Editing 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 <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>

==References==
{{reflist}}
*{{cite arXiv |eprint=math/0110047 |last1=Ginzburg |first1=Viktor L. |last2=Gurel |first2=Basak Z. |title=A C<sup>2</sup>-smooth counterexample to the Hamiltonian Seifert conjecture in R<sup>4</sup> |date=2001 }}
*{{cite journal|first=Jenny|last= Harrison|authorlink=Jenny Harrison|title=  <math>C^2</math> counterexamples to the Seifert conjecture|journal=[[Topology (journal)| Topology]] | volume=  27  |year=1988|issue= 3|pages= 249–278|doi=10.1016/0040-9383(88)90009-2|mr=0963630|doi-access=}}
*{{cite journal|first=Greg|last=Kuperberg|authorlink=Greg Kuperberg|title=A volume-preserving counterexample to the Seifert conjecture|journal= [[Commentarii Mathematici Helvetici]] |volume= 71  |year=1996|issue= 1|pages= 70–97|doi=10.1007/BF02566410|mr=1371679|arxiv=alg-geom/9405012|s2cid=18212778 }}
*{{cite journal|first1=Greg|last1=Kuperberg|authorlink1=Greg Kuperberg |first2=Krystyna|last2= Kuperberg|authorlink2=Krystyna Kuperberg|title=Generalized counterexamples to the Seifert conjecture|journal=[[Annals of Mathematics]] |series=Second series |volume=  143  |year=1996|number= 3|pages= 547–576|mr=1394969|doi=10.2307/2118536|jstor=2118536 |arxiv=math/9802040|s2cid=16309410 }}
*{{cite journal|first=Krystyna|last= Kuperberg|authorlink=Krystyna Kuperberg|title=A smooth counterexample to the Seifert conjecture|journal=[[Annals of Mathematics]] |series=Second series |volume=140 |year=1994|number=3|pages= 723–732|mr=1307902|doi=10.2307/2118623|jstor= 2118623}}
*{{cite journal |jstor=1971077 |last1=Schweitzer |first1=Paul A. |title=Counterexamples to the Seifert Conjecture and Opening Closed Leaves of Foliations |journal=Annals of Mathematics |date=1974 |volume=100 |issue=2 |pages=386–400 |doi=10.2307/1971077 }}
*{{cite journal |jstor=2032372 |last1=Seifert |first1=Herbert |title=Closed Integral Curves in 3-Space and Isotopic Two-Dimensional Deformations |journal=Proceedings of the American Mathematical Society |date=1950 |volume=1 |issue=3 |pages=287–302 |doi=10.2307/2032372 }}


==Further reading==
*{{cite journal |last=Kuperberg |first=Krystyna  |year=1999 |url=http://www.ams.org/notices/199909/fea-kuperberg.pdf |title=Aperiodic dynamical systems |journal=[[Notices of the AMS]] |volume=46 |issue=9 |pages=1035–1040}}


[[Category:Differential topology]]
[[Category:Disproved conjectures]]