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.^[1]

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 ${\displaystyle O(n+1)\to {\text{Diff}}(S^{n})}$ is a weak equivalence for all ${\displaystyle n}$ is meant when referring to the Smale conjecture. For ${\displaystyle n=1}$, this is easy, for ${\displaystyle n=2}$, Smale proved it himself.^[2]

For ${\displaystyle n\geq 5}$ the conjecture is false due to the failure of ${\displaystyle {\text{Diff}}(S^{n})}$ to be contractible^[3]

In late 2018, Tadayuki Watanabe released a preprint that proves the failure of Smale's conjecture in the remaining 4-dimensional case^[4] relying on work around the Kontsevich integral, a generalization of the Gauss linking integral. As of 2021, the proof remains unpublished in a mathematical journal.

See also[]

• Sphere bundle

References[]

External links[]

• Hartnett, Kevin (2021-10-26). "How Tadayuki Watanabe Disproved a Major Conjecture About Spheres". Quanta Magazine.