“Seifert conjecture”的意思、由来-开放百科全书

词条

Seifert conjecture

释义

References
Further reading

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
{{cite journal|first=Jenny|last= Harrison|authorlink=Jenny Harrison|title= counterexamples to the Seifert conjecture|journal=Topology | volume= 27 |year=1988|issue= 3|pages= 249–278|doi=10.1016/0040-9383(88)90009-2|mr=0963630}}
{{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}}
{{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= (2) |volume= 143 |year=1996|number= 3|pages= 547–576|mr=1394969|doi=10.2307/2118536}}
{{cite journal|first=Krystyna|last= Kuperberg|authorlink=Krystyna Kuperberg|title=A smooth counterexample to the Seifert conjecture|journal=Annals of Mathematics|series= (2) |volume=140 |year=1994|number=3|pages= 723–732|mr=1307902|doi=10.2307/2118623}}
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.

References

Further reading