Smith conjecture
Encyclopedia
In mathematics
, the Smith conjecture states that if f is a diffeomorphism
of the 3-sphere
, of finite order
then the fixed point set of f cannot be a nontrivial knot.
showed that a non-trivial orientation-preserving diffeomorphism of finite order must have fixed point set equal to a circle, and asked in if the fixed point set can be knotted. proved the Smith conjecture for the special case of diffeomorphisms of order 2 (and hence any even order). The proof of the general case was described by and depended on several major advances in 3-manifold
theory, in particular the work of William Thurston
on hyperbolic structures
on 3-manifolds, and results by William Meeks and Shing-Tung Yau
on minimal surface
s in 3-manifolds, with some additional help from Hyman Bass
, Cameron Gordon
, Shalen, and Litherland.
gave an example of a continuous involution of the 3-sphere whose fixed point set is a wildly embedded circle, so the Smith conjecture is false in the topological (rather than the smooth or PL) category.
showed that the analogue of the Smith conjecture in higher dimensions is false: the fixed point set of a periodic diffeomorphism of a sphere of dimension at least 4 can be a knotted sphere of codimension 2.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Smith conjecture states that if f is a diffeomorphism
Diffeomorphism
In mathematics, a diffeomorphism is an isomorphism in the category of smooth manifolds. It is an invertible function that maps one differentiable manifold to another, such that both the function and its inverse are smooth.- Definition :...
of the 3-sphere
3-sphere
In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. It consists of the set of points equidistant from a fixed central point in 4-dimensional Euclidean space...
, of finite order
Order (group theory)
In group theory, a branch of mathematics, the term order is used in two closely related senses:* The order of a group is its cardinality, i.e., the number of its elements....
then the fixed point set of f cannot be a nontrivial knot.
showed that a non-trivial orientation-preserving diffeomorphism of finite order must have fixed point set equal to a circle, and asked in if the fixed point set can be knotted. proved the Smith conjecture for the special case of diffeomorphisms of order 2 (and hence any even order). The proof of the general case was described by and depended on several major advances in 3-manifold
3-manifold
In mathematics, a 3-manifold is a 3-dimensional manifold. The topological, piecewise-linear, and smooth categories are all equivalent in three dimensions, so little distinction is made in whether we are dealing with say, topological 3-manifolds, or smooth 3-manifolds.Phenomena in three dimensions...
theory, in particular the work of William Thurston
William Thurston
William Paul Thurston is an American mathematician. He is a pioneer in the field of low-dimensional topology. In 1982, he was awarded the Fields Medal for his contributions to the study of 3-manifolds...
on hyperbolic structures
Hyperbolic 3-manifold
A hyperbolic 3-manifold is a 3-manifold equipped with a complete Riemannian metric of constant sectional curvature -1. In other words, it is the quotient of three-dimensional hyperbolic space by a subgroup of hyperbolic isometries acting freely and properly discontinuously...
on 3-manifolds, and results by William Meeks and Shing-Tung Yau
Shing-Tung Yau
Shing-Tung Yau is a Chinese American mathematician working in differential geometry. He was born in Shantou, Guangdong Province, China into a family of scholars from Jiaoling, Guangdong Province....
on minimal surface
Minimal surface
In mathematics, a minimal surface is a surface with a mean curvature of zero.These include, but are not limited to, surfaces of minimum area subject to various constraints....
s in 3-manifolds, with some additional help from Hyman Bass
Hyman Bass
Hyman Bass is an American mathematician, known for work in algebra and in mathematics education. From 1959-1998 he was Professor in the Mathematics Department at Columbia University, where he is now professor emeritus...
, Cameron Gordon
Cameron Gordon (mathematician)
Cameron Gordon is a Professor and Sid W. Richardson Foundation Regents Chair in the Department of mathematics at the University of Texas at Austin, known for his work in knot theory. Among his notable results is his work with Marc Culler, John Luecke, and Peter Shalen on the cyclic surgery theorem...
, Shalen, and Litherland.
gave an example of a continuous involution of the 3-sphere whose fixed point set is a wildly embedded circle, so the Smith conjecture is false in the topological (rather than the smooth or PL) category.
showed that the analogue of the Smith conjecture in higher dimensions is false: the fixed point set of a periodic diffeomorphism of a sphere of dimension at least 4 can be a knotted sphere of codimension 2.