II - Algebraic Topology for Two Dimensional Complexes

Summary

This chapter presents homotopy classifications of two dimensional CW complexes and maps between them. Cases of these abstract classifications are detailed in Chapter III. Simplicial techniques are invoked in Section 1 to analyze maps of balls and spheres into 2-complexes. This analysis is applied in Section 2 to study the long exact sequence of homotopy groups for a 2-complex and to derive J. H. C. Whitehead's equivalence of the homotopy theory of 2-complexes with the purely algebraic theory of free crossed modules. Cellular chain complexes of universal coverings of 2-complexes are developed in Section 3. This equivariant world provides the foundation for the treatment in Section 4 of an abelianized version of Whitehead's equivalence, namely, the theory of algebraic 2-type of 2-complexes due to S. Mac Lane and Whitehead.

Techniques in Homotopy

In this section, we use simplicial approximations of maps between simplicial complexes to construct combinatorial approximations of maps between CW complexes, at least in dimensions one and two.

Simplicial Techniques

We view real m-space ℝ^m as a real vector space and we assume that the reader is familiar with the concepts of finite simplicial complexes K in ℝ^m and simplicial maps φ: K → M between such complexes. We don't distinguish notationally between a simplicial complex and the associated topological subspace of ℝ^m, and let context convey the object under consideration.