5 - Topological methods in group theory

Summary

Introduction

This article is a revised version of notes on an advanced course given in Liverpool from January to March 1977 in preparation for the symposium. The lectures given by Terry Wall at the symposium were mainly taken from Sections 3 and 4, and much of the material in John Stallings' lectures is in Sections 5 and 6. It seemed worth publishing the whole, as a rather full introduction to the area for those with a background in topology. Originality is not claimed for the results in the earlier sections (though full references have not always been given), but the uniqueness results in Section 7 and most of Section 8 are due to Peter Scott.

BASIC NOTIONS

The link between topology and group theory comes from the fundamental group. I shall make no attempt to present this: almost every introductory topology text does so. Particularly suitable for this course is Massey's book [18]. An equivalent account, from a different viewpoint, is given by Brown [2], Let us recall the basic properties of the fundamental group.

(1) For every topological space X and point x ϵ X we have a group π₁ (X; x). This depends only on the path component of X containing x. A path from x to y induces an isomorphism π₁ (X; x) → π₁ (X; y); a closed path induces an inner automorphism.