Stephen Smale
Stephen Smale (born July 15, 1930) is an American mathematician, known for his research in topology, dynamical systems and mathematical economics. He was awarded the Fields Medal in 1966 and spent more than three decades on the mathematics facult ...
3-sphere
In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. It may be embedded in 4-dimensional Euclidean space as the set of points equidistant from a fixed central point. Analogous to how the boundary of a ball in three dimens ...
has the homotopy-type of its isometry group, the orthogonal group O(4). It was proved in 1983 by
Allen Hatcher Allen, Allen's or Allens may refer to:
Buildings
* Allen Arena, an indoor arena at Lipscomb University in Nashville, Tennessee
* Allen Center, a skyscraper complex in downtown Houston, Texas
* Allen Fieldhouse, an indoor sports arena on the Unive ...
.
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). 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
Three-ball (or "3-ball", colloquially) is a folk game of pool played with any three standard pool and . The game is frequently gambled upon. The goal is to () the three object balls in as few shots as possible.Kontsevich integral, a generalization of the
Gauss linking integral
In mathematics, the linking number is a numerical invariant that describes the linking of two closed curves in three-dimensional space. Intuitively, the linking number represents the number of times that each curve winds around the other. In ...
. As of 2021, the proof remains unpublished in a mathematical journal.
See also
*
Sphere bundle In the mathematical field of topology, a sphere bundle is a fiber bundle in which the fibers are spheres S^n of some dimension ''n''. Similarly, in a disk bundle, the fibers are disks D^n. From a topological perspective, there is no difference betw ...
References
External links
* {{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
''Quanta Magazine'' is an editorially independent online publication of the Simons Foundation covering developments in physics, mathematics, biology and computer science
Computer science is the study of computation, automation, and i ...