On groups of diffeomorphisms

Abstract

I. We consider here the groups of homeomorphisms on Euclidean n-space and the n-sphere Sn. Chiefly we will be concerned with the question of whether or not these groups reduce in an homotopy sense to the ordinary orthogonal group acting on these spaces. Such questions are intimately connected with the theory of fibre bundles in which these spaces occur as fibres. We will restrict ourselves to the case where the homeomorphisms are of class C' and will topologize the various groups taking account of the differentiability. We first consider Euclidean n-space En. We denote by K the group of all homeomorphisms f of En such that f and f'l are of class C'. K becomes a topological group by demanding uniform convergence of f and its derivatives on compact sets,' i.e. a typical neighborhood of the identity function is given by