TITLE: Diffeomorphisms vs homeomorphisms of 3-manifolds
QUESTION [18 upvotes]: For a smooth 3-manifold $M$, is the natural map from the space of diffeomorphisms of $M$ to the space of homeomorphisms of $M$,
$${\sf Diff}(M) \longrightarrow {\sf Top}(M),$$
a weak homotopy equivalence? Equivalently, is the space of smooth structures on a topological 3-manifold contractible? (This is as opposed to just connected, which is the usual statement of Moise's theorem.)

REPLY [26 votes]: $\mathsf{Diff}(S^3)$ is homotopy equivalent to $O(4)$, by Hatcher's solution to the Smale Conjecture: Hatcher, Allen E. A proof of the Smale conjecture, Diff(S3)≃O(4). Ann. of Math. (2) 117 (1983), no. 3, 553–607.  
Cerf proved that the Smale conjecture implies that $\mathsf{Diff}(M) \to \mathsf{Homeo}(M)$ is a weak equivalence for all $3$--manifolds $M$, in J. Cerf, Sur les difféomorphismes de la sphère de dimension trois  ( $\Gamma_4$ = 0 ), Lecture Notes in 
Math., vol. 53, Springer-Verlag, Berlin and New York, 1968, I think.  See Linearization in 3-Dimensional Topology by Hatcher.