Topological Methods in Group Theory

by Ross Geoghegan

Overview

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 Poincare duality (via cells and dual cells) to bring out the topological meaning of Poincare 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 themathematics presented here has not appeared in book form before.

About the Author:
Ross Geoghegan is Professor of Mathematics at the State University of New York at Binghamton (Binghamton University)

Product Details

ISBN-13:	9781441925640
Publisher:	Springer New York
Publication date:	11/29/2010
Series:	Graduate Texts in Mathematics Series , #243
Edition description:	Softcover reprint of hardcover 1st ed. 2008
Pages:	489
Product dimensions:	6.10(w) x 9.25(h) x 0.99(d)

Table of Contents

Algebraic Topology for Group Theory     1
CW Complexes and Homotopy     3
Review of general topology     3
CW complexes     10
Homotopy     23
Maps between CW complexes     28
Neighborhoods and complements     31
Cellular Homology     35
Review of chain complexes     35
Review of singular homology     37
Cellular homology: the abstract theory     40
The degree of a map from a sphere to itself     43
Orientation and incidence number     52
The geometric cellular chain complex     60
Some properties of cellular homology     62
Further properties of cellular homology     65
Reduced homology     70
Fundamental Group and Tietze Transformations     73
Fundamental group, Tietze transformations, Van Kampen Theorem     73
Combinatorial description of covering spaces     84
Review of the topologically defined fundamental group     94
Equivalence of the two definitions     96
Some Techniques in Homotopy Theory     101
Altering a CW complex within its homotopy type     101
Cell trading     110
Domination, mappingtori, and mapping telescopes     112
Review of homotopy groups     116
Geometric proof of the Hurewicz Theorem     119
Elementary Geometric Topology     125
Review of topological manifolds     125
Simplicial complexes and combinatorial manifolds     129
Regular CW complexes     135
Incidence numbers in simplicial complexes     139
Finiteness Properties of Groups     141
The Borel Construction and Bass-Serre Theory     143
The Borel construction, stacks, and rebuilding     143
Decomposing groups which act on trees (Bass-Serre Theory)     148
Topological Finiteness Properties and Dimension of Groups     161
K(G, 1) complexes     161
Finiteness properties and dimensions of groups     169
Recognizing the finiteness properties and dimension of a group     176
Brown's Criterion for finiteness     177
Homological Finiteness Properties of Groups     181
Homology of groups     181
Homological finiteness properties     185
Synthetic Morse theory and the Bestvina-Brady Theorem     187
Finiteness Properties of Some Important Groups     197
Finiteness properties of Coxeter groups      197
Thompson's group F and homotopy idempotents     201
Finiteness properties of Thompson's Group F     206
Thompson's simple group T     212
The outer automorphism group of a free group     214
Locally Finite Algebraic Topology for Group Theory     217
Locally Finite CW Complexes and Proper Homotopy     219
Proper maps and proper homotopy theory     219
CW-proper maps     227
Locally Finite Homology     229
Infinite cellular homology     229
Review of inverse and direct systems     235
The derived limit     241
Homology of ends     248
Cohomology of CW Complexes     259
Cohomology based on infinite and finite (co)chains     259
Cohomology of ends     265
A special case: Orientation of pseudomanifolds and manifolds     267
Review of more homological algebra     273
Comparison of the various homology and cohomology theories     277
Homology and cohomology of products     281
Topics in the Cohomology of Infinite Groups     283
Cohomology of Groups and Ends Of Covering Spaces     285
Cohomology of groups     285
Homology and cohomology of highly connected covering spaces     286
Topological interpretation of H*(G, RG)     293
Ends of spaces     295
Ends of groups and the structure of H[superscript 1](G, RG)     300
Proof of Stallings' Theorem     308
The structure of H[superscript 2](G, RG)     314
Asphericalization and an example of H[superscript 3](G, ZG)     321
Coxeter group examples of H[superscript n](G, ZG)     324
The case H*(G, RG) = 0     330
An example of H*(G, RG) = 0     331
Filtered Ends of Pairs of Groups     333
Filtered homotopy theory     333
Filtered chains     333
Filtered ends of spaces     341
Filtered cohomology of pairs of groups     344
Filtered ends of pairs of groups     346
Poincare Duality in Manifolds and Groups     353
CW manifolds and dual cells     353
Poincare and Lefschetz Duality     356
Poincare Duality groups and duality groups     362
Homotopical Group Theory     367
The Fundamental Group At Infinity     369
Connectedness at infinity     369
Analogs of the fundamental group     379
Necessary conditions for a free Z-action      383
Example: Whitehead's contractible 3-manifold     387
Group invariants: simple connectivity, stability, and semistability     393
Example: Coxeter groups and Davis manifolds     396
Free topological groups     397
Products and group extensions     399
Sample theorems on simple connectivity and semistability     401
Higher homotopy theory of groups     411
Higher proper homotopy     411
Higher connectivity invariants of groups     413
Higher invariants of group extensions     415
The space of proper rays     418
Z-set compactifications     421
Compactifiability at infinity as a group invariant     425
Strong shape theory     426
Three Essays     431
Three Essays     433
l[subscript 2]-Poincare duality     433
Quasi-isometry invariants     435
The Bieri-Neumann-Strebel invariant     441
References     453
Index     463

Editorial Reviews

From the reviews:

"The author of this book has done a great service to the geometric group theory community by writing a very useful and well-written book on many topics in geometric group theory that every neophyte and researcher in the field should know. … This book is suitable as a textbook for a graduate course, with many good examples and exercises. The reviewer highly recommends this book as a basic reference book for topological methods in group theory." (John G. Ratcliffe, Mathematical Reviews, Issue 2008 j)

"This is an interesting book on the interplay between algebraic topology and the theory of infinite discrete groups written for graduate students and group theorists who need to learn more in geometric and homological group theory. … It is a beautiful text in algebraic topology, with modern topics and which points the reader towards new research directions." (Corina Mohorianu, Zentralblatt MATH, Vol. 1142, 2008)

“This book is an invaluable resource for anyone wanting a deep understanding of topics related to the ends of groups. … there is a good deal of material in this book that does not appear anywhere else in the literature. … Geoghegan’s book provides a well-presented, concrete development of geometric group theory focused on a topological approach.” (John Meier, Bulletin of the American Mathematical Society, July, 2012)

From the Publisher