One of the unresolved questions about 3-manifolds is the generalized Smale conjecture, which roughly interpreted asks for the homotopy type of the space of diffeomorphisms of a 3-manifold. Smale originally conjectured that $Diff(S^3)\simeq O(4)$, and this was proven by Hatcher. He also worked out the homotopy type of diffeomorphisms of Haken 3-manifolds. Another interpretation of Smale's question is that the space of round (constant sectional curvature $=1$) metrics on $S^3$ is contractible. Gabai proved the analogous statement that the space of hyperbolic metrics on a hyperbolic 3-manifold is contractible, and recently McCullough and Soma have dealt many small (non-Haken) Seifert-fibered spaces. However, the case of the generalized Smale conjecture for elliptic manifolds is still open (see however the work of Hong et. al.). I think this is an important open question, and it would be useful to have a unified proof of these results (in particular, Gabai's results makes use of a computer-aided proof of the existence of "non-coalescable insulator families").
One of the unresolved questions about 3-manifolds is the generalized Smale conjecture, which roughly interpreted asks for the homotopy type of the space of diffeomorphisms of a 3-manifold. Smale originally conjectured that $Diff(S^3)\simeq O(4)$, and this was proven by Hatcher. He also worked out the homotopy type of diffeomorphisms of Haken 3-manifolds. Another interpretation of Smale's question is that the space of round (constant sectional curvature $=1$) metrics on $S^3$ is contractible. Gabai proved the analogous statement that the space of hyperbolic metrics on a hyperbolic 3-manifold is contractible, and recently McCullough and Soma have dealt many small (non-Haken) Seifert-fibered spaces. However, the case of the generalized Smale conjecture for elliptic manifolds is still open. I think this is an important open question, and it would be useful to have a unified proof of these results (in particular, Gabai's results makes use of a computer-aided proof of the existence of "non-coalescable insulator families").