{
  "id": "1110.6001",
  "version": "v3",
  "published": "2011-10-27T07:54:47.000Z",
  "updated": "2017-01-26T08:47:51.000Z",
  "title": "$\\mathrm{G}$-Theory of $\\mathbb{F}_1$-Algebras I: the Equivariant Nishida Problem",
  "authors": [
    "Snigdhayan Mahanta"
  ],
  "comment": "24 pages; v2 introduction rewritten, references added; v3 revised according to the referee's comments (to appear in J. Homotopy Relat. Struct.)",
  "categories": [
    "math.AT",
    "math.KT",
    "math.RA"
  ],
  "abstract": "We develop a version of $\\mathrm{G}$-theory for an $\\mathbb{F}_1$-algebra (i.e., the $\\mathrm{K}$-theory of pointed $G$-sets for a pointed monoid $G$) and establish its first properties. We construct a Cartan assembly map to compare the Chu--Morava $\\mathrm{K}$-theory for finite pointed groups with our $\\mathrm{G}$-theory. We compute the $\\mathrm{G}$-theory groups for finite pointed groups in terms of stable homotopy of some classifying spaces. We introduce certain Loday--Whitehead groups over $\\mathbb{F}_1$ that admit functorial maps into classical Whitehead groups under some reasonable hypotheses. We initiate a conjectural formalism using combinatorial Grayson operations to address the Equivariant Nishida Problem - it asks whether $\\mathbb{S}^G$ admits operations that endow $\\oplus_n\\pi_{2n}(\\mathbb{S}^G)$ with a pre-$\\lambda$-ring structure, where $G$ is a finite group and $\\mathbb{S}^G$ is the $G$-fixed point spectrum of the equivariant sphere spectrum.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2011-11-10T04:31:51.000Z",
      "title": "G-Theory of \\F_1-Algebras I: the Equivariant Nishida Problem",
      "abstract": "We develop a version of G-theory for \\F_1-algebras and establish its first properties. We construct a Cartan assembly map to compare the Chu-Morava K-theory for finite pointed groups with our G-theory. We compute the G-theory groups for finite pointed groups in terms of stable homotopy of some classifying spaces. We also construct combinatorial Grayson operations on them. We discuss how our formalism is relevant to the Equivariant Nishida Problem - it asks whether there are operations on \\S^G that endow \\oplus_n\\pi_{2n}(\\S^G) with a pre-\\lambda-ring structure, where G is a finite group and \\S^G is the G-fixed point spectrum of the equivariant sphere spectrum.",
      "comment": "24 pages; v2 introduction rewritten, references added",
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2017-01-26T08:47:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20Mxx",
      "19Dxx",
      "55Pxx"
    ],
    "keywords": [
      "equivariant nishida problem",
      "finite pointed groups",
      "construct combinatorial grayson operations",
      "equivariant sphere spectrum",
      "finite group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1110.6001M"
    }
  }
}