Seifert Conjecture Overthrown Part 1

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Article: 174 of geometry.research
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From: sander@geom.umn.edu (Evelyn Sander)
Subject: Seifert Conjecture Overthrown
Organization: University of Minnesota, Twin Cities
Date: Tue, 12 Apr 1994 22:49:11 GMT
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One of the long-standing problems of dynamical systems is to prove or disprove the following 1950 postulate of Seifert:

Seifert Conjecture: Every non-vanishing vector field on the three-sphere has a periodic orbit.

After many years, this conjecture is now known to be false. In 1974, P.A. Schweitzer constructed a C^1-smooth counterexample (i.e. the vector field in this counterexample is once continuously differentiable). In 1988, J. Harrison constructed a C^2 counterexample. Finally, just this past summer, K.M. Kuperberg solved the problem conclusively, when she constructed an infinitely continuously differentiable counterexample. This article gives some of the background of the problem as it arises in Hamiltonian systems, differential geometry, and topology. Part 2 gives a heuristic description of Kuperberg's counterexample. Parts 1 and 2 are both based on a recent seminar by Richard McGehee, University of Minnesota mathematics professor and interim director of the Geometry Center. Notation: for the rest of this article, I refer to the two-sphere as S^2 and the three-sphere as S^3.

The Hamiltonian Case

Consider the following Hamiltonian system of differential equations on R^4 (x and y are vectors in R^2):

  x'=y
  y'=-x

Solutions of this differential equation stay within a fixed level set of its Hamiltonian H(x,y) = |x|^2+|y|^2. The level sets, {(x,y): H(x,y)=h, where h is constant}, are diffeomorphic to S^3 for positive h. Thus we can restrict to a vector field on S^3. The simplest kinds of solutions to understand are fixed points and periodic orbits. Thus we are interested in whether there are any periodic orbits. In other words, does the Seifert conjecture hold for this example? In fact in this particular example, all solutions are periodic. What happens if we perturb this system by a Hamiltonian perturbation? For small enough perturbation, Weinstein showed that the system will still have at least two periodic orbits.

We do not have to restrict our attention to perturbations of this specific system. More generally, consider any Hamiltonian system in R^4. If a level set for the Hamiltonian of the system is diffeomorphic to S^3, is there a periodic solution on this level set? In other words, does the Seifert conjecture hold if the vector field comes from a Hamiltonian system? Rabinowitz showed that if the level set is star-like, in other words every ray from the origin intersects it in exactly one point, then the answer is yes. The answer is not known for a general level set.

Riemannian Geometry

In the case of Riemannian geometry, one version of the Seifert conjecture is a question about closed geodesics. Namely, for a given Riemannian metric on S^2, are there any closed geodesics? The unit tangent bundle of S^2 is diffeomorphic to S^3, which means that a geodesic flow on S^2 corresponds to a flow on S^3. A periodic orbit for this flow in S^3 corresponds to a closed geodesic. In this very restricted case, the Seifert conjecture does hold. Bangert and Franks proved the much stronger result that every Riemannian metric on S^2 has infinitely many closed geodesics.

Topology

In the topological context, we generalize the question:

Generalized Seifert Question: Let M be a compact n-dimensional manifold with Euler characteristic zero. This last condition guarantees that there exist non-vanishing vector fields on M. Does every non-vanishing C^r vector field on M have a periodic orbit?

In 1966, W. Wilson showed that for a manifold described above of dimension four or larger, the answer to this question is no for arbitrarily smooth vector fields. In other words, Wilson found an infinitely differentiable non-vanishing vector field on M which contains no periodic orbits. I stated the result for S^3, but in fact Kuperberg's example also shows that for any manifold as above of dimension three, the answer to the Seifert question is no.

As the above discussion shows, the Seifert conjecture comes up in a variety of fields. It is an exciting result that Kuperberg has disproved it. In part two of this article, I will give an outline of the ideas used Kuperberg's infinitely differentiable counterexample.

References:

Krystyna M. Kuperberg, "A C^infinity counterexample to the Seifert conjecture in dimension three," preprint, 1993.

Paul Rabinowitz, "Periodic Solutions of Hamiltonian Systems," Communications of Pure and Applied Math. 31(1978), 157-184.

Alan Weinstein, "Symplectic V-Manifolds, Periodic Orbits of Hamiltonian Systems, and the Volume of Certain Riemannian Manifolds," Communications of Pure and Applied Math. 30(1977), 265-271.

Wilson, "On the minimal sets of non-singular vector fields," Annals of Math. 84(1966), 529-536.

The following two references are less technical than this article:

Barry Cipra, "Collaboration Closes in on Closed Geodesics," What's Happening in the Mathematical Sciences, AMS, (1)1993, 27-30.

Barry Cipra, "Smoothing Out the Seifert Conjecture," What's Happening in the Mathematical Sciences, AMS, (to appear).

McGehee's lecture took place February 24, 1994, as part of the Dynamics and Mechanics seminars at the University of Minnesota.

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