Residual finiteness of some automorphism groups of high dimensional manifolds

We show that for a smooth, closed 2-connected manifold $M$ of dimension $d\ge 6$, the topological mapping class group ${\pi }_0\mathrm{H}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{o}(M)$ is residually finite, in contrast to the situation for the smooth mapping class group ${\pi }_0\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}(M)$. Combined with a result of Sullivan, this implies that ${\pi }_0\mathrm{H}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{o}(M)$ is an arithmetic group. The proof uses embedding calculus, and is of independent interest: we show that the $T_k$-mapping class group, ${\pi }_0{T}_k\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}(M)$, is residually finite, for all $k\in \mathbb{N}$. The statement on the topological mapping class group is then deduced from the Weiss fibre sequence, convergence of the embedding calculus tower and smoothing theory.