A Duality in Function Spaces

Abstract

0. Introduction. James Eells, in his paper [2], has shown a fresh approach to the computation of cohomology groups of functions spaces. Roughly speaking, he proves that, if X is an infinite-dimensional manifold and if Y is a submanifold of codimension n, then there is an isomorphism relating Hp+n(X, X Y) to HP(Y) (for details, see Example 1.4 [B] below). Eells then proceeds to show that many important function spaces are indeed infinite-dimensional manifolds, and, in many instances, there are useful submanifolds of finite codimension. By this method, Eells is able to obtain many new results on function spaces as well as to give new insights to known facts. A drawback of his approach is noted by Eells himself: should be remarked that although Alexander-Pontrjagin duality is a theory of topological character, our applications of it require the differentiable structure of our function spaces (e.g. to establish that certain subspaces are in fact finite-codimensional submanifolds). It is the purpose of the present paper to give a purely topological foundation to the 'duality theorem' in function spaces. It will be seen, for instance, that in the applications discussed in [2] the differentiable structures are completely irrelevant. We believe that the new setting for the 'duality theorem' makes it much easier to apply even to a situation where a differentiability is readily available. The hybrid nature of our approach should be pointed out: We study local properties of a function space by means of fiber maps and global properties by sheaf theory. In the present paper, we shall deal exclusively with the additive cohomology. The multiplicative structure of the cohomology of function spaces will be considered in a subsequent paper. Finally, I should like to express my gratitude to J. Eells for many lively discussions we had on the subject of this paper.