The Nielsen Realization Problem for Non-Orientable Surfaces

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 $g$.