A homology valued invariant for trivalent fatgraph spines

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.