## On the geometric and topological rigidity of hyperbolic 3-manifolds (1997)

Venue: | J. Amer. Math. Soc |

Citations: | 31 - 2 self |

### BibTeX

@ARTICLE{Gabai97onthe,

author = {David Gabai},

title = {On the geometric and topological rigidity of hyperbolic 3-manifolds},

journal = {J. Amer. Math. Soc},

year = {1997},

pages = {37--74}

}

### Years of Citing Articles

### OpenURL

### Abstract

Abstract. A homotopy equivalence between a hyperbolic 3-manifold and a closed irreducible 3-manifold is homotopic to a homeomorphsim provided the hyperbolic manifold satisfies a purely geometric condition. There are no known examples of hyperbolic 3-manifolds which do not satisfy this condition. One of the central problems of 3-manifold topology is to determine when a homotopy equivalence between two closed orientable irreducible 3-manifolds is homotopic to a homeomorphism. If one of these manifolds is S 3, then this is Poincaré’s problem. The results of [Re], [Fr], [Ru], [Bo], and [HR] (see also [Ol]) completely solve this problem for maps between lens spaces. In particular there exist nonhomeomorphic but homotopy equivalent lens spaces (e.g. L(7,1) and L(7,2)), and there exist self-homotopy equivalences not homotopic to homeomorphisms (e.g. the self-homotopy equivalence of L(8,1) whose π1-map is multiplication by 3). By Waldhausen [W] (resp. Scott [S]) a homotopy equivalence between a closed Haken 3-manifold (resp. a Seifert-fibred space with infinite π1) and an irreducible 3-manifold can be homotoped to a homeomorphism. By Mostow [M] a homotopy equivalence between two closed hyperbolic 3-manifolds can be homotoped to a homeomorphism and in fact an isometry. However, the general case of homotopy equivalence between a hyperbolic 3-manifold and an irreducible 3-manifold remains to be investigated. These problems and results should be contrasted with the conjecture [T] that a closed irreducible orientable 3-manifold is either Haken, or Seifert fibred with infinite π1, or the quotient of S 3 by an orthogonal action, or the quotient of H 3 via a cocompact group of hyperbolic isometries. Theorem 1 [G2]. Let N be a closed, orientable, hyperbolic 3-manifold containing an embedded hyperbolic tube of radius (log 3)/2 =.549306... about a closed geodesic. Then: (i) If f: M → N is a homotopy equivalence where M is an irreducible 3-manifold, then f is homotopic to a homeomorphism. (ii) If f, g: M → N are homotopic homeomorphisms, then f is isotopic to g. (iii) The space of hyperbolic metrics on N is path connected.