Editing Seifert conjecture (section)

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 <math>C^1</math> counterexample.  Schweitzer's construction was then modified by [[Jenny Harrison]] in 1988 to make a <math>C^{2+\delta}</math> [[counterexample]] for some <math>\delta > 0</math>.  The existence of smoother counterexamples remained an open question until 1993 when [[Krystyna Kuperberg]] constructed a very different <math>C^\infty</math> 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 <math>C^\omega</math> steady state flows on <math>S^3</math> possess closed flowlines<ref>{{Cite arXiv  |last1=Etnyre |first1=J. |last2=Ghrist |first2=R. |date=1997 |title=Contact Topology and Hydrodynamics |arxiv=dg-ga/9708011 }}</ref> based on similar results for [[Beltrami vector field|Beltrami flows]] on the [[Weinstein conjecture]].<ref>{{Cite journal |last=Hofer |first=H. |date=1993 |title=Pseudoholomorphic curves in symplectizations with applications to the Weinstein conjecture in dimension three. |url=https://eudml.org/doc/144157 |journal=Inventiones Mathematicae |volume=114 |issue=3 |pages=515–564 |doi=10.1007/BF01232679 |bibcode=1993InMat.114..515H |s2cid=123618375 |issn=0020-9910}}</ref>