Free simplicial groups and the second relative homotopy group of an adjunction space

Abstract

(If (YX) is a pair of spaces, then 7rnz(YX) is a ;rrtX-crossed module.) Suppose that X is a connected CW-complex, and that Y is obtained from X by attaching 2-cells. Whitehead [9] showed that in this case, x2(xX) is a free x,X-crossed module, i.e., if {c,} are the elements of ;71#‘,X) corresponding to the 2-cells of Y-X, and if (g,} are elements of another n,X-crossed module G with ag,=ac,, then there is a unique homomorphism of crossed modules h : 7t2(Y,X) + G for which h(c,) =g,. Whitehead’s proof of this theorem is rather difficult and geometric (see [l] for a more modern exposition of Whitehead’s proof). Brown and Higgins [2] have shown that this theorem follows from their 2-dimensional generalization of the van Kampen theorem, in the context of ‘double groupoids’. Ratcliffe [7], using his study of free and projective crossed modules, was able to give a more algebraic proof. The purpose of the present paper is to give a completely algebraic proof of this theorem. We will obtain the theorem through a study offree simplicial groups, which are models for loop spaces (and are therefore useful for computing (absolute) homotopy groups).