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
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.
Kuperberg, Greg; Kuperberg, Krystyna (1996). "Generalized counterexamples to the Seifert conjecture". Annals of Mathematics. (2). 143 (3): 547–576. arXiv:math/9802040. doi:10.2307/2118536. MR 1394969.
Kuperberg, Krystyna (1994). "A smooth counterexample to the Seifert conjecture". Annals of Mathematics. (2). 140 (3): 723–732. doi:10.2307/2118623. MR 1307902.
P. A. Schweitzer, Counterexamples to the Seifert conjecture and opening closed leaves of foliations, Annals of Mathematics (2) 100 (1974), 386–400.
H. Seifert, Closed integral curves in 3-space and isotopic two-dimensional deformations, Proc. Amer. Math. Soc. 1, (1950). 287–302.

References

Further reading