Homotopy Equivalences of 3-Manifolds and Deformation Theory of Kleinian Groups

It is well known that some compact $3$-manifolds with boundary admit homotopy equivalences that are not homotopic to homeomorphisms. This title investigates a natural question arising in the topological theory of $3$-manifolds, and applies the results to give new information about the deformation theory of hyperbolic $3$-manifolds.