# Topological field theory on r-spin surfaces and the Arf invariant

## TL;DR

This paper develops a combinatorial model for r-spin surfaces with boundary, enabling the construction of 2D topological field theories that compute the Arf-invariant, and demonstrates its invariance under the mapping class group.

## Contribution

It introduces a novel combinatorial model for r-spin surfaces that facilitates state sum constructions of topological field theories and relates these to the Arf-invariant.

## Key findings

- The model computes the r-spin Arf-invariant.
- The Arf-invariant is shown to be constant on mapping class group orbits.
- Provides an alternative proof of the invariance of the Arf-invariant.

## Abstract

We give a combinatorial model for r-spin surfaces with parametrised boundary based on Novak (2015). The r-spin structure is encoded in terms of $\mathbb{Z}_r$-valued indices assigned to the edges of a polygonal decomposition. This combinatorial model is designed for our state sum construction of two-dimensional topological field theories on r-spin surfaces. We show that an example of such a topological field theory computes the Arf-invariant of an r-spin surface as introduced in Geiges, Gonzalo (2012) and Randal-Williams (2014). This implies in particular that the r-spin Arf-invariant is constant on orbits of the mapping class group, providing an alternative proof of that fact.