{
  "id": "1808.00604",
  "version": "v1",
  "published": "2018-08-02T00:10:14.000Z",
  "updated": "2018-08-02T00:10:14.000Z",
  "title": "The $\\mathrm{RO}(G)$-Graded Cohomology of the Equivariant Classifying Space $B_G\\mathrm{SU}(2)$",
  "authors": [
    "Zev Chonoles"
  ],
  "categories": [
    "math.AT"
  ],
  "abstract": "We compute the additive structure of the $\\mathrm{RO}(C_n)$-graded Bredon equivariant cohomology of the equivariant classifying space $B_{C_n}\\mathrm{SU}(2)$, for any $n$ that is either prime or a product of distinct odd primes, and we also compute its multiplicative structure for $n=2$. In particular, as an algebra over the cohomology of a point, we show that the cohomology of $B_{C_2}\\mathrm{SU}(2)$ is generated by two elements subject to a single relation: writing $\\sigma$ for the sign representation of $C_2$ in $\\mathrm{RO}(C_2)$, the generators are an element $c$ in dimension $4\\sigma$ and an element $C$ in dimension $4+4\\sigma$, satisfying the relation $c^2 = \\epsilon^4 c + \\xi^2 C$, where $\\epsilon$ and $\\xi$ are elements of the cohomology of a point. Throughout, we take coefficients in the Burnside ring Mackey functor $A$. The key tools used are equivariant \"even-dimensional freeness\" and \"multiplicative comparison\" theorems for $G$-cell complexes, both proven by Lewis in [Lew88] and subsequently refined by Shulman in [Shu10], and with the former theorem extended by Basu and Ghosh in [BG16]. The latter theorem enables us to compute the multiplicative structure of the cohomology of $B_{C_2}\\mathrm{SU}(2)$ by embedding it in a direct sum of cohomology rings whose structure is more easily understood. Both theorems require the cells of the $G$-cell complex to be attached in a well-behaved order, and a significant step in our work is to give $B_{C_n}\\mathrm{SU}(2)$ a satisfactory $C_n$-cell complex structure.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-08-02T00:10:14.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "55N91",
      "55R40"
    ],
    "keywords": [
      "equivariant classifying space",
      "graded cohomology",
      "cell complex structure",
      "multiplicative structure",
      "graded bredon equivariant cohomology"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}