V - Calculating Generators of Π₂

Summary

This article discusses combinatorial geometric techniques that determine explicit generators for the second homotopy module of a 2-complex in terms of its cell structure. Applications of the techniques are also presented. The discussion focuses on the theory of pictures. Pictures have been used for many purposes; we shall be concerned only with their application to π₂ calculations.

There are three sections, each of which is divided into several subsections. The first section provides an overview of the theory of pictures from a homotopytheoretic perspective. The second section deals with generalities related to the generation of π₂. Some proofs are included in these first two sections, and there are several exercises. The third section is devoted to a summary description without proofs of various calculations and applications that have been obtained in the study of π₂.

This paper may be taken as a companion and sequel to [Pr91], where pictures are treated within the purely combinatorial context of identity sequences.

The Theory of Pictures

As is customary, 2-complexes will be specified by means of group presentations. Let P = 〈x | r〉 be a presentation for a group G. Thus, G = F/N where F is the free group with basis x and N is the normal closure in F of the set r of (not necessarily reduced) words in x ∪ x^-1. We allow the occurrence of trivial and repeated relators r ∈ r, so r should be treated as an indexed set of words.