Algebraic & Geometric Topology

A homology-valued invariant for trivalent fatgraph spines

Yusuke Kuno

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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

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

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

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Zentralblatt MATH identifier

Primary: 20F34: Fundamental groups and their automorphisms [See also 57M05, 57Sxx] 32G15: Moduli of Riemann surfaces, Teichmüller theory [See also 14H15, 30Fxx] 57N05: Topology of $E^2$ , 2-manifolds

fatgraphs Teichmüller space Johnson homomorphism spin structures


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.

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