Smith conjecture
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In mathematics, the Smith conjecture was a problem open for many years, and proved at the end of the 1970s. It states that if f is an orientation-preserving diffeomorphism (not the identity) of the 3-sphere, of finite order and having some fixed point, then the fixed point set of f is an unknot.
This conjecture was conceived during the 1930s by the American topologist Paul A. Smith, who showed that such a diffeomorphism must have fixed point set equal to a knot. The proof depended on several major advances in 3-manifold theory, in particular the work of William Thurston on hyperbolic structures on 3-manifolds, results by Meeks and Yau on minimal surfaces in 3-manifolds, and work by Bass on finitely generated subgroups of GL(2,C). Cameron Gordon, around 1978, upon hearing of the then-new results by Thurston, Meeks-Yau, et al, completed the proof of the Smith conjecture.
[edit] See also
[edit] Reference
- The Smith conjecture. Papers presented at the symposium held at Columbia University, New York, 1979. Edited by John W. Morgan and Hyman Bass. Pure and Applied Mathematics, Academic Press, 1984.
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