Smale conjecture

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). Interestingly, this statement is not equivalent to the generalized Smale Conjecture, in higher dimensions.

Another equivalent statement is that the group of diffeomorphisms of the 3-ball which restrict to the identity on the boundary is contractible.

Yet another equivalent statement is that the space of constant-curvature Riemann metrics on the 3-sphere is contractible.

The (false) statement that the inclusion $O(n+1)\; \backslash to\; \backslash text\{Diff\}(S^n)$ is a weak equivalence for all $n$ is sometimes meant when referring to the generalized Smale conjecture. For $n\; =\; 1$, this is classical, for $n\; =\; 2$, Smale proved it himself.{{Cite journal|last=Smale|first=Stephen|date=August 1959|title=Diffeomorphisms of the 2-Sphere|journal=Proceedings of the American Mathematical Society|volume=10|issue=4|pages=621–626|doi=10.2307/2033664|jstor=2033664}}

For $n\backslash ge5$ the conjecture is false due to the failure of $\backslash text\{Diff\}(S^n)/O(n+1)$ to be contractible.{{Cite web|first=Allen|last=Hatcher|date=2012|title=A 50 -Year View of Diffeomorphism Groups|url=https://pi.math.cornell.edu/~hatcher/Papers/Diff%28M%292012.pdf|journal=}}

In late 2018, Tadayuki Watanabe released a preprint that proves the failure of Smale's conjecture in the remaining 4-dimensional case{{cite arXiv|last=Watanabe|first=Tadayuki|date=2019-08-19|title=Some exotic nontrivial elements of the rational homotopy groups of Diff(S⁴)|class=math.GT |eprint=1812.02448}} 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 As of 2021, the proof remains unpublished in a mathematical journal.

• Sphere bundle

{{Reflist}}

• {{Cite web|last=Hartnett|first=Kevin|date=2021-10-26|title=How Tadayuki Watanabe Disproved a Major Conjecture About Spheres|url=https://www.quantamagazine.org/how-tadayuki-watanabe-solved-a-topological-mystery-about-spheres-20211026/|website=Quanta Magazine|language=en}}

Category:Smooth manifolds

Category:Low-dimensional topology

Category:Theorems in topology

Category:Conjectures that have been proved