This text 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. It is well known
that some compact $3$-manifolds with boundary admit homotopy
equivalences that are not homotopic to homeomorphisms. We investigate when the
subgroup $\mathcal{R}(M)$ of outer automorphisms of
$\pi_1(M)$ which are induced by homeomorphisms of a compact
$3$-manifold $M$ has finite index in the group
$\operatorname{Out}(\pi_1(M))$ of all outer automorphisms. This
question is completely resolved for Haken $3$-manifolds. It is also
resolved for many classes of reducible $3$-manifolds and
$3$-manifolds with boundary patterns, including all pared
$3$-manifolds.

The components of the interior $\operatorname{GF}(\pi_1(M))$ of the
space $\operatorname{AH}(\pi_1(M))$ of all (marked) hyperbolic
$3$-manifolds homotopy equivalent to $M$ are enumerated by
the marked homeomorphism types of manifolds homotopy equivalent to
$M$, so one may apply the topological results above to study the
topology of this deformation space. We show that
$\operatorname{GF}(\pi_1(M))$ has finitely many components if and only
if either $M$ has incompressible boundary, but no “double trouble,”
or $M$ has compressible boundary and is “small.” (A
hyperbolizable $3$-manifold with incompressible boundary has double
trouble if and only if there is a thickened torus component of its
characteristic submanifold which intersects the boundary in at least two
annuli.) More generally, the deformation theory of hyperbolic structures on
pared manifolds is analyzed.

Some expository sections detail Johannson's formulation of the
Jaco-Shalen-Johannson characteristic submanifold theory, the topology of
pared $3$-manifolds, and the deformation theory of hyperbolic
$3$-manifolds. An epilogue discusses related open problems and recent
progress in the deformation theory of hyperbolic $3$-manifolds.

Readership

Graduate students and research mathematicians interested in
geometry and topology.