SKK groups of manifolds and non-unitary invertible TQFTs

This work considers the computation of controllable cut-and-paste groups $\mathrm{SKK}^{\xi}_n$ of manifolds with tangential structure $\xi:B_n\to BO_n$. To this end, we apply the work of Galatius-Madsen-Tillman-Weiss, Genauer and Schommer-Pries, who showed that for a wide range of structures $\xi$ these groups fit into a short exact sequence that relates them to bordism groups of $\xi$-manifolds with kernel generated by the disc-bounding $\xi$-sphere. The order of this sphere can be computed by knowing the possible values of the Euler characteristic of $\xi$-manifolds. We are thus led to address two key questions: the existence of $\xi$-manifolds with odd Euler characteristic of a given dimension and conditions for the exact sequence to admit a splitting. We resolve these questions in a wide range of cases. $\mathrm{SKK}$ groups are of interest in physics as they play a role in the classification of non-unitary invertible topological quantum field theories, which classify anomalies and symmetry protected topological (SPT) phases of matter. Applying our topological results, we give a complete classification of non-unitary invertible topological quantum field theories in the tenfold way in dimensions 1-5.