Diffeomorphisms of a product of spheres

Abstract

O. Introduction Much progress has been made recently on the manifold classification problem in the case of highly connected manifolds, notably by Wall [16] and [17]. In terms of the methods involved, a natural next step is to consider manifolds whose homology is free and (except for the top and bottom groups) is concentrated in two separated dimensions. We say a manifold M is of type (n, k, r) ifM is a simply connected smooth n-manifold with 3<2k+l<n and Hk(M)~-Hn_k(M)~--7Z r the only non-trivial homology groups. In w 1 we represent such manifolds as the union of two handlebodies along their boundaries; the case of interest is stated as: 1.7. Corollary. Let M be of type (n,k, 1) with k-3, 5,6, 7 (mod 8). Then M'~S k  D n-k U Sk X O n-k for some heDiff(S k x S"-k-1). h 2 and 3 are devoted to an analysis of the group 7zo(Diff(S k  St)) of isotopy classes of diffeomorphisms of S k  S t. Let Diff + (S k  S ~) denote those elements of Diff(S k  S ~) which induce the identity on all homology groups. Our main result is: