Continuous mappings into nonsimple spaces

Abstract

equivariant with respect to these operators into equivariant homotopy classes. The set of classes so obtained, for all h, bear a definite relation to the homotopy classes of the base spaces, so that the problem is reduced to the classification of equivariant maps into a space simple in all dimensions. The classification scheme and theorem of Eilenberg [2 ] are shown to go over completely for the equivariant maps considered (even with proofs formally unaltered) whenever X is a polytope. It follows from this that Eilenberg's technique for simple Y has a purely mechanical generalization for maps into nonsimple spaces, the only additional requirement being that the set of homomorphisms 2ri(X)--ri( Y) induced by continuous maps of X into Y be known. In similar fashion, the generalization by Olum [3] of the Eilenberg scheme for maps of arbitrary X, when given for simple Y formally generalizes to apply to the nonsimple case. The paper is divided in three parts. Part I deals with covering spaces, and with the existence, homotopy, and extension properties of equivariant maps; an important role is played by a condition under which a covering space of a subset A CX can be embedded into a covering space of X in such a way that the image still covers A. In Part II, the relation between the equivariant homotopy classes of maps of suitable covering spaces and the homotopy classes of the base spaces (both the relative and free homotopy cases) are given. Part III shows the Eilenberg procedure and theorem formally valid for the equivariant maps arising in Part II. Two simple applications, for illustrative purposes, are given in this paper.