On the Second Relative Homotopy Group of an Adjunction Space: An Exposition of a Theorem of J. H. C. Whitehead

In his 1949 paper [6], a different exposition of part of the proof was given, and also the result was codified finally in saying that the group A is the free crossed nx{XQ, x0)-module on the 2-cells. One difficulty in obtaining this theorem by standard methods of algebraic topology is that it is a non-abelian result. A proof has recently been given by Ratcliffe, using a homological characterisation of free crossed modules [3]. The theorem also is a special case of the generalisation to dimension 2 of the Seifert-van Kampen Theorem [1], where free crossed modules arise as very special cases of pushouts of crossed modules. Indeed one aim of the program completed in [1] was to forpulate a generalisation of Whitehead's theorem, and to prove it by verification of a universal property. A list of papers which apply the theorem is also given in [1]. In spite of these other proofs and generalisations, Whitehead's proof still has interest. It has a curious structure, a modern-looking use of transversality, and a clever interplay of the relations for a knot group and the rules of a crossed module. Also the /ideas have some relevance to difficult problems in combinatorial group theory involving identities between relations. (I hope to give elsewhere an exposition of some ideas of Peter Stefan in this area.t) But the proof is difficult to read, for reasons which include its originality of conception, and the change in notation and formulation over the years 1941-49. I hope therefore that it will prove useful to present a straightforward account of Whitehead's proof; my own contribution is simply that of presentation in a modern, and uniform notation and terminology. Thanks are due to the late Peter Stefan for discussions on some of this material, and to Johannes Huebschmann for helpful comments on an earlier draft.