On the group of homotopy equivalences of a manifold

Abstract

We consider the group of homotopy equivalences E ( M ) \mathcal E(M) of a simply connected manifold M M which is part of the fundamental extension of groups due to Barcus-Barratt. We show that the kernel of this extension is always a finite group and we compute this kernel for various examples. This leads to computations of the group E ( M ) \mathcal E(M) for special manifolds M M , for example if M M is a connected sum of products S n × S m S^n\times S^m of spheres. In particular the group E ( S n × S n ) \mathcal E(S^n\times S^n) is determined completely. Also the connection of E ( M ) \mathcal E(M) with the group of isotopy classes of diffeomorphisms of M M is studied.