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 $g \geq 5$, $n \geq 0$. Let $\mathcal{T}(N)$ be the two-sided curve complex of $N$. If $λ:\mathcal{T}(N) \rightarrow \mathcal{T}(N)$ is a superinjective simplicial map, then there exists a homeomorphism $h : N \rightarrow N$ unique up to isotopy such that $H(α) = λ(α)$ for every vertex $α$ in $\mathcal{T}(N)$ where $H=[h]$.

51 pages, 36 figures