Superinjective Simplicial Maps of the Complexes of Curves on Nonorientable Surfaces

TL;DR

This paper proves that for certain nonorientable surfaces, every superinjective simplicial map of the curve complex corresponds to a surface homeomorphism, extending understanding of surface symmetries and automorphisms.

Contribution

It establishes that superinjective simplicial maps of the curve complex are induced by homeomorphisms for a broad class of nonorientable surfaces, including some with boundary.

Findings

Superinjective maps correspond to surface homeomorphisms.

Results apply to surfaces with specific genus and boundary conditions.

Extends known results from orientable to nonorientable surfaces.

Abstract

We prove that each superinjective simplicial map of the complex of curves of a compact, connected, nonorientable surface is induced by a homeomorphism of the surface, if $(g, n) \in \{(1, 0), (1, 1), (2, 0), (2, 1), (3, 0)\}$ or $g + n \geq 5$, where $g$ is the genus of the surface and $n$ is the number of the boundary components.