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

TL;DR

This paper demonstrates that for certain smooth manifolds homeomorphic to products of spheres, the diffeomorphism group maps onto a subgroup of automorphisms containing a congruence subgroup of SL_n(Z), revealing deep algebraic structure.

Contribution

It establishes that the image of the diffeomorphism group homomorphism includes a congruence subgroup of SL_n(Z) for manifolds homeomorphic to products of spheres with odd dimension.

Findings

The image contains a congruence subgroup of SL_n(Z).

The result applies when n ≥ 3.

It links diffeomorphism groups to arithmetic groups.

Abstract

Let M be a smooth manifold which is homeomorphic to the n-fold product of S^k, 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 SL_n (Z) whenever n is at least 3.