The Nielsen Realization Problem for Non-Orientable Surfaces

TL;DR

This paper explores the relationship between Teichmüller spaces and mapping class groups of non-orientable surfaces, demonstrating how finite subgroups can be lifted to diffeomorphisms but the projection to the mapping class group lacks a section for large genus.

Contribution

It establishes a connection between Teichmüller spaces of non-orientable surfaces and their orientable double covers, and proves lifting properties of finite subgroups in the mapping class group.

Findings

Finite subgroups of the mapping class group can be lifted to diffeomorphisms.

The projection from diffeomorphisms to the mapping class group does not admit a section for large genus.

Teichmüller space of a non-orientable surface embeds into that of its orientable double cover.

Abstract

We show the Teichm\"uller space of a non-orientable surface with marked points (considered as a Klein surface) can be identified with a subspace of the Teichm\"uller 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 $g$.