论文标题
Nielsen实现问题的不可取向表面
The Nielsen Realization Problem for Non-Orientable Surfaces
论文作者
论文摘要
我们可以通过其可定向的双层盖的Teichmüller空间的子空间来识别出不可取向表面的Teichmüller空间。同样，众所周知，可以使用$ \ text {mod}（s_ {g-1}; 2k）$的子组（映射类$ \ text {mod}（mod}（n_g; k）$，可以通过$ \ text {mod}（s_ {g-1}; 2k）$识别，其定向双盖的映射类组。这些事实与经典的尼尔森实现定理一起证明，$ \ text {mod}（n_g; k）$的每个有限亚组可以同态抬高至diffefemormormormormormormormorphists $ \ text {diff}（diff}（diff}（n_g; k）$）的子组。相比之下，我们显示投影$ \ text {diff}（n_g）\ to \ text {mod}（n_g）$不接受大型$ g $的部分。
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. Also, it is well known that the mapping class group $\text{Mod} (N_g; k)$ of a non-orientable surface can be identified with a subgroup of $\text{Mod} (S_{g-1}; 2k)$, 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 $\text{Mod}(N_g; k)$ can be lifted isomorphically to a subgroup of the group of diffeomorphisms $\text{Diff}(N_g; k)$. In contrast, we show the projection $\text{Diff}(N_g) \to \text{Mod}(N_g)$ does not admit a section for large $g$.
