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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.