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- Smith_conjecture abstract "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.Smith (1939, remark after theorem 4) showed that a non-trivial orientation-preserving diffeomorphism of finite order with fixed points must have fixed point set equal to a circle, and asked in (Eilenberg 1949, Problem 36) if the fixed point set can be knotted. Waldhausen (1969) 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 Morgan & Bass (1984) 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 surfaces in 3-manifolds, with some additional help from Hyman Bass, Cameron Gordon, Shalen, and Litherland.Montgomery & Zippin (1954) 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. Giffen (1966) 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.".
- Smith_conjecture wikiPageExternalLink books?id=sXcwg4zi1_EC.
- Smith_conjecture wikiPageExternalLink 1968950.
- Smith_conjecture wikiPageExternalLink 1969448.
- Smith_conjecture wikiPageExternalLink 2031959.
- Smith_conjecture wikiPageExternalLink 2373054.
- Smith_conjecture wikiPageID "4476828".
- Smith_conjecture wikiPageRevisionID "606444250".
- Smith_conjecture authorlink "Paul A. Smith".
- Smith_conjecture hasPhotoCollection Smith_conjecture.
- Smith_conjecture last "Smith".
- Smith_conjecture loc "remark after theorem 4".
- Smith_conjecture year "1939".
- Smith_conjecture subject Category:3-manifolds.
- Smith_conjecture subject Category:Conjectures.
- Smith_conjecture subject Category:Diffeomorphisms.
- Smith_conjecture subject Category:Theorems_in_topology.
- Smith_conjecture type Abstraction100002137.
- Smith_conjecture type Cognition100023271.
- Smith_conjecture type Communication100033020.
- Smith_conjecture type Concept105835747.
- Smith_conjecture type Conjectures.
- Smith_conjecture type Content105809192.
- Smith_conjecture type Hypothesis105888929.
- Smith_conjecture type Idea105833840.
- Smith_conjecture type Message106598915.
- Smith_conjecture type Proposition106750804.
- Smith_conjecture type PsychologicalFeature100023100.
- Smith_conjecture type Speculation105891783.
- Smith_conjecture type Statement106722453.
- Smith_conjecture type Theorem106752293.
- Smith_conjecture type TheoremsInTopology.
- Smith_conjecture comment "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.Smith (1939, remark after theorem 4) showed that a non-trivial orientation-preserving diffeomorphism of finite order with fixed points must have fixed point set equal to a circle, and asked in (Eilenberg 1949, Problem 36) if the fixed point set can be knotted.".
- Smith_conjecture label "Smith conjecture".
- Smith_conjecture sameAs m.0c4m5s.
- Smith_conjecture sameAs Q7545379.
- Smith_conjecture sameAs Q7545379.
- Smith_conjecture sameAs Smith_conjecture.
- Smith_conjecture wasDerivedFrom Smith_conjecture?oldid=606444250.
- Smith_conjecture isPrimaryTopicOf Smith_conjecture.