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

TL;DR

This paper proves that any superinjective simplicial map of the two-sided curve complex on a nonorientable surface of genus at least 5 is induced by a homeomorphism, establishing a strong correspondence between combinatorial and geometric structures.

Contribution

It demonstrates that superinjective simplicial maps of the two-sided curve complex correspond to homeomorphisms of the surface, extending known results to nonorientable surfaces.

Findings

Superinjective maps are induced by homeomorphisms.

Uniqueness of the homeomorphism up to isotopy.

Applicable to nonorientable surfaces of genus ≥ 5.

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 $\lambda :\mathcal{T}(N) \rightarrow \mathcal{T}(N)$ is a superinjective simplicial map, then there exists a homeomorphism $h : N \rightarrow N$ unique up to isotopy such that $H(\alpha) = \lambda(\alpha)$ for every vertex $\alpha$ in $\mathcal{T}(N)$ where $H=[h]$.