The Nielsen realization problem for non-orientable surfaces

IF 0.6 4区数学 Q3 MATHEMATICS Topology and its Applications Pub Date : 2024-05-16 DOI:10.1016/j.topol.2024.108957

Nestor Colin , Miguel A. Xicoténcatl

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Abstract

We show the Teichmüller space of a non-orientable surface with marked points (considered as a Klein surface) can be identified with a subspace of the Teichmüller space of its orientable double cover. It is also well known that the mapping class group $Mod (N_{g}; k)$ of a non-orientable surface can be identified with a subgroup of $Mod (S_{g - 1}; 2 k)$ , the mapping class group of its orientable double cover. These facts, together with the classical Nielsen realization theorem, are used to prove that every finite subgroup of $Mod (N_{g}; k)$ can be lifted isomorphically to a subgroup of the group of diffeomorphisms $Diff (N_{g}; k)$ . In contrast, we show the projection $Diff (N_{g}) \to Mod (N_{g})$ does not admit a section for large g.

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不可定向曲面的尼尔森实现问题

我们证明了有标记点的不可定向曲面（视为克莱因曲面）的泰希缪勒空间可以与其可定向双盖的泰希缪勒空间的子空间相识别。同样众所周知的是，不可定向曲面的映射类群 Mod(Ng;k) 可以与其可定向双盖的映射类群 Mod(Sg-1;2k) 的一个子群相识别。这些事实以及经典的尼尔森实现定理被用来证明，Mod(Ng;k) 的每个有限子群都可以同构地提升到衍射群 Diff(Ng;k) 的一个子群。与此相反，我们证明了投影 Diff(Ng)→Mod(Ng) 在大 g 时不允许分段。

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来源期刊

Topology and its Applications 数学-数学

CiteScore

1.20

自引率

33.30%

发文量

251

审稿时长

6 months

期刊介绍： Topology and its Applications is primarily concerned with publishing original research papers of moderate length. However, a limited number of carefully selected survey or expository papers are also included. The mathematical focus of the journal is that suggested by the title: Research in Topology. It is felt that it is inadvisable to attempt a definitive description of topology as understood for this journal. Certainly the subject includes the algebraic, general, geometric, and set-theoretic facets of topology as well as areas of interactions between topology and other mathematical disciplines, e.g. topological algebra, topological dynamics, functional analysis, category theory. Since the roles of various aspects of topology continue to change, the non-specific delineation of topics serves to reflect the current state of research in topology. At regular intervals, the journal publishes a section entitled Open Problems in Topology, edited by J. van Mill and G.M. Reed. This is a status report on the 1100 problems listed in the book of the same name published by North-Holland in 1990, edited by van Mill and Reed.

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