{
  "id": "2103.04393",
  "version": "v1",
  "published": "2021-03-07T16:30:19.000Z",
  "updated": "2021-03-07T16:30:19.000Z",
  "title": "A note on the hit problem for the Steenrod algebra and its applications",
  "authors": [
    "Nguyen Khac Tin"
  ],
  "comment": "9 pages. arXiv admin note: substantial text overlap with arXiv:1609.02250; substantial text overlap with arXiv:1609.03006 by other authors",
  "categories": [
    "math.AT"
  ],
  "abstract": "Let $P_{k}=H^{*}((\\mathbb{R}P^{\\infty})^{k})$ be the modulo-$2$ cohomology algebra of the direct product of $k$ copies of infinite dimensional real projective spaces $\\mathbb{R}P^{\\infty}$. Then, $P_{k}$ is isomorphic to the graded polynomial algebra $\\mathbb{F}_{2}[x_{1},\\ldots,x_{k}]$ of $k$ variables, in which each $x_{j}$ is of degree 1, and let $GL_k$ be the general linear group over the prime field $\\mathbb{F}_2$ which acts naturally on $P_k$. Here the cohomology is taken with coefficients in the prime field $\\mathbb F_2$ of two elements. We study the {\\it hit problem}, set up by Frank Peterson, of finding a minimal set of generators for the polynomial algebra $P_k$ as a module over the mod-2 Steenrod algebra, $\\mathcal{A}$. In this Note, we explicitly compute the hit problem for $k = 5$ and the degree $5(2^s-1)+24.2^s$ with $s$ an arbitrary non-negative integer. These results are used to study the Singer algebraic transfer which is a homomorphism from the homology of the mod-$2$ Steenrod algebra, $\\mbox{Tor}^{\\mathcal{A}}_{k, k+n}(\\mathbb{F}_2, \\mathbb{F}_2),$ to the subspace of $\\mathbb{F}_2\\otimes_{\\mathcal{A}}P_k$ consisting of all the $GL_k$-invariant classes of degree $n.$ We show that Singer's conjecture for the algebraic transfer is true in the case $k=5$ and the above degrees. This method is different from that of Singer in studying the image of the algebraic transfer. Moreover, as a consequence, we get the dimension results for polynomial algebra in some generic degrees in the case $k=6.$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-03-07T16:30:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "55S10",
      "55S05",
      "55T15"
    ],
    "keywords": [
      "steenrod algebra",
      "hit problem",
      "polynomial algebra",
      "prime field",
      "applications"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}