{
  "id": "math/0211104",
  "version": "v1",
  "published": "2002-11-06T09:55:05.000Z",
  "updated": "2002-11-06T09:55:05.000Z",
  "title": "Homotopy types of the components of spaces of embeddings of compact polyhedra into 2-manifolds",
  "authors": [
    "Tatsuhiko Yagasaki"
  ],
  "comment": "31 pages, 3 figures",
  "categories": [
    "math.GT",
    "math.GN"
  ],
  "abstract": "Suppose M is a connected PL 2-manifold and X is a compact connected subpolyhedron of M (X \\neq 1pt, a closed 2-manifold). Let E(X, M) denote the space of topological embeddings of X into M with the compact-open topology and let E(X, M)_0 denote the connected component of the inclusion i_X : X \\subset M in E(X, M). In this paper we classify the homotopy type of E(X, M)_0 in term of the subgroup G = Im[{i_X}_\\ast : \\pi_1(X) \\to \\pi_1(M)]. We show that if G is not a cyclic group and M \\neq T^2, T^2 then E(X, M)_0 \\simeq \\ast, if G is a nontrivial cyclic group and M \\neq P^2, T^2, K^2 then E(X, M)_0 \\simeq S^1, and when G = 1, if X is an arc or M is orientable then E(X, M)_0 \\simeq ST(M) and if X is not an arc and M is nonorientable then E(X, M)_0 \\simeq ST(\\tilde{M}). Here S^1 is the circle, T^2 is the torus, P^2 is the projective plane and K^2 is the Klein bottle. The symbol ST(M) denotes the tangent unit circle bundle of M with respect to any Riemannian metric of M and \\tilde{M} denotes the orientation double cover of M.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2002-11-06T09:55:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57N05",
      "57N20",
      "57N35"
    ],
    "keywords": [
      "homotopy type",
      "compact polyhedra",
      "embeddings",
      "tangent unit circle bundle",
      "nontrivial cyclic group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 31,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2002math.....11104Y"
    }
  }
}