IV - Crossed Modules and Π₂ Homotopy Modules

Summary

Introduction

This chapter is partly an introduction to crossed modules, with emphasis on the role that they play in the study of 2-complexes, and an introduction to various identity properties.

Recall from Chapter II that, if X ⊆ Y are topological spaces, then the boundary map ∂ : π₂(Y, X) → π₁(X) is well known to be a π₁(X)-crossed module. We will call this the crossed module associated with the pair (X, Y).

In Section 2, we will study projective and free crossed modules, in particular J. Ratcliffe's characterization of these modules. We also show that, if (Y, X) is a pair of 2-complexes, then the crossed module associated to it is projective. This is definitely not true if the pair is not 2-dimensional. Further, we characterize when the kernel of a projective crossed module is trivial in homological terms.

In Section 3, we study the coproduct of crossed modules. The purpose here is to demonstrate how the second homotopy module of a 2-complex can be built up from subcomplexes.

In Chapter II, it was shown that a 2-complex is aspherical iff it satisfies the identity property. Let X be a 2-complex with fundamental group G. If N is a subgroup of G, let X_N denote the covering of X corresponding to N. In

3.3, we study a generalized iV-identity property and show how it is equivalent to the vanishing of the Hurewicz map h : π₂(X) → H₂(X_N), the so-called N-Cockcroft property.