Seifert conjecture

In mathematics, the Seifert conjecture states that every nonsingular, continuous vector field on the 3-sphere has a closed orbit. It is named after Herbert Seifert. In a 1950 paper, Seifert asked if such a vector field exists, but did not phrase non-existence as a conjecture. He also established the conjecture for perturbations of the Hopf fibration.

The conjecture was disproven in 1974 by Paul Schweitzer, who exhibited a $C^{1}$ counterexample. Schweitzer's construction was then modified by Jenny Harrison in 1988 to make a $C^{2+\delta }$ counterexample for some $\delta >0$ . The existence of smoother counterexamples remained an open question until 1993 when Krystyna Kuperberg constructed a very different $C^{\infty }$ counterexample. Later this construction was shown to have real analytic and piecewise linear versions. In 1997 for the particular case of incompressible fluids it was shown that all $C^{\omega }$ steady state flows on $S^{3}$ possess closed flowlines^[1] based on similar results for Beltrami flows on the Weinstein conjecture.^[2]

References

^ Etnyre, J.; Ghrist, R. (1997). "Contact Topology and Hydrodynamics". arXiv:dg-ga/9708011.
^ Hofer, H. (1993). "Pseudoholomorphic curves in symplectizations with applications to the Weinstein conjecture in dimension three". Inventiones Mathematicae. 114 (3): 515–564. Bibcode:1993InMat.114..515H. doi:10.1007/BF01232679. ISSN 0020-9910. S2CID 123618375.

Ginzburg, Viktor L.; Gurel, Basak Z. (2001). "A C²-smooth counterexample to the Hamiltonian Seifert conjecture in R⁴". arXiv:math/0110047.
Harrison, Jenny (1988). " $C^{2}$ counterexamples to the Seifert conjecture". Topology. 27 (3): 249–278. doi:10.1016/0040-9383(88)90009-2. MR 0963630.
Kuperberg, Greg (1996). "A volume-preserving counterexample to the Seifert conjecture". Commentarii Mathematici Helvetici. 71 (1): 70–97. arXiv:alg-geom/9405012. doi:10.1007/BF02566410. MR 1371679. S2CID 18212778.
Kuperberg, Greg; Kuperberg, Krystyna (1996). "Generalized counterexamples to the Seifert conjecture". Annals of Mathematics. Second series. 143 (3): 547–576. arXiv:math/9802040. doi:10.2307/2118536. JSTOR 2118536. MR 1394969. S2CID 16309410.
Kuperberg, Krystyna (1994). "A smooth counterexample to the Seifert conjecture". Annals of Mathematics. Second series. 140 (3): 723–732. doi:10.2307/2118623. JSTOR 2118623. MR 1307902.
Schweitzer, Paul A. (1974). "Counterexamples to the Seifert Conjecture and Opening Closed Leaves of Foliations". Annals of Mathematics. 100 (2): 386–400. doi:10.2307/1971077. JSTOR 1971077.
Seifert, Herbert (1950). "Closed Integral Curves in 3-Space and Isotopic Two-Dimensional Deformations". Proceedings of the American Mathematical Society. 1 (3): 287–302. doi:10.2307/2032372. JSTOR 2032372.

References

Further reading