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.)



Diffeomorphisms vs homeomorphisms of 3-manifolds
$\mathsf{Diff}(S^3)$ 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](http://www.mathunion.org/ICM/ICM1978.1/Main/icm1978.1.0463.0468.ocr.pdf) by Hatcher.


