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- Q7545379 subject Q10129919.
- Q7545379 subject Q6248542.
- Q7545379 subject Q8376416.
- Q7545379 subject Q8981903.
- Q7545379 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.".
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- Q7545379 wikiPageWikiLink Q6248542.
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- Q7545379 wikiPageWikiLink Q8376416.
- Q7545379 wikiPageWikiLink Q8981903.
- Q7545379 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.".
- Q7545379 label "Smith conjecture".