181-195
Finiteness Properties of Some Important Groups….Pages 197-216
Front Matter….Pages 217-217
Locally Finite CW Complexes and Proper Homotopy….Pages 219-228
Locally Finite Homology….Pages 229-257
Cohomology of CW Complexes….Pages 259-282
Front Matter….Pages 283-283
Cohomology of Groups and Ends of Covering Spaces….Pages 285-331
Filtered Ends of Pairs of Groups….Pages 333-351
Poincaré Duality in Manifolds and Groups….Pages 353-365
Front Matter….Pages 367-367
The Fundamental Group At Infinity….Pages 369-409
Higher homotopy theory of groups….Pages 411-429
Front Matter….Pages 431-431
Three Essays….Pages 433-452
Back Matter….Pages 453-477