# 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 counterexample. Schweitzer's construction was then modified by Jenny Harrison in 1988 to make a counterexample for some . The existence of smoother counterexamples remained an open question until 1993 when Krystyna Kuperberg constructed a very different counterexample. Later this construction was shown to have real analytic and piecewise linear versions.

## References

- V. Ginzburg and B. Gürel,
*A -smooth counterexample to the Hamiltonian Seifert conjecture in*, Ann. of Math. (2) 158 (2003), no. 3, 953–976 - Harrison, Jenny (1988). "
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.

## Further reading

- K. Kuperberg,
*Aperiodic dynamical systems*. Notices Amer. Math. Soc. 46 (1999), no. 9, 1035–1040.