{
  "id": "2003.06033",
  "version": "v1",
  "published": "2020-03-12T21:56:23.000Z",
  "updated": "2020-03-12T21:56:23.000Z",
  "title": "Smooth approximation for classifying spaces of diffeomorphism groups",
  "authors": [
    "Yong-Geun Oh",
    "Hiro Lee Tanaka"
  ],
  "comment": "17 pages. Comments welcome! Portions of this work previously appeared in arXiv:1911.00349v2; that previous work has been split into multiple papers (including this one) to better explicate the ingredients",
  "categories": [
    "math.AT",
    "math.DG",
    "math.SG"
  ],
  "abstract": "We prove a smooth approximation theorem for classifying spaces of certain infinite-dimensional smooth groups. More precisely, using the framework of diffeological spaces, we show that the smooth singular complex of a classifying space BG is weakly homotopy equivalent to the (continuous) singular complex of BG when G is a diffeomorphism group of a compact smooth manifold. In particular, the smooth homotopy groups of BG are naturally isomorphic to the usual (continuous) homotopy groups of BG. On top of a computation of homotopy groups, our methods yield a way to construct homotopically coherent actions of G using infinity-categorical techniques. We discuss some generalizations and consequences of this result with an eye toward [OT19], where we show that higher homotopy groups of symplectic automorphism groups map to Fukaya-categorical invariants, and where we prove a conjecture of Teleman from the 2014 ICM in the Liouville and monotone settings.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-03-12T21:56:23.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "classifying space",
      "diffeomorphism group",
      "symplectic automorphism groups map",
      "infinite-dimensional smooth groups",
      "smooth homotopy groups"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}