Parametrized scissors congruence $K$-theory of manifolds and cobordism categories
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866912996854333440 |
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| author | Merling, Mona Raptis, George Semikina, Julia |
| author_facet | Merling, Mona Raptis, George Semikina, Julia |
| contents | We introduce a parametrized version of scissors congruence $K$-theory of manifolds with tangential structure, which includes a topologized version of the scissors congruence $K$-theory of oriented manifolds as a special case. We examine the relation of this $K$-theory spectrum with cut-and-paste invariants, the (parametrized) cobordism category and with (bivariant) algebraic $K$-theory of spaces. We show that the scissors congruence $K$-theory of oriented manifolds agrees on $π_0$ with a version of the oriented cobordism category where we allow cobordisms to have free boundaries. Lastly, we show that the spectrum level refinement of the Euler characteristic from the scissors congruence $K$-theory to $K(\mathbb{Z})$ detects on $π_1$ the Kervaire semicharacteristic. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_01810 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Parametrized scissors congruence $K$-theory of manifolds and cobordism categories Merling, Mona Raptis, George Semikina, Julia Algebraic Topology K-Theory and Homology We introduce a parametrized version of scissors congruence $K$-theory of manifolds with tangential structure, which includes a topologized version of the scissors congruence $K$-theory of oriented manifolds as a special case. We examine the relation of this $K$-theory spectrum with cut-and-paste invariants, the (parametrized) cobordism category and with (bivariant) algebraic $K$-theory of spaces. We show that the scissors congruence $K$-theory of oriented manifolds agrees on $π_0$ with a version of the oriented cobordism category where we allow cobordisms to have free boundaries. Lastly, we show that the spectrum level refinement of the Euler characteristic from the scissors congruence $K$-theory to $K(\mathbb{Z})$ detects on $π_1$ the Kervaire semicharacteristic. |
| title | Parametrized scissors congruence $K$-theory of manifolds and cobordism categories |
| topic | Algebraic Topology K-Theory and Homology |
| url | https://arxiv.org/abs/2504.01810 |