Residual finiteness of some automorphism groups of high dimensional manifolds

TL;DR

This paper proves that the topological mapping class group of certain high-dimensional manifolds is residually finite and arithmetic, contrasting with the smooth case, using embedding calculus and smoothing theory.

Contribution

It establishes residual finiteness of topological mapping class groups for high-dimensional manifolds and introduces new results on the residual finiteness of the $T_k$-mapping class groups.

Findings

Topological mapping class groups are residually finite for d ≥ 6.

The $T_k$-mapping class groups are residually finite for all k.

The topological mapping class group is an arithmetic group.

Abstract

We show that for a smooth, closed 2-connected manifold $M$ of dimension $d \geq 6$, the topological mapping class group $\pi_0 \mathrm{Homeo}(M)$ is residually finite, in contrast to the situation for the smooth mapping class group $\pi_0 \mathrm{Diff}(M)$. Combined with a result of Sullivan, this implies that $\pi_0 \mathrm{Homeo}(M)$ is an arithmetic group. The proof uses embedding calculus, and is of independent interest: we show that the $T_k$-mapping class group, $\pi_0 T_k \mathrm{Diff}(M)$, 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.