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

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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 H k (M; Z). We prove that the image of this homomorphism contains a congruence subgroup of SLₙ (Z) whenever n is at least 3.