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 ${\displaystyle C^{1}}$ counterexample. Schweitzer's construction was then modified by Jenny Harrison in 1988 to make a ${\displaystyle C^{2+\delta }}$ counterexample for some ${\displaystyle \delta >0}$. The existence of smoother counterexamples remained an open question until 1993 when Krystyna Kuperberg constructed a very different ${\displaystyle C^{\infty }}$ counterexample. Later this construction was shown to have real analytic and piecewise linear versions. In 1997 for the particular case of incompressible fluids it was shown that all ${\displaystyle C^{\omega }}$ steady state flows on ${\displaystyle S^{3}}$ possess closed flowlines¹ based on similar results for Beltrami flows on the Weinstein conjecture.²