On the mapping class groups of # r (S p × S p ) for p = 3, 7

Abstract

For \({M_r := \sharp_r(S^p \times S^p),\,p=3, 7}\), we calculate \({\pi_0{\rm Diff}(M_r)/\Theta_{2p+1}}\) and \({\mathcal{E}(M_r)}\), respectively the group of isotopy classes of orientation preserving diffeomorphisms of M r modulo isotopy classes with representatives which are the identity outside a 2p-disc and the group of homotopy classes of orientation preserving homotopy equivalences of M r .