Automatic continuity in homeomorphism groups of compact 2-manifolds

Automatic continuity for homeomorphism groups and applications

Let M be a compact manifold, possibly with boundary. We show that the group of homeomorphisms of M has the automatic continuity property: any homomorphism from Homeo(M) to any separable group is necessarily continuous. This answers a question of C. Rosendal. If N is a submanifold of M, the group of homeomorphisms of M that preserve N also has this property. Various applications of automatic continuity are discussed, including applications to the topology and structure of groups of germs of homeomorphisms. In an appendix with Frederic Le Roux we also show, using related techniques, that the gro…

关于自动连续性的稳定表面分类

Classification of Stable Surfaces with respect to Automatic Continuity

本文研究了稳定表面的同胚群是否具有自动连续性，通过一般框架证明了同胚群的自动连续性，并回答了Mann提出的两个问题。

Automatic continuity for homeomorphism groups of noncompact manifolds

We extend the proof of automatic continuity for homeomorphism groups of manifolds to non-compact manifolds and manifolds with marked points and their mapping class groups. Specifically, we show that, for any manifold $M$ homeomorphic to the interior of a compact manifold, and a set $X \subset M$ homeomorphic to the union of a Cantor set and finite set, the relative homeomorphism group $\mathrm{Homeo}(M, X)$ and the mapping class group $\mathrm{Homeo}(M, X)/\mathrm{Homeo}_0(M,X)$ have the property that any homomorphism from such a group to any separable topological group is necessarily continuo…

纯映射类群的自动连续性

Automatic continuity of pure mapping class groups

本文完整分类了使纯映射类群及其紧支撑逼近闭包具有自动连续性的可定向无限型曲面，并证明有限端、无非紧边界时 Map(S) 不具该性质。

自相似2维流形的同胚群

Homeomorphism groups of self-similar 2-manifolds

该研究围绕自相似2维流形的同胚群展开，通过综述新旧成果、研究稳定集结构，证明了三项强化/推广型定理，为无限亏格2维流形同胚群的Steinhaus性质提供了首批实例。

The compact-open topology on the homeomorphism group of a surface without boundary is minimal

We show that the homeomorphism group of a surface without boundary does not admit a Hausdorff group topology strictly coarser than the compact-open topology. In combination with known automatic continuity results, this implies that the compact-open topology is the unique Hausdorff separable group topology on the group if the surface is closed or the complement in a closed surface of either a finite set or the union of a finite set and a Cantor set.