Superinjective Simplicial Maps of the Two-sided Curve Complexes on Nonorientable Surfaces

Abstract

Let $N$ be a compact, connected, nonorientable surface of genus $g$ with $n$ boundary components with $g\ge 5$, $n\ge 0$. Let $\mathcal{T}(N)$ be the two-sided curve complex of $N$. If $\lambda :\mathcal{T}(N)\to \mathcal{T}(N)$ is a superinjective simplicial map, then there exists a homeomorphism $h:N\to N$ unique up to isotopy such that $H(\alpha )=\lambda (\alpha )$ for every vertex $\alpha$ in $\mathcal{T}(N)$ where $H=[h]$.

51 pages, 36 figures