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.
- Eilenberg, Samuel (1949), "On the Problems of Topology", Annals of Mathematics, Second Series (Annals of Mathematics) 50 (2): 247–260, doi:10.2307/1969448, ISSN 0003-486X, MR 0030189
- Giffen, Charles H. (1966), "The generalized Smith conjecture", American Journal of Mathematics 88: 187–198, doi:10.2307/2373054, ISSN 0002-9327, MR 0198462
- Montgomery, Deane; Zippin, Leo (1954), "Examples of transformation groups", Proceedings of the American Mathematical Society 5: 460–465, doi:10.2307/2031959, ISSN 0002-9939, MR 0062436
- 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, MR 758459
- Smith, P. A. (1939), "Transformations of finite period. II", Annals of Mathematics. Second Series 40: 690–711, doi:10.2307/1968950, ISSN 0003-486X, MR 0000177
- 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, MR 0236916
|This topology-related article is a stub. You can help Wikipedia by expanding it.|