About: Smith conjecture

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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. Paul A. Smith showed that a non-trivial orientation-preserving diffeomorphism of finite order with fixed points must have a fixed point set equal to a circle, and asked in if the fixed point set could be knotted. Friedhelm Waldhausen 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 John Morgan and Hyman Bass 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 Mee

Attributes	Values
rdf:type	yago:WikicatConjectures yago:WikicatTheoremsInTopology yago:Abstraction100002137 yago:Cognition100023271 yago:Communication100033020 yago:Concept105835747 yago:Content105809192 yago:Hypothesis105888929 yago:Idea105833840 yago:Message106598915 yago:Proposition106750804 yago:PsychologicalFeature100023100 yago:Speculation105891783 yago:Statement106722453 yago:Theorem106752293
rdfs:label	Smith conjecture (en)
rdfs: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. Paul A. Smith showed that a non-trivial orientation-preserving diffeomorphism of finite order with fixed points must have a fixed point set equal to a circle, and asked in if the fixed point set could be knotted. Friedhelm Waldhausen 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 John Morgan and Hyman Bass 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 Mee (en)
dcterms:subject	Diffeomorphisms 3-manifolds Conjectures Theorems in topology
Wikipage page ID	4476828 (xsd:integer)
Wikipage revision ID	1117894337 (xsd:integer)
Link from a Wikipage to another Wikipage	American Journal of Mathematics Annals of Mathematics Diffeomorphisms 3-manifolds Conjectures Theorems in topology Mathematics Proceedings of the American Mathematical Society William Thurston Minimal surface 3-sphere 3-manifold Diffeomorphism Hilbert–Smith conjecture Hyperbolic 3-manifold Academic Press Topology (journal) Knot (mathematics) Order (group theory) Cameron Gordon (mathematician) Shing-Tung Yau William Hamilton Meeks, III Peter Shalen Orientation-preserving Fixed point set
Link from a Wikipage to an external page	https://books.google.com/books%3Fid=sXcwg4zi1_EC https://pub.uni-bielefeld.de/record/1782176
sameAs	Smith conjecture Smith conjecture Smith conjecture Smith conjecture
dbp:wikiPageUsesTemplate	dbt:Citation dbt:Harv dbt:No_footnotes dbt:Short_description dbt:Harvs dbt:Topology-stub
authorlink	Friedhelm Waldhausen (en) Paul Althaus Smith (en)
first	John (en) Leo (en) Hyman (en) Deane (en) Paul A. (en) Friedhelm (en)
last	Bass (en) Montgomery (en) Morgan (en) Smith (en) Waldhausen (en) Zippin (en)
year	1939 (xsd:integer) 1954 (xsd:integer) 1969 (xsd:integer) 1984 (xsd:integer)
loc	remark after theorem 4 (en)
has 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. Paul A. Smith showed that a non-trivial orientation-preserving diffeomorphism of finite order with fixed points must have a fixed point set equal to a circle, and asked in if the fixed point set could be knotted. Friedhelm Waldhausen 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 John Morgan and Hyman Bass 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 Bass, Cameron Gordon, Peter Shalen, and Rick Litherland. Deane Montgomery and Leo Zippin 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. Charles Giffen 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. (en)
author1-link	John Morgan (en) Deane Montgomery (en)
author2-link	Hyman Bass (en) Leo Zippin (en)
gold:hypernym	Diffeomorphism
prov:wasDerivedFrom	wikipedia-en:Smith_conjecture?oldid=1117894337&ns=0
page length (characters) of wiki page	4322 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Smith_conjecture
is Link from a Wikipage to another Wikipage of	List of conjectures History of knot theory List of knot theory topics Spherical space form conjecture Friedhelm Waldhausen Minimal surface Cameron Gordon (mathematician) R. H. Bing Paul Althaus Smith
is foaf:primaryTopic of	wikipedia-en:Smith_conjecture

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