{
  "id": "1711.05708",
  "version": "v1",
  "published": "2017-11-15T18:07:32.000Z",
  "updated": "2017-11-15T18:07:32.000Z",
  "title": "Steinberg Summands and Symmetric Powers of the G-Sphere",
  "authors": [
    "Krishanu Roy Sankar"
  ],
  "categories": [
    "math.AT"
  ],
  "abstract": "Let $G$ be a finite abelian $p$-group. We use the symmetric powers of the $G$-equivariant sphere spectrum to obtain a filtration for $H\\underline{\\mathbb{F}}_p$, the Eilenberg-Maclane spectrum for the constant Mackey functor $\\underline{\\mathbb{F}}_p$. Our main theorem is that there is an equivalence between the $k$-th cofiber of this filtration and the Steinberg summand of the $G$-equivariant classifying space of $(\\mathbb{Z}/p)^k$. We also show that when one smashes with $H\\underline{\\mathbb{F}}_p$, the filtration splits into its associated graded. In a future paper, we will use this result to compute the equivariant dual Steenrod algebra $H\\underline{\\mathbb{F}}_p\\wedge H\\underline{\\mathbb{F}}_p$ at odd primes via explicit cellular constructions of these equivariant classifying spaces.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-11-15T18:07:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "55P91",
      "55S15",
      "55S10",
      "55P42",
      "55N91"
    ],
    "keywords": [
      "steinberg summand",
      "symmetric powers",
      "equivariant classifying space",
      "equivariant dual steenrod algebra",
      "equivariant sphere spectrum"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}