A homology valued invariant for trivalent fatgraph spines

TL;DR

This paper introduces a new homology-valued invariant for trivalent fatgraph spines of bordered surfaces, derived from existing 1-cocycles, with explicit formulas and connections to spin structures.

Contribution

It presents a novel invariant based on two 1-cocycles, providing explicit formulas and linking to spin structures, advancing understanding of surface invariants.

Findings

Invariant takes values in the first homology of the surface.

Explicit formula for the invariant is provided.

Mod 2 reduction relates to spin structures.

Abstract

We introduce an invariant for trivalent fatgraph spines of a once bordered surface, which takes values in the first homology of the surface. This invariant is the secondary object coming from two 1-cocycles on the dual fatgraph complex, one introduced by Morita and Penner in 2008, and the other by Penner, Turaev, and the author in 2013. We present an explicit formula for this invariant and investigate its properties. We also show that the mod 2 reduction of the invariant is the difference of naturally defined two spin structures on the surface.