Smale conjecture

The Smale conjecture, named after Stephen Smale, is the statement that the diffeomorphism group of the 3-sphere has the homotopy-type of its isometry group, the orthogonal group O(4). It was proved in 1983 by Allen Hatcher.

Equivalent statements

There are several equivalent statements of the Smale conjecture. One is that the component of the unknot in the space of smooth embeddings of the circle in 3-space has the homotopy-type of the round circles, equivalently, O(3). Another equivalent statement is that the group of diffeomorphisms of the 3-ball which restrict to the identity on the boundary is contractible.

Higher dimensions

Sometimes also the (false) statement that the inclusion $O(n+1)\to {\text{Dif}}(S^{n})$ is a weak equivalence for all $n$ is meant when referring to the Smale conjecture. For $n=1$ , this is easy, for $n=2$ , Smale proved it himself.

Mostly by famous work of Kervaire and Milnor on exotic spheres, it has long been known that this fails in all dimensions at least 5.

In late 2018, Tadayuki Watanabe released a preprint that proves the failure of Smale's conjecture in the remaining 4-dimensional case.[1]

References

"Watanabe's proof of the failure of the 4-dimensional Smale conjecture" (PDF).

Stephen Smale, "Diffeomorphisms of the 2-sphere", Proceedings of the American Mathematical Society 10 (1959), 621–626. doi:10.2307/2033664 MR0112149
Allen Hatcher, "A proof of the Smale conjecture, $\scriptstyle {\mathrm {Diff} }(S^{3})\simeq {\mathrm {O} }(4)$ ", Annals of Mathematics (2) 117 (1983), no. 3, 553–607. doi:10.2307/2007035 MR0701256

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[1] "Watanabe's proof of the failure of the 4-dimensional Smale conjecture" (PDF).