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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 wikiPageLength "3770".
- Smith_conjecture wikiPageOutDegree "26".
- Smith_conjecture wikiPageRevisionID "627082419".
- Smith_conjecture wikiPageWikiLink 3-manifold.
- Smith_conjecture wikiPageWikiLink 3-sphere.
- Smith_conjecture wikiPageWikiLink Academic_Press.
- Smith_conjecture wikiPageWikiLink American_Journal_of_Mathematics.
- Smith_conjecture wikiPageWikiLink Annals_of_Mathematics.
- Smith_conjecture wikiPageWikiLink Cameron_Gordon_(mathematician).
- Smith_conjecture wikiPageWikiLink Category:3-manifolds.
- Smith_conjecture wikiPageWikiLink Category:Conjectures.
- Smith_conjecture wikiPageWikiLink Category:Diffeomorphisms.
- Smith_conjecture wikiPageWikiLink Category:Theorems_in_topology.
- Smith_conjecture wikiPageWikiLink Diffeomorphism.
- Smith_conjecture wikiPageWikiLink Fixed_point_(mathematics).
- Smith_conjecture wikiPageWikiLink Hilbert–Smith_conjecture.
- Smith_conjecture wikiPageWikiLink Hyman_Bass.
- Smith_conjecture wikiPageWikiLink Hyperbolic_3-manifold.
- Smith_conjecture wikiPageWikiLink Knot_(mathematics).
- Smith_conjecture wikiPageWikiLink Mathematics.
- Smith_conjecture wikiPageWikiLink Minimal_surface.
- Smith_conjecture wikiPageWikiLink Order_(group_theory).
- Smith_conjecture wikiPageWikiLink Orientation_(vector_space).
- Smith_conjecture wikiPageWikiLink Proceedings_of_the_American_Mathematical_Society.
- Smith_conjecture wikiPageWikiLink Shing-Tung_Yau.
- Smith_conjecture wikiPageWikiLink Topology_(journal).
- Smith_conjecture wikiPageWikiLink William_B._Meeks,_Jr.
- Smith_conjecture wikiPageWikiLink William_Thurston.
- Smith_conjecture wikiPageWikiLinkText "Smith conjecture".
- Smith_conjecture authorlink "Paul A. Smith".
- Smith_conjecture last "Smith".
- Smith_conjecture loc "remark after theorem 4".
- Smith_conjecture wikiPageUsesTemplate Template:Citation.
- Smith_conjecture wikiPageUsesTemplate Template:Harv.
- Smith_conjecture wikiPageUsesTemplate Template:Harvs.
- Smith_conjecture wikiPageUsesTemplate Template:Harvtxt.
- Smith_conjecture wikiPageUsesTemplate Template:Topology-stub.
- 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 hypernym Diffeomorphism.
- Smith_conjecture type Conjecture.
- Smith_conjecture type Function.
- Smith_conjecture type Homeomorphism.
- Smith_conjecture type Statement.
- Smith_conjecture type Theorem.
- Smith_conjecture type Statement.
- 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 Q7545379.
- Smith_conjecture sameAs m.0c4m5s.
- Smith_conjecture sameAs Q7545379.
- Smith_conjecture wasDerivedFrom Smith_conjecture?oldid=627082419.
- Smith_conjecture isPrimaryTopicOf Smith_conjecture.