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. The zeroth homotopy group of a bordism category is the usual Thom bordism group, the first homotopy group can be identified with a Reinhart vector field bordism group, or the so called SKK group as observed by Ebert, B\"ockstedt-Svane and Kreck-Stolz-Teichner. We present the computation of SKK groups for stable tangential structures. Then we consider non-stable examples: the 2-dimensional framed and $r$-spin SKK groups and compute them explicitly using the combinatorial model of framed and $r$-spin surfaces of Novak, Runkel and the author.