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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 ( 1939, remark after theorem 4) 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 ( Eilenberg 1949, Problem 36) if the fixed point set could be knotted. Friedhelm 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 John Morgan and Hyman 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 Bass, Cameron Gordon, Peter Shalen, and Rick Litherland.
Deane Montgomery and Leo 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. Charles 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.
See also
References
- Eilenberg, Samuel (1949), "On the Problems of Topology", Annals of Mathematics, Second Series, 50 (2): 247–260, doi: 10.2307/1969448, ISSN 0003-486X, JSTOR 1969448, MR 0030189
- Giffen, Charles H. (1966), "The generalized Smith conjecture", American Journal of Mathematics, 88 (1): 187–198, doi: 10.2307/2373054, ISSN 0002-9327, JSTOR 2373054, MR 0198462
- Montgomery, Deane; Zippin, Leo (1954), "Examples of transformation groups", Proceedings of the American Mathematical Society, 5 (3): 460–465, doi: 10.2307/2031959, ISSN 0002-9939, JSTOR 2031959, MR 0062436
- Morgan, John W.; Bass, Hyman, eds. (1984), The Smith conjecture, Pure and Applied Mathematics, vol. 112, Boston, MA: Academic Press, ISBN 978-0-12-506980-9, MR 0758459
- Smith, Paul A. (1939), "Transformations of finite period. II", Annals of Mathematics, Second Series, 40 (3): 690–711, Bibcode: 1939AnMat..40..690S, doi: 10.2307/1968950, ISSN 0003-486X, JSTOR 1968950, MR 0000177
- Waldhausen, Friedhelm (1969), "Über Involutionen der 3-Sphäre", Topology, 8: 81–91, doi: 10.1016/0040-9383(69)90033-0, ISSN 0040-9383, MR 0236916
 | This topology-related article is a stub. You can help Wikipedia by expanding it. |
| This article includes a
list of references,
related reading, or
external links, but its sources remain unclear because it lacks
inline citations. (November 2021) |
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 ( 1939, remark after theorem 4) 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 ( Eilenberg 1949, Problem 36) if the fixed point set could be knotted. Friedhelm 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 John Morgan and Hyman 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 Bass, Cameron Gordon, Peter Shalen, and Rick Litherland.
Deane Montgomery and Leo 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. Charles 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.
See also
References
- Eilenberg, Samuel (1949), "On the Problems of Topology", Annals of Mathematics, Second Series, 50 (2): 247–260, doi: 10.2307/1969448, ISSN 0003-486X, JSTOR 1969448, MR 0030189
- Giffen, Charles H. (1966), "The generalized Smith conjecture", American Journal of Mathematics, 88 (1): 187–198, doi: 10.2307/2373054, ISSN 0002-9327, JSTOR 2373054, MR 0198462
- Montgomery, Deane; Zippin, Leo (1954), "Examples of transformation groups", Proceedings of the American Mathematical Society, 5 (3): 460–465, doi: 10.2307/2031959, ISSN 0002-9939, JSTOR 2031959, MR 0062436
- Morgan, John W.; Bass, Hyman, eds. (1984), The Smith conjecture, Pure and Applied Mathematics, vol. 112, Boston, MA: Academic Press, ISBN 978-0-12-506980-9, MR 0758459
- Smith, Paul A. (1939), "Transformations of finite period. II", Annals of Mathematics, Second Series, 40 (3): 690–711, Bibcode: 1939AnMat..40..690S, doi: 10.2307/1968950, ISSN 0003-486X, JSTOR 1968950, MR 0000177
- Waldhausen, Friedhelm (1969), "Über Involutionen der 3-Sphäre", Topology, 8: 81–91, doi: 10.1016/0040-9383(69)90033-0, ISSN 0040-9383, MR 0236916
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