  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}
