{
  "id": "2203.03703",
  "version": "v1",
  "published": "2021-12-01T09:03:05.000Z",
  "updated": "2021-12-01T09:03:05.000Z",
  "title": "On the dimension of the \"cohits\" space $\\mathbb Z_2\\otimes_{\\mathcal A_2} H^{*}((\\mathbb RP(\\infty))^{\\times t}, \\mathbb Z_2)$ and some applications",
  "authors": [
    "Dang Vo Phuc"
  ],
  "comment": "6 pages. This paper is an announcement whose details will appear elsewhere. Comments are welcome! arXiv admin note: text overlap with arXiv:1907.08768",
  "categories": [
    "math.AT",
    "math.RA",
    "math.RT"
  ],
  "abstract": "We denote by $\\mathbb Z_2$ the prime field of two elements and by $P_t = \\mathbb Z_2[x_1, \\ldots, x_t]$ the polynomial algebra of $t$ generators $x_1, \\ldots, x_t$ with the degree of each $x_i$ being one. Let $\\mathcal A_2$ be the Steenrod algebra over $\\mathbb Z_2.$ A central problem of homotopy theory is to determine a minimal set of generators for the \"cohits\" space $\\mathbb Z_2\\otimes_{\\mathcal A_2} P_t.$ This problem, which is called the \"hit\" problem for Steenrod algebra, has been systematically studied for $t\\leq 4.$ The present paper is devoted to the investigation of the structure of $\\mathbb Z_2\\otimes_{\\mathcal A_2} P_t$ in some certain \"generic\" degrees. More specifically, we explicitly determine a monomial basis of $\\mathbb Z_2\\otimes_{\\mathcal A_2} P_5$ in degree $n_s=5(2^{s}-1) + 42.2^{s}$ for every non-negative integer $s.$ As a result, it confirms Sum's conjecture [14] for a relation between the minimal sets of $\\mathcal A_2$-generators of the algebras $P_{t-1}$ and $P_{t}$ in the case $t=5$ and degree $n_s$. As applications, we obtain the dimension of $\\mathbb Z_2\\otimes_{\\mathcal A_2} P_6$ in the generic degree $5(2^{s+5}-1) + n_0.2^{s+5}$ for all $s\\geq 0,$ and show that the Singer's cohomological transfer [11] is an isomorphism in bidegree $(5, 5+n_s)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-12-01T09:03:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "55Q45",
      "55S10",
      "55S05",
      "55T15",
      "55R12"
    ],
    "keywords": [
      "applications",
      "steenrod algebra",
      "minimal set",
      "generators",
      "confirms sums conjecture"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 6,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}