Residual finiteness of some automorphism groups of high dimensional manifolds

2024/10/11 by Fadi Mezher, Mezher, Fadi · 1 citation
Mathematics · #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #Advanced Differential Equations and Dynamical Systems

paper · pdf · doi:10.48550/arxiv.2410.08902

Abstract

We show that for a smooth, closed 2-connected manifold M of dimension d ≥ 6, the topological mapping class group π₀ Homeo(M) is residually finite, in contrast to the situation for the smooth mapping class group π₀ Diff(M). Combined with a result of Sullivan, this implies that π₀ 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, π₀ T_k Diff(M), is residually finite, for all k ∈ 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.

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