{
  "id": "1904.01708",
  "version": "v1",
  "published": "2019-04-02T23:28:12.000Z",
  "updated": "2019-04-02T23:28:12.000Z",
  "title": "Symmetric Powers and Eilenberg--Maclane Spectra",
  "authors": [
    "Krishanu Sankar"
  ],
  "comment": "47 pages",
  "categories": [
    "math.AT"
  ],
  "abstract": "We filter the equivariant Eilenberg Maclane spectrum $H\\underline{\\mathbb{F}}_p$ using the mod $p$ symmetric powers of the equivariant sphere spectrum, $\\mathrm{Sp}_{\\mathbb{Z}/p}^{\\infty}(\\Sigma^{\\infty G}S^0)$. When $G$ is a $p$-group, we show that the layers in the filtration are the Steinberg summands of the equivariant classifying spaces of $(\\mathbb{Z}/p)^n$ for $n=0, 1, 2, \\ldots$. We show that the layers of the filtration split after smashing with $H\\underline{\\mathbb{F}}_p$. Along the way, we produced a general computation of the geometric fixed points of $H\\underline{\\mathbb{Z}}$ and $H\\underline{\\mathbb{F}}_p$ by using symmetric powers.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-04-02T23:28:12.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "symmetric powers",
      "eilenberg-maclane spectra",
      "equivariant eilenberg maclane spectrum",
      "equivariant sphere spectrum",
      "filtration split"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 47,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}