{
  "id": "1608.02185",
  "version": "v1",
  "published": "2016-08-07T06:03:34.000Z",
  "updated": "2016-08-07T06:03:34.000Z",
  "title": "Half dimensional collapse of ends of manifolds of nonpositive curvature",
  "authors": [
    "Grigori Avramidi",
    "T. Tam Nguyen Phan"
  ],
  "comment": "39 pages, 13 figures",
  "categories": [
    "math.GT",
    "math.DG",
    "math.MG"
  ],
  "abstract": "We study the topology of ends of noncompact, complete Riemannian $n$-manifolds $M$ with bounded nonpositive sectional curvature and finite volume. It is known that if such a manifold is negatively curved or does not contain arbitrarily small geodesic loops, then it is tame in the sense that it is homeomorphic to the interior of a compact manifold $\\overline{M}$ with boundary $\\partial\\overline{M}$ since the thin part $M_{<\\epsilon}$, i.e. the end of $M$, is topologically a product of a closed manifold with a ray. Let $\\widetilde{M}_{<\\epsilon}$ be a lift of $M_{<\\epsilon}$ in the universal cover $\\widetilde{M}$. We show that in this case, any finite polyhedron $P$ in $\\widetilde{M}_{<\\epsilon}$ can be homotoped within $\\widetilde{M}_{<\\epsilon}$ to factor through a polyhedron $Q$ of dimension less than $ \\lfloor n/2\\rfloor$. A corollary of this is that the homology of $\\widetilde{M}_{<\\epsilon}$ vanishes in dimension greater or equal to $ \\lfloor n/2\\rfloor$. Another corollary is that when $M$ has dimension less than $6$ each component of the boundary $\\partial \\overline{M}$ is aspherical. A third corollary is that any complex homotopy equivalent to $M$ has dimension greater or equal to $\\lceil n/2\\rceil$. These bounds are sharp by examples such as products of noncompact hyperbolic surfaces.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-08-07T06:03:34.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "half dimensional collapse",
      "nonpositive curvature",
      "dimension greater",
      "contain arbitrarily small geodesic loops",
      "complex homotopy equivalent"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 39,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}