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 ${\displaystyle C^{1}}$ counterexample. Schweitzer's construction was then modified by Jenny Harrison in 1988 to make a ${\displaystyle C^{2+\delta }}$ counterexample for some ${\displaystyle \delta >0}$. The existence of smoother counterexamples remained an open question until 1993 when Krystyna Kuperberg constructed a very different ${\displaystyle C^{\infty }}$ 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 ${\displaystyle C^{\omega }}$ steady state flows on ${\displaystyle S^{3}}$ possess closed flowlines^[1] based on similar results for Beltrami flows on the Weinstein conjecture.^[2]

References

• Ginzburg, Viktor L.; Gurel, Basak Z. (2001). "A C²-smooth counterexample to the Hamiltonian Seifert conjecture in R⁴". arXiv:math/0110047.
• Harrison, Jenny (1988). "${\displaystyle C^{2}}$ 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.

Further reading

• Kuperberg, Krystyna (1999). "Aperiodic dynamical systems" (PDF). Notices of the AMS. 46 (9): 1035–1040.