Fadi Mezher: On the arithmeticity of automorphism groups of high dimensional manifolds

Abstract.

Given a smooth manifold of dimension at least 5, are the groups of isotopy classes of isomorphisms (both in the smooth and topological categories) arithmetic groups? Building on work of Sullivan, and combined with embedding calculus and profinite homotopy theory, we answer this question positively in the case of the topological mapping class group, and isolate the obstruction for arithmeticity in terms of exotic spheres for the smooth mapping class group. Time permitting, I will discuss applications of the above result to the question of detectability of exotic spheres in embedding calculus, as part of ongoing joint work with M. Krannich and A. Kupers.