Topological field theory on r-spin surfaces and 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.

Figures (29)

• Figure 1: Glueing a torus from a rectangle. Each step is a regular cell map and each generalised cell decomposition is a PLCW decomposition.
• Figure 2: $a)$ A generalised cell decomposition which is not a PLCW decomposition. There are one 2-cell, four 1-cells and four 0-cells. One can visualise it by folding a paper and glueing it only along the bottom edge. $b)$ A triangle with two sides identified and a 1-gon, both PLCW decompositions. The map between them is not a regular cell map as the edge in the middle has no image. $c)$ A PLCW decomposition of a sphere into two faces, one edge (red line) and one vertex.
• Figure 3: Figure of a face with adjacent edges and vertices in a marked PLCW decomposition. The orientation of the face is that of the paper plane, the orientation of the edges is indicated by an arrow on them. The half-dot indicates the marked edge of the face the half-dot lies in. The arrow in the middle shows the clockwise direction along the marked edge $e$ and $v$ is the vertex sitting on the boundary of $e$ in clockwise direction. Note that the clockwise vertex $v$ of the edge $e$ is determined by the orientation of the face and not by the orientation of the edge $e$.
• Figure 4: Moves of Lemma \ref{['lem:moves']} for a face of $\Sigma$. All edge orientations and markings are arbitrary unless shown explicitly. $(1)$ Flipping the edge orientation of $e$. $(2a)$, $(2b)$ Moving the edge marking for a face. $(3)$ Shifting the edge indices for a face. The dotted edges $e_3$ and $e_5$ are identified, hence the edge index remains unchanged. The edges $e_1$ and $e_2$ are counterclockwise oriented, hence the $+k$ shift of the corresponding edge indices $s_1$ and $s_2$, the edge $e_4$ is clockwise oriented, hence the $-k$ shift of $s_4$.
• Figure 5: Two arcs $p,q\in A(\gamma)$ of a curve $\gamma$ on a face $f$. Here $f_p=f_q=f$, $\hat{s}_{e_p}=-s_{e_p}-1$, $\hat{s}_{e_q}=s_{e_q}$, $\hat{\delta}_{f_p}^p=+1$ and $\hat{\delta}_{f_q}^q=0$.

Theorems & Definitions (63)

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