Simplicial Maps of the Complexes of Curves on Nonorientable Surfaces

TL;DR

This paper proves that certain simplicial maps of the complex of curves on nonorientable surfaces are induced by homeomorphisms, extending understanding of the automorphisms of these complexes.

Contribution

It establishes that simplicial maps preserving adjacency are always induced by homeomorphisms for specific nonorientable surfaces.

Findings

Simplicial maps are induced by homeomorphisms for specified surface types.

Superinjective maps correspond to homeomorphisms.

Automorphisms of the complex are induced by surface homeomorphisms.

Abstract

Let $N$ be a compact, connected, nonorientable surface of genus $g$ with $n$ boundary components. Let $\lambda$ be a simplicial map of the complex of curves, $\mathcal{C}(N)$, on $N$ which satisfies the following: $[a]$ and $[b]$ are connected by an edge in $\mathcal{C}(N)$ if and only if $\lambda([a])$ and $\lambda([b])$ are connected by an edge in $\mathcal{C}(N)$ for every pair of vertices $[a], [b]$ in $\mathcal{C}(N)$. We prove that $\lambda$ is induced by a homeomorphism of $N$ if $(g, n) \in \{(1, 0), (1, 1), (2, 0)$, $(2, 1), (3, 0)\}$ or $g + n \geq 5$. Our result implies that superinjective simplicial maps and automorphisms of $\mathcal{C}(N)$ are induced by homeomorphisms of $N$.