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

We present a state-sum construction of topological field theories (TFT)
on r-spin surfaces which uses a combinatorial model of r-spin structures
based on the work of Novak. We give an example of such a TFT which
computes the Arf invariant for r even. We use the combinatorial model
and this TFT to calculate diffeomorphism classes of r-spin surfaces with
parametrized boundary, extending results of Geiges and Gonzalo and of
Randal-Williams.