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, 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, 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.

Seifert conjecture

References

Further reading