Injective Simplicial Maps of the Complexes of Curves of Nonorientable Surfaces

TL;DR

This paper proves that for certain nonorientable surfaces, any injective simplicial map of the curve complex is induced by a homeomorphism, extending understanding of the automorphisms of these complexes.

Contribution

It establishes that injective simplicial maps of the curve complexes of specific nonorientable surfaces are always induced by homeomorphisms, generalizing known results.

Findings

Injective simplicial maps are induced by homeomorphisms for surfaces with g + n ≤ 3 or g + n ≥ 5.

The result applies to nonorientable surfaces of certain genus and boundary configurations.

Provides a classification of automorphisms of the curve complex in these cases.

Abstract

Let $N$ be a compact, connected, nonorientable surface of genus $g$ with $n$ boundary components, and $\mathcal{C}(N)$ be the complex of curves of $N$. Suppose that $g + n \leq 3$ or $g + n \geq 5$. If $\lambda : \mathcal{C}(N) \rightarrow \mathcal{C}(N)$ is an injective simplicial map, then $\lambda$ is induced by a homeomorphism of $N$.