## 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

### Keywords:

3-sphere; group action; spherical 3-manifold; lens space

Zbl 0096.17302
Full Text:

### References:

 [1] B Evans, J Maxwell, Quaternion actions on $$S^3$$, Amer. J. Math. 101 (1979) 1123 · Zbl 0417.57023 · doi:10.2307/2374129 [2] H Hopf, Zum Clifford-Kleinschen Raumproblem, Math. Ann. 95 (1926) 313 · doi:10.1007/BF01206614 [3] R Lee, Semicharacteristic classes, Topology 12 (1973) 183 · Zbl 0264.57012 · doi:10.1016/0040-9383(73)90006-2 [4] G R Livesay, Fixed point free involutions on the 3-sphere, Ann. of Math. $$(2)$$ 72 (1960) 603 · Zbl 0096.17302 · doi:10.2307/1970232 [5] J Milnor, Groups which act on $$S^n$$ without fixed points, Amer. J. Math. 79 (1957) 623 · Zbl 0078.16304 · doi:10.2307/2372566 [6] R Myers, Free involutions on lens spaces, Topology 20 (1981) 313 · Zbl 0508.57032 · doi:10.1016/0040-9383(81)90005-7 [7] P M Rice, Free actions of $$Z_4$$ on $$S^3$$, Duke Math. J. 36 (1969) 749 [8] G X Ritter, Free $$Z_8$$ actions on $$S^3$$, Trans. Amer. Math. Soc. 181 (1973) 195 · Zbl 0264.57016 · doi:10.2307/1996629 [9] J H Rubinstein, On 3-manifolds that have finite fundamental group and contain Klein bottles, Trans. Amer. Math. Soc. 251 (1979) 129 · Zbl 0414.57005 · doi:10.2307/1998686 [10] J H Rubinstein, Free actions of some finite groups on $$S^3$$ I, Math. Ann. 240 (1979) 165 · Zbl 0382.57019 · doi:10.1007/BF01364631 [11] P Scott, The geometries of 3-manifolds, Bull. London Math. Soc. 15 (1983) 401 · Zbl 0561.57001 · doi:10.1112/blms/15.5.401 [12] W Threlfall, H Seifert, Topologische Untersuchung der Diskontinuitätsbereiche endlicher Bewegungsgruppen des dreidimensionalen sphärischen Raumes, Math. Ann. 104 (1931) 1 · Zbl 0006.03403 · doi:10.1007/BF01457920 [13] W P Thurston, Three-dimensional geometry and topology Vol. 1, Princeton Mathematical Series 35, Princeton University Press (1997) · Zbl 0873.57001 [14] C B Thomas, Elliptic structures on 3-manifolds, London Mathematical Society Lecture Note Series 104, Cambridge University Press (1986) · Zbl 0595.57001 [15] J A Wolf, Spaces of constant curvature, Publish or Perish (1984)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.