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

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, Bökstedt–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.

Keywords

invertible topological field theory, spin, SKK group, bordism group

2010 Mathematics Subject Classification

57R15, 57R56

Full Text (PDF format)

Received 23 September 2019

Received revised 8 March 2022

Accepted 8 March 2022

Published 22 March 2023