  We complete the proof of the Generalized Smale Conjecture, apart from the
case of $RP^3$, and give a new proof of Gabai's theorem for hyperbolic
3-manifolds. We use an approach based on Ricci flow through singularities,
which applies uniformly to spherical space forms other than $S^3$ and $RP^3$
and hyperbolic manifolds, to prove that the moduli space of metrics of constant
sectional curvature is contractible. As a corollary, for such a 3-manifold $X$,
the inclusion $\text{Isom} (X,g)\to \text{Diff}(X)$ is a homotopy equivalence
for any Riemannian metric $g$ of constant sectional curvature.
