Foams with flat connections and algebraic K-theory
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| Format: | Preprint |
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2024
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| _version_ | 1866910457801998336 |
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| author | Gepner, David Im, Mee Seong Khovanov, Mikhail Kitchloo, Nitu |
| author_facet | Gepner, David Im, Mee Seong Khovanov, Mikhail Kitchloo, Nitu |
| contents | This paper proposes a connection between algebraic K-theory and foam cobordisms, where foams are stratified manifolds with singularities of a prescribed form. We consider $n$-dimensional foams equipped with a flat bundle of finitely-generated projective $R$-modules over each facet of the foam, together with gluing conditions along the subfoam of singular points. In a suitable sense which will become clear, a vertex (or the smallest stratum) of an $n$-dimensional foam replaces an $(n+1)$-simplex with a total ordering of vertices. We show that the first K-theory group of a ring $R$ can be identified with the cobordism group of decorated 1-foams embedded in the plane. A similar relation between the $n$-th algebraic K-theory group of a ring $R$ and the cobordism group of decorated $n$-foams embedded in $\mathbb{R}^{n+1}$ is expected for $n>1$. An analogous correspondence is proposed for arbitrary exact categories. Modifying the embedding and other conditions on the foams may lead to new flavors of K-theory groups. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_14465 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Foams with flat connections and algebraic K-theory Gepner, David Im, Mee Seong Khovanov, Mikhail Kitchloo, Nitu K-Theory and Homology Algebraic Topology Geometric Topology Primary: 57R90, 19B99, 19D06, 18M05, Secondary: 19A99, 55N22 This paper proposes a connection between algebraic K-theory and foam cobordisms, where foams are stratified manifolds with singularities of a prescribed form. We consider $n$-dimensional foams equipped with a flat bundle of finitely-generated projective $R$-modules over each facet of the foam, together with gluing conditions along the subfoam of singular points. In a suitable sense which will become clear, a vertex (or the smallest stratum) of an $n$-dimensional foam replaces an $(n+1)$-simplex with a total ordering of vertices. We show that the first K-theory group of a ring $R$ can be identified with the cobordism group of decorated 1-foams embedded in the plane. A similar relation between the $n$-th algebraic K-theory group of a ring $R$ and the cobordism group of decorated $n$-foams embedded in $\mathbb{R}^{n+1}$ is expected for $n>1$. An analogous correspondence is proposed for arbitrary exact categories. Modifying the embedding and other conditions on the foams may lead to new flavors of K-theory groups. |
| title | Foams with flat connections and algebraic K-theory |
| topic | K-Theory and Homology Algebraic Topology Geometric Topology Primary: 57R90, 19B99, 19D06, 18M05, Secondary: 19A99, 55N22 |
| url | https://arxiv.org/abs/2405.14465 |