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 C 1 counterexample. Schweitzer's construction was then modified by Jenny Harrison in 1988 to make a C 2 + δ counterexample for some δ > 0 . The existence of smoother counterexamples remained an open question until 1993 when Krystyna Kuperberg constructed a very different C ∞ counterexample. Later this construction was shown to have real analytic and piecewise linear versions.