{
  "id": "2405.14465",
  "version": "v1",
  "published": "2024-05-23T11:54:04.000Z",
  "updated": "2024-05-23T11:54:04.000Z",
  "title": "Foams with flat connections and algebraic K-theory",
  "authors": [
    "David Gepner",
    "Mee Seong Im",
    "Mikhail Khovanov",
    "Nitu Kitchloo"
  ],
  "comment": "57 pages, many figures",
  "categories": [
    "math.KT",
    "math.AT",
    "math.GT"
  ],
  "abstract": "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-05-23T11:54:04.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57R90",
      "19B99",
      "19D06",
      "18M05",
      "19A99",
      "55N22"
    ],
    "keywords": [
      "flat connections",
      "th algebraic k-theory group",
      "cobordism group",
      "dimensional foam replaces",
      "arbitrary exact categories"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 57,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}