Topological Methods in Group Theory

  • Ross Geoghegan
Part of the Graduate Texts in Mathematics book series (GTM, volume 243)

Table of contents

  1. Front Matter
    Pages I-XIV
  2. Algebraic Topology for Group Theory

  3. Finiteness Properties of Groups

  4. Locally Finite Algebraic Topology for Group Theory

    1. Front Matter
      Pages 217-217
    2. Pages 229-257
  5. Topics in the Cohomology of Infinite Groups

About this book

Introduction

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.

Keywords

Algebraic topology CW complex Cohomology Fundamental group Group theory Homotopy algebra homology homotopy theory set

Authors and affiliations

  • Ross Geoghegan
    • 1
  1. 1.Department of Mathematical SciencesBinghamton University (SUNY)BinghamtonUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-0-387-74614-2
  • Copyright Information Springer-Verlag New York 2008
  • Publisher Name Springer, New York, NY
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-0-387-74611-1
  • Online ISBN 978-0-387-74614-2
  • Series Print ISSN 0072-5285
  • About this book