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. 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. 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 statement that the inclusion is a weak equivalence for all is meant when referring to the Smale conjecture. For, this is easy, for, 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.