Realizing congruence subgroups inside the diffeomorphism group of a product of homotopy spheres

Abstract

Let M be a smooth manifold which is homeomorphic to the n-fold product of Sk, where k is odd. There is an induced homomorphism from the group of diffeomorphisms of M to the automorphism group of Hk(M;Z). We prove that the image of this homomorphism contains a congruence subgroup of SLn(Z) whenever n is at least 3.