{
  "id": "math/0209077",
  "version": "v3",
  "published": "2002-09-07T04:56:27.000Z",
  "updated": "2002-12-17T22:32:28.000Z",
  "title": "Finite subset spaces of S^1",
  "authors": [
    "Christopher Tuffley"
  ],
  "comment": "Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol2/agt-2-43.abs.html",
  "journal": "Algebr. Geom. Topol. 2 (2002) 1119-1145",
  "categories": [
    "math.GT"
  ],
  "abstract": "Given a topological space X denote by exp_k(X) the space of non-empty subsets of X of size at most k, topologised as a quotient of X^k. This space may be regarded as a union over 0 < l < k+1 of configuration spaces of l distinct unordered points in X. In the special case X=S^1 we show that: (1) exp_k(S^1) has the homotopy type of an odd dimensional sphere of dimension k or k-1; (2) the natural inclusion of exp_{2k-1}(S^1) h.e. S^{2k-1} into exp_2k(S^1) h.e. S^{2k-1} is multiplication by two on homology; (3) the complement exp_k(S^1)-exp_{k-2}(S^1) of the codimension two strata in exp_k(S^1) has the homotopy type of a (k-1,k)-torus knot complement; and (4) the degree of an induced map exp_k(f): exp_k(S^1)-->exp_k(S^1) is (deg f)^[(k+1)/2] for f: S^1-->S^1. The first three results generalise known facts that exp_2(S^1) is a Moebius strip with boundary exp_1(S^1), and that exp_3(S^1) is the three-sphere with exp_1(S^1) inside it forming a trefoil knot.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2002-12-17T22:32:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "54B20",
      "55Q52",
      "57M25"
    ],
    "keywords": [
      "finite subset spaces",
      "non-empty subsets",
      "results generalise",
      "moebius strip",
      "trefoil knot"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2002math......9077T"
    }
  }
}