A K-theory spectrum for cobordism cut and paste groups
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866911209489432576 |
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| author | Hoekzema, Renee S. Rovi, Carmen Semikina, Julia |
| author_facet | Hoekzema, Renee S. Rovi, Carmen Semikina, Julia |
| contents | Cobordism groups and cut-and-paste groups of manifolds arise from imposing two different relations on the monoid of manifolds under disjoint union. By imposing both relations simultaneously, a cobordism cut and paste group $\overline{\mathrm{SK}}_n$ is defined. In this paper, we extend this definition to manifolds with boundary obtaining $\overline{\mathrm{SK}}^{\partial}_n$ and study the relationship of this group to an appropriately defined cobordism group of manifolds with boundary. The main results are the construction of a spectrum that recovers on $π_0$ the cobordism cut and paste groups of manifolds with boundary, $\overline{\mathrm{SK}}^{\partial}_n$, and a map of spectra that lifts the canonical quotient map $\mathrm{SK}^{\partial}_n \rightarrow \overline{\mathrm{SK}}^{\partial}_n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_00682 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A K-theory spectrum for cobordism cut and paste groups Hoekzema, Renee S. Rovi, Carmen Semikina, Julia Algebraic Topology Category Theory Geometric Topology K-Theory and Homology 19D99, 18D99, 18F25, 18F30, 57N70, 55P15, 57R75, 57R90, 57N40 Cobordism groups and cut-and-paste groups of manifolds arise from imposing two different relations on the monoid of manifolds under disjoint union. By imposing both relations simultaneously, a cobordism cut and paste group $\overline{\mathrm{SK}}_n$ is defined. In this paper, we extend this definition to manifolds with boundary obtaining $\overline{\mathrm{SK}}^{\partial}_n$ and study the relationship of this group to an appropriately defined cobordism group of manifolds with boundary. The main results are the construction of a spectrum that recovers on $π_0$ the cobordism cut and paste groups of manifolds with boundary, $\overline{\mathrm{SK}}^{\partial}_n$, and a map of spectra that lifts the canonical quotient map $\mathrm{SK}^{\partial}_n \rightarrow \overline{\mathrm{SK}}^{\partial}_n$. |
| title | A K-theory spectrum for cobordism cut and paste groups |
| topic | Algebraic Topology Category Theory Geometric Topology K-Theory and Homology 19D99, 18D99, 18F25, 18F30, 57N70, 55P15, 57R75, 57R90, 57N40 |
| url | https://arxiv.org/abs/2210.00682 |