Residual finiteness of some automorphism groups of high dimensional manifolds
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866908388472913920 |
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| author | Mezher, Fadi |
| author_facet | Mezher, Fadi |
| contents | We show that for a smooth, closed 2-connected manifold $M$ of dimension $d \geq 6$, the topological mapping class group $π_0 \mathrm{Homeo}(M)$ is residually finite, in contrast to the situation for the smooth mapping class group $π_0 \mathrm{Diff}(M)$. Combined with a result of Sullivan, this implies that $π_0 \mathrm{Homeo}(M)$ is an arithmetic group. The proof uses embedding calculus, and is of independent interest: we show that the $T_k$-mapping class group, $π_0 T_k \mathrm{Diff}(M)$, 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_08902 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Residual finiteness of some automorphism groups of high dimensional manifolds Mezher, Fadi Geometric Topology Algebraic Topology We show that for a smooth, closed 2-connected manifold $M$ of dimension $d \geq 6$, the topological mapping class group $π_0 \mathrm{Homeo}(M)$ is residually finite, in contrast to the situation for the smooth mapping class group $π_0 \mathrm{Diff}(M)$. Combined with a result of Sullivan, this implies that $π_0 \mathrm{Homeo}(M)$ is an arithmetic group. The proof uses embedding calculus, and is of independent interest: we show that the $T_k$-mapping class group, $π_0 T_k \mathrm{Diff}(M)$, 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. |
| title | Residual finiteness of some automorphism groups of high dimensional manifolds |
| topic | Geometric Topology Algebraic Topology |
| url | https://arxiv.org/abs/2410.08902 |