SKK groups of manifolds and non-unitary invertible TQFTs
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| Format: | Preprint |
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2025
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| author | Hoekzema, Renee S. Stehouwer, Luuk Veselá, Simona |
| author_facet | Hoekzema, Renee S. Stehouwer, Luuk Veselá, Simona |
| contents | This work considers the computation of controllable cut-and-paste groups $\mathrm{SKK}^ξ_n$ of manifolds with tangential structure $ξ:B_n\to BO_n$. 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 $ξ$ these groups fit into a short exact sequence that relates them to bordism groups of $ξ$-manifolds with kernel generated by the disc-bounding $ξ$-sphere. The order of this sphere can be computed by knowing the possible values of the Euler characteristic of $ξ$-manifolds. We are thus led to address two key questions: the existence of $ξ$-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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_07917 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | SKK groups of manifolds and non-unitary invertible TQFTs Hoekzema, Renee S. Stehouwer, Luuk Veselá, Simona Algebraic Topology Mathematical Physics Geometric Topology 57N65 (Primary), 55N22, 57R56, 18A05 (Secondary) This work considers the computation of controllable cut-and-paste groups $\mathrm{SKK}^ξ_n$ of manifolds with tangential structure $ξ:B_n\to BO_n$. 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 $ξ$ these groups fit into a short exact sequence that relates them to bordism groups of $ξ$-manifolds with kernel generated by the disc-bounding $ξ$-sphere. The order of this sphere can be computed by knowing the possible values of the Euler characteristic of $ξ$-manifolds. We are thus led to address two key questions: the existence of $ξ$-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. |
| title | SKK groups of manifolds and non-unitary invertible TQFTs |
| topic | Algebraic Topology Mathematical Physics Geometric Topology 57N65 (Primary), 55N22, 57R56, 18A05 (Secondary) |
| url | https://arxiv.org/abs/2504.07917 |