Smith conjecture

Template:Short descriptionTemplate:No footnotes 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.

Template:Harvs 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 if the fixed point set could be knotted. Template:Harvs 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 Template:Harvs 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.

Template:Harvs 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. Template:Harvs 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

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