{
  "id": "1908.04420",
  "version": "v1",
  "published": "2019-08-12T22:07:36.000Z",
  "updated": "2019-08-12T22:07:36.000Z",
  "title": "Positive scalar curvature on stratified spaces, I: the simply connected case",
  "authors": [
    "Boris Botvinnik",
    "Paolo Piazza",
    "Jonathan Rosenberg"
  ],
  "comment": "27 pages",
  "categories": [
    "math.DG",
    "math.KT"
  ],
  "abstract": "Let $M_\\Sigma$ be an $n$-dimensional Thom-Mather stratified space of depth $1$. We denote by $\\beta M$ the singular locus and by $L$ the associated link. In this paper we study the problem of when such a space can be endowed with a wedge metric of positive scalar curvature. We relate this problem to recent work on index theory on stratified spaces, giving first an obstruction to the existence of such a metric in terms of a wedge $\\alpha$-class $\\alpha_w (M_\\Sigma)\\in KO_n$. In order to establish a sufficient condition we need to assume additional structure: we assume that the link of $M_\\Sigma$ is a homogeneous space of positive scalar curvature, $L=G/K$, where the semisimple compact Lie group $G$ acts transitively on $L$ by isometries. Examples of such manifolds include compact semisimple Lie groups and Riemannian symmetric spaces of compact type. Under these assumptions, when $M_\\Sigma$ and $\\beta M$ are spin, we reinterpret our obstruction in terms of two $\\alpha$-classes associated to the resolution of $M_\\Sigma$, $M$, and to the singular locus $\\beta M$. Finally, when $M_\\Sigma$, $\\beta M$, $L$, and $G$ are simply connected and $\\dim M$ is big enough, and when some other conditions on $L$ (satisfied in a large number of cases) hold, we establish the main result of this article, showing that the vanishing of these two $\\alpha$-classes is also sufficient for the existence of a well-adapted wedge metric of positive scalar curvature.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-08-12T22:07:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53C21",
      "58J22",
      "53C27",
      "19L41",
      "55N22"
    ],
    "keywords": [
      "positive scalar curvature",
      "simply connected case",
      "singular locus",
      "semisimple compact lie group",
      "wedge metric"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}