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.

References

*V. Ginzburg and B. Gürel, " [http://front.math.ucdavis.edu/math.DG/0110047 A $C^2$-smooth counterexample to the Hamiltonian Seifert conjecture in $R^4$] ", Ann. of Math. (2) 158 (2003), no. 3, 953--976
*J. Harrison, "$C^2$ counterexamples to the Seifert conjecture", Topology 27 (1988), no. 3, 249--278.
*G. Kuperberg "A volume-preserving counterexample to the Seifert conjecture", Comment. Math. Helv. 71 (1996), no. 1, 70--97.
*K. Kuperberg "A smooth counterexample to the Seifert conjecture", Ann. of Math. (2) 140 (1994), no. 3, 723--732.
*G. Kuperberg and K. Kuperberg, " [http://front.math.ucdavis.edu/math.DS/9802040 Generalized counterexamples to the Seifert conjecture] ", Ann. of Math. (2) 143 (1996), no. 3, 547--576.
*H. Seifert, "Closed integral curves in 3-space and isotopic two-dimensional deformations", Proc. Amer. Math. Soc. 1, (1950). 287--302.
*P. A. Schweitzer, "Counterexamples to the Seifert conjecture and opening closed leaves of foliations", Ann. of Math. (2) 100 (1974), 386--400.

Further reading

*K. Kuperberg, " [http://www.ams.org/notices/199909/fea-kuperberg.pdf Aperiodic dynamical systems] ". Notices Amer. Math. Soc. 46 (1999), no. 9, 1035--1040.