Automorphisms of curve complexes on nonorientable surfaces

Abstract

For a compact connected nonorientable surface $N$ of genus $g$ with $n$ boundary components, we prove that the natural map from the mapping class group of $N$ to the automorphism group of the curve complex of $N$ is an isomorphism provided that $g+n\ge 5$. We also prove that two curve complexes are isomorphic if and only if the underlying surfaces are diffeomorphic.