Homotopy Hyperbolic 3-Manifolds are Hyperbolic

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: \it\noindent Let $N$ be a closed hyperbolic 3-manifold. Then \begin{enumerate} \item[(1)] If $f\colon M \to N$ is a homotopy equivalence where $M$ is a closed irreducible 3-manifold, then $f$ is homotopic to a homeomorphism. \item[(2)] If $f,g\colon M\to N$ are homotopic homeomorphisms, then $f$ is isotopic to $g$. \item[(3)] The space of hyperbolic metrics on $N$ is path connected. \end{enumerate}