On simplicial maps of the complexes of curves of nonorientable surfaces

Abstract

Let $N$ be a compact, connected, nonorientable surface of genus $g$ with $n$ boundary components and $\mathcal{C}\left(N\right)$ be the complex of curves of $N$. Suppose that $g+n\le 3$ or $g+n\ge 5$. If $\lambda :\mathcal{C}\left(N\right)\to \mathcal{C}\left(N\right)$ is an injective simplicial map, then $\lambda$ is induced by a homeomorphism of $N$. If $\left(g,n\right)\ne \left(1,2\right)$ and $\lambda :\mathcal{C}\left(N\right)\to \mathcal{C}\left(N\right)$ is a simplicial map that satisfies the connectivity property, then $\lambda$ is induced by a homeomorphism of $N$.