Seifert conjecture explained

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

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

Further reading

• Kuperberg . Krystyna . 1999 . Aperiodic dynamical systems . . 46 . 9 . 1035–1040.

Notes and References

1. Etnyre . J. . Ghrist . R. . 1997 . Contact Topology and Hydrodynamics . dg-ga/9708011 .
2. Hofer . H. . 1993 . Pseudoholomorphic curves in symplectizations with applications to the Weinstein conjecture in dimension three. . Inventiones Mathematicae . 114 . 3 . 515–564 . 10.1007/BF01232679 . 1993InMat.114..515H . 123618375 . 0020-9910.