{
  "id": "math/0406081",
  "version": "v2",
  "published": "2004-06-04T12:05:21.000Z",
  "updated": "2005-07-14T12:35:01.000Z",
  "title": "Differentials in the homological homotopy fixed point spectral sequence",
  "authors": [
    "Robert R. Bruner",
    "John Rognes"
  ],
  "comment": "Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol5/agt-5-27.abs.html",
  "journal": "Algebr. Geom. Topol. 5 (2005) 653-690",
  "categories": [
    "math.AT"
  ],
  "abstract": "We analyze in homological terms the homotopy fixed point spectrum of a T-equivariant commutative S-algebra R. There is a homological homotopy fixed point spectral sequence with E^2_{s,t} = H^{-s}_{gp}(T; H_t(R; F_p)), converging conditionally to the continuous homology H^c_{s+t}(R^{hT}; F_p) of the homotopy fixed point spectrum. We show that there are Dyer-Lashof operations beta^epsilon Q^i acting on this algebra spectral sequence, and that its differentials are completely determined by those originating on the vertical axis. More surprisingly, we show that for each class x in the $^{2r}-term of the spectral sequence there are 2r other classes in the E^{2r}-term (obtained mostly by Dyer-Lashof operations on x) that are infinite cycles, i.e., survive to the E^infty-term. We apply this to completely determine the differentials in the homological homotopy fixed point spectral sequences for the topological Hochschild homology spectra R = THH(B) of many S-algebras, including B = MU, BP, ku, ko and tmf. Similar results apply for all finite subgroups C of T, and for the Tate- and homotopy orbit spectral sequences. This work is part of a homological approach to calculating topological cyclic homology and algebraic K-theory of commutative S-algebras.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2005-07-14T12:35:01.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "19D55",
      "55S12",
      "55T05",
      "55P43",
      "55P91"
    ],
    "keywords": [
      "homotopy fixed point spectral sequence",
      "homological homotopy fixed point spectral",
      "homotopy fixed point spectrum",
      "differentials",
      "homotopy orbit spectral sequences"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}