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

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