A Degree Theorem for the Space of Ribbon Graphs

TL;DR

This paper generalizes a classical theorem to the space of ribbon graphs on surfaces, showing it has a layered simplicial complex structure with high connectivity and symmetry properties.

Contribution

It extends Hatcher and Vogtmann's Cerf Theory to ribbon graphs on surfaces, establishing a filtered simplicial complex structure with invariance under the mapping class group.

Findings

The space of ribbon graphs is filtered by simplicial complexes.

Each simplicial complex is (k-1)-dimensional and (k-2)-connected.

The complexes are invariant under the surface's mapping class group.

Abstract

This paper extends results of Hatcher and Vogtmann's work "Cerf Theory for Graphs" to ribbon graphs. Given an orientable, punctured and basepointed surface Sigma, we prove that the space of ribbon graphs that can be drawn in Sigma is filtered by simplicial complexes. The k-th simplicial complex is (k-1)-dimensional, (k-2)-connected and invariant under the action of the basepoint preserving mapping class group of Sigma.