## Algebraic & Geometric Topology

### A homology-valued invariant for trivalent fatgraph spines

Yusuke Kuno

#### 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 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.

#### Article information

Source
Algebr. Geom. Topol., Volume 17, Number 3 (2017), 1785-1811.

Dates
Received: 25 May 2016
Revised: 20 October 2016
Accepted: 30 October 2016
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1510841410

Digital Object Identifier
doi:10.2140/agt.2017.17.1785

Mathematical Reviews number (MathSciNet)
MR3677940

Zentralblatt MATH identifier
06762601

#### Citation

Kuno, Yusuke. A homology-valued invariant for trivalent fatgraph spines. Algebr. Geom. Topol. 17 (2017), no. 3, 1785--1811. doi:10.2140/agt.2017.17.1785. https://projecteuclid.org/euclid.agt/1510841410

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