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We classify invertible 2-dimensional framed and $r$-spin topological fieldtheories by computing the homotopy groups and the $k$-invariant of thecorresponding bordism categories. The zeroth homotopy group of a bordismcategory is the usual Thom bordism group, the first homotopy group can beidentified with a Reinhart vector field bordism group, or the so called SKKgroup as observed by Ebert, B\"ockstedt-Svane and Kreck-Stolz-Teichner. Wepresent the computation of SKK groups for stable tangential structures. Then weconsider non-stable examples: the 2-dimensional framed and $r$-spin SKK groupsand compute them explicitly using the combinatorial model of framed and$r$-spin surfaces of Novak, Runkel and the author.

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