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Period three actions on the three-sphere. (English) Zbl 1037.57012

The authors prove that a period three action on the three-sphere is standard, i.e. smooth free actions of the cyclic group of three elements on the three-sphere are conjugate to an orthogonal group action. Orthogonality is conjectured to hold for every finite group acting on \(S^3\), as a particular case of Thurston’s geometrization, but it is only known in some cases. To the reviewer’s knowledge, the first case to be proved was the cyclic group with two elements, by G. R. Livesay [Ann. Math. (2) 72, 603-611 (1960; Zbl 0096.17302)]. Other cases where proved by Rice, Ritter, Evans and Maxwell, Rubinstein and Meyers.
To prove the theorem, the authors find an unknotted curve in \(S^3\) invariant under the action. The main tool to find such a curve is the so called sweepout. A special case of a sweepout is the foliation of \(S^3\) by two-spheres with two singular leaves, which was used by Livesay in his proof for the group with two elements. In general, a sweepout is a manifold \(M\) with a height function so that generic level sets are unions of spheres, and a degree one map from \(M\) to \(S^3\) which is an embedding on each level set of the height function.
One has to look at the orbit of the sweepout by the action of the cyclic group of order three. In general, the generic intersection of the level sets of these sweepouts is transverse. The authors define a complexity for the generic level sets of the sweepout, counting triple points and double curves of the intersection. They work with a minimax sweepout (i.e., one that minimizes the maximum of the complexities along generic level sets). The basic idea is to show that a minimax sweepout must have an unknotted invariant curve. This is a nice argument that can possibly be extended to other three-manifolds.

MSC:

57M60 Group actions on manifolds and cell complexes in low dimensions
57M50 General geometric structures on low-dimensional manifolds
57S17 Finite transformation groups

Citations:

Zbl 0096.17302
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References:

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