Diffeomorphisms vs homeomorphisms of 3-manifolds - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T00:01:54Z http://mathoverflow.net/feeds/question/84532 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/84532/diffeomorphisms-vs-homeomorphisms-of-3-manifolds Diffeomorphisms vs homeomorphisms of 3-manifolds John Francis 2011-12-29T18:08:13Z 2011-12-30T05:12:05Z <p>For a smooth 3-manifold $M$, is the natural map from the space of diffeomorphisms of $M$ to the space of homeomorphisms of $M$,</p> <p>$${\sf Diff}(M) \longrightarrow {\sf Top}(M),$$</p> <p>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.)</p> http://mathoverflow.net/questions/84532/diffeomorphisms-vs-homeomorphisms-of-3-manifolds/84541#84541 Answer by Richard Kent for Diffeomorphisms vs homeomorphisms of 3-manifolds Richard Kent 2011-12-29T19:22:47Z 2011-12-29T19:31:05Z <p>$\mathsf{Diff}(S^3)$ 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. </p> <p>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 <a href="http://www.mathunion.org/ICM/ICM1978.1/Main/icm1978.1.0463.0468.ocr.pdf" rel="nofollow">Linearization in 3-Dimensional Topology</a> by Hatcher.</p>