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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.

Smith (1939, remark after theorem 4) showed that a non-trivial orientation-preserving diffeomorphism of finite order 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 & Zippen (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.
See also

Hilbert-Smith conjecture

References

Eilenberg, Samuel (1949), "On the Problems of Topology", Annals of Mathematics, Second Series (Annals of Mathematics) 50 (2): 247–260, ISSN 0003-486X, MR0030189
Giffen, Charles H. (1966), "The generalized Smith conjecture", American Journal of Mathematics 88: 187–198, ISSN 0002-9327, MR0198462
Montgomery, Deane; Zippin, Leo (1954), "Examples of transformation groups", Proceedings of the American Mathematical Society 5: 460–465, ISSN 0002-9939, MR0062436
Morgan, John W.; Bass, Hyman, eds. (1984), The Smith conjecture, Pure and Applied Mathematics, 112, Boston, MA: Academic Press, ISBN 978-0-12-506980-9, MR758459
Smith, P. A. (1939), "Transformations of finite period. II", Annals of Mathematics. Second Series 40: 690–711, ISSN 0003-486X, MR0000177
Waldhausen, Friedhelm (1969), "Über Involutionen der 3-Sphäre", Topology. An International Journal of Mathematics 8: 81–91, doi:10.1016/0040-9383(69)90033-0, ISSN 0040-9383, MR0236916

Mathematics Encyclopedia

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