Injective Simplicial Maps of the Arc Complex on Nonorientable Surfaces

TL;DR

This paper proves that injective simplicial maps of the arc complex on nonorientable surfaces are induced by homeomorphisms, and characterizes the automorphism group of the arc complex in relation to the surface's mapping class group.

Contribution

It establishes a rigidity result for the arc complex on nonorientable surfaces and describes its automorphism group in terms of the surface's mapping class group.

Findings

Injective simplicial maps are induced by homeomorphisms.

Automorphism group of the arc complex is isomorphic to the quotient of the mapping class group.

Provides a classification of symmetries of the arc complex.

Abstract

We prove that each injective simplicial map from the arc complex of a compact, connected, nonorientable surface with nonempty boundary to itself is induced by a homeomorphism of the surface. We also prove that the automorphism group of the arc complex is isomorphic to the quotient of the mapping class group of the surface by its center.