On invertible $2$-dimensional framed and $r$-spin topological field theories

Abstract

We classify invertible 2-dimensional framed and $r$-spin topological field theories by computing the homotopy groups and the $k$-invariant of the corresponding bordism categories. By a recent result of Kreck, Stolz and Teichner the first homotopy groups are given by the so called SKK groups. We compute them explicitly using the combinatorial model of framed and $r$-spin surfaces of Novak, Runkel and the author.