A low-valence ribbon graph complex computing the cohomology of $M_{g,m}$

TL;DR

This paper proves that all cohomology classes of the moduli space M_{g,m} can be represented by specific combinatorial ribbon quiver graphs with vertices of valence at most four, establishing a sharp bound.

Contribution

It introduces a combinatorial representation of cohomology classes in M_{g,m} using ribbon quivers with a maximum valence of four, which is proven to be optimal.

Findings

Every cohomology class of M_{g,m} can be represented by a ribbon quiver with vertices of at most four valence.

The four-valent bound for vertices is proven to be sharp.

The combinatorial representation simplifies understanding the cohomology of M_{g,m}.

Abstract

It is proven that every cohomology class of the moduli space $M_{g,m}$ for any $2g+m\geq 3$, $m\geq 1$ can be represented combinatorially by a ribbon quiver with at most four-valent vertices. The "at most four"-valency condition is sharp.