Smith conjecture

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. In the 1960s Friedhelm Waldhausen proved the Smith conjecture for the special case of diffeomorphisms of order 2 (and hence any even order). The proof of the general case given in 1979 depended on several major advances in 3-manifold theory, in particular the work of William Thurston on hyperbolic structures on 3-manifolds, results by William Meeks and Shing-Tung Yau on minimal surfaces in 3-manifolds, and work by Hyman 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.

ee also

*Hilbert-Smith conjecture

References

*"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. ISBN 0-12-506980-4