{
  "id": "1407.7201",
  "version": "v2",
  "published": "2014-07-27T08:45:48.000Z",
  "updated": "2015-01-31T08:42:56.000Z",
  "title": "Projective Spaces and Splitting of Madsen-Tillmann Spectra",
  "authors": [
    "Takuji Kashiwabara",
    "Hadi Zare"
  ],
  "comment": "Comments are welcome",
  "categories": [
    "math.AT",
    "math.GT"
  ],
  "abstract": "We show that $BSO(2n+1)_+$ splits off $MTO(2n)$, which after localisation away from $2$, refines to a homotopy equivalence $MTO(2n)\\simeq BO(2n)_+\\simeq BSO(2n+1)_+\\simeq BSp(n)_+$ as well as $MTO(2n+1)\\simeq *$ for all $n\\geqslant0$. This reduces the problem of studying $MTO(n)$, with $n\\geqslant0$, to the study of its $2$-local version; at the prime $2$ our splitting allows to identify some algebraically independent torsion classes in homology of $\\Omega^\\infty MTO(2n)$. The classes arising from $BSO(2n+1)_+$ have been known to exist in the case of $n=1$ due to work of Randal-Williams. We also show that $BSpin(2n+1)_+$ splits off $MTPin_{-}(2n)$ which leads to existence of some algebraically independent classes in homology of $\\Omega^\\infty MTPin_{-}(2n)$, and that $BSU(n+1)_+$ splits off $MTU(n)$ when localised at $p$ with $p\\nmid(n-1)(n+1)$, hence locating some torsion classes in homology $\\Omega^\\infty MTU(n)$. As a corollary, $S^0$ splits off $MTO(2n)$, $MTU(n)$ and $MTSp(n)$ at certain primes; we have sharpened the conditions using Madsen-Tillmann-Weiss map. As an application, we find a family of algebraically independent elements among universally defined characteristic classes discussed by Randal-Williams. We also provide a nonexistence result of short exact sequences of Hopf algebras for an infinite family of Madsen-Tillmann spectra. For $n=1$, localised at the prime $2$, $MTO(2)$ splits as $BSO(3)_+\\vee\\Sigma^{-2}D(2)$, where the other summand is obtained using Steinberg idempotents which will be discussed in a subsequent work.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-07-27T08:45:48.000Z",
      "abstract": "We refine the classical splitting of $BSO(2n+1)_+$ off $BO(n)_+$ to show that $BSO(2n+1)_+$ splits off $MTO(2n)$. In complex case, we show that $BSU(n+1)_+$ splits off $MTU(n)$ whenever $p$ does not divide $(n-1)(n+1)$. As an immediate corollary, we show that $S^0$ splits off $MTO(2n)$, and $MTU(n)$ at these primes. When $n=1$, this is a special case of the splittings obtained by Steinberg idempotents which will be discussed in a subsequent work.",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-01-31T08:42:56.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "madsen-tillmann spectra",
      "projective spaces",
      "complex case",
      "immediate corollary",
      "special case"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1407.7201K"
    }
  }
}