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 a 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 two naturally defined spin structures on the surface.
"A homology-valued invariant for trivalent fatgraph spines." Algebr. Geom. Topol. 17 (3) 1785 - 1811, 2017. https://doi.org/10.2140/agt.2017.17.1785