HOMOTOPY HYPERBOLIC 3-MANIFOLDS ARE HYPERBOLIC

DAVID GABAI G. ROBERT MEYERHOFF NATHANIEL THURSTON

Abstract. This paper introduces a rigorous computer-assisted procedure for analyzing hyperbolic 3-manifolds. This technique is used to complete the proof of several long-standing rigidity conjectures in 3-manifold theory as well as to provide a new lower bound for the volume of a closed orientable hyperbolic 3-manifold.

We prove the following result: Let N be a closed hyperbolic 3-manifold. Then

(1) If f: M → N is a homotopy equivalence where M is a closed irreducible 3-manifold, then f is homotopic to a homeomorphism.

(2) If f, g: M → N are homotopic homeomorphisms, then f is isotopic to g.

(3) The space of hyperbolic metrics on N is path connected.