View source for Seifert conjecture

The assertion that every non-singular (i.e. everywhere non-zero) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130200/s1302001.png" /> [[Vector field|vector field]] on the three-dimensional sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130200/s1302002.png" /> possesses a circular orbit.

The conjecture is a three-dimensional analogue of the well-known hairy ball theorem, stating that there is no continuous non-singular vector field on the two-dimensional sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130200/s1302003.png" />.

Integrating a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130200/s1302004.png" /> vector field results in a flow, which on a closed manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130200/s1302005.png" /> is a dynamical system, i.e. a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130200/s1302006.png" /> with the properties:

1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130200/s1302007.png" />; and

2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130200/s1302008.png" /> (the parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130200/s1302009.png" /> is usually interpreted as time; cf. also [[Dynamical system|Dynamical system]]; [[Flow (continuous-time dynamical system)|Flow (continuous-time dynamical system)]]).

An orbit, or a trajectory, of a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130200/s13020010.png" /> is the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130200/s13020011.png" />. If an orbit is simple closed curve, then it is called circular, closed or periodic.

The [[Hopf fibration|Hopf fibration]] is an [[Essential mapping|essential mapping]] from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130200/s13020012.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130200/s13020013.png" /> whose fibres, the inverse images of single points, are simple closed curves. The Seifert conjecture has its roots in a 1950 paper of H. Seifert [[#References|[a8]]], who proved that a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130200/s13020014.png" /> non-singular vector field on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130200/s13020015.png" /> possesses a periodic orbit if it is "almost parallel" to the fibres of the Hopf fibration.

The even-dimensional spheres do not admit non-singular vector fields, and a higher-dimensional version of the Seifert conjecture for the odd-dimensional spheres has been established in 1966 by F.W. Wilson [[#References|[a9]]] as follows: Any non-singular vector field on a smooth <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130200/s13020016.png" />-dimensional manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130200/s13020017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130200/s13020018.png" />, can be modified to a vector field with a set of isolated invariant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130200/s13020019.png" />-tori, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130200/s13020020.png" />, so that for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130200/s13020021.png" />:

a) in both cases as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130200/s13020022.png" /> and as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130200/s13020023.png" />, the orbit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130200/s13020024.png" /> limits on one of the tori; and

b) every orbit contained in one of the tori is dense in that torus. Thus, each of the spheres <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130200/s13020025.png" /> admits a non-singular vector field with no circular orbits.

For his construction, Wilson introduced a plug, a special non-singular vector field on the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130200/s13020026.png" />-dimensional disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130200/s13020027.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130200/s13020028.png" /> is the unit interval. The plug is constant and parallel to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130200/s13020029.png" /> on the boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130200/s13020030.png" />, and satisfies the trapped-orbit condition and the matched-ends condition (see below). The plug can be inserted in a non-singular vector field on an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130200/s13020031.png" />-dimensional manifold (the mechanism of insertion is illustrated in Fig.a1).

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/s130200a.gif" />

Figure: s130200a

Inserting a plug

The trapped-orbit condition guarantees that at least one orbit enters the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130200/s13020032.png" /> at the bottom, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130200/s13020033.png" />, but never leaves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130200/s13020034.png" />. The matched-ends condition means that if an orbit enters the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130200/s13020035.png" /> at the bottom and leaves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130200/s13020036.png" />, then the exit point is the point on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130200/s13020037.png" /> exactly above the entry point. By appropriately inserting a number of copies of a plug in a vector field on a manifold, Wilson changed the flow so that each orbit starts inside a plug and ends inside one, too.

In dimension three, Wilson's theorem yields isolated circular orbits and does not resolve the Seifert conjecture. The conjecture remained unsolved until a remarkable construction by P.A. Schweitzer in 1972. His 1974 paper [[#References|[a7]]] describes a three-dimensional plug without periodic orbits, which Schweitzer used to break the isolated periodic orbits, see Fig.a2 and Fig.a3. Inside the plug, instead of circular orbits, there are invariant Denjoy sets to trap the entering orbits. This, initially <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130200/s13020038.png" />, construction was later improved to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130200/s13020039.png" /> by J.M. Harrison [[#References|[a3]]].

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/s130200b.gif" />

Figure: s130200b

Schweitzer's plug

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/s130200c.gif" />

Figure: s130200c

Breaking an orbit

Significant changes to the status of the Seifert conjecture came about in 1993 when H. Hofer [[#References|[a4]]] proved that the Seifert conjecture holds for the Reeb vector field of a contact form on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130200/s13020040.png" /> (cf. also [[Contact structure|Contact structure]]). It was the next, after Seifert, advancement in the spirit of the conjecture.

On the other hand, a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130200/s13020041.png" /> counterexample to the Seifert conjecture (in its original formulation) was found by K. Kuperberg [[#References|[a5]]] the same year. This aperiodic vector field on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130200/s13020042.png" /> also employs a plug. A partial self-insertion performed on a Wilson-type plug breaks the periodic orbits in the plug itself in a recursive process, see Fig.a4.

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/s130200d.gif" />

Figure: s130200d

The K-plug — a self-inserted Wilson plug

The following year, G. Kuperberg [[#References|[a1]]] modified Schweitzer's vector field to a volume-preserving counterexample to the Seifert conjecture, thereby narrowing the gap between the counterexamples and Hofer's theorem.

The above constructions generalize to higher dimensions, but counterexamples with stronger properties exist in dimensions above three. The Hamiltonian version of the Seifert conjecture is false for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130200/s13020043.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130200/s13020044.png" />, as V.L. Ginzburg [[#References|[a2]]] proved that there is a smooth function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130200/s13020045.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130200/s13020046.png" />, such that the Hamiltonian flow of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130200/s13020047.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130200/s13020048.png" /> has no closed orbits (cf. also [[Hamiltonian system|Hamiltonian system]]).

==Modified Seifert conjecture.==
A minimal set of a dynamical system is an invariant, non-empty, compact set containing no proper invariant, non-empty, compact subsets. The modified Seifert conjecture [[#References|[a7]]], [[#References|[a9]]] asserts that every non-singular <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130200/s13020049.png" /> vector field on an odd-dimensional sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130200/s13020050.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130200/s13020051.png" />, has a minimal set of codimension at least two, i.e. of dimension at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130200/s13020052.png" />. The invariant sets in the three-dimensional plugs of Wilson and Schweitzer are one-dimensional. In 1996 it was shown [[#References|[a6]]] that the modified Seifert conjecture is false for real-analytic as well as for piecewise-linear flows, for all odd-dimensional spheres: Every non-singular vector field on any manifold can be modified in the given smoothness category so that every minimal set is of codimension one.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> G. Kuperberg, "A volume-preserving counterexample to the Seifert conjecture" ''Comment. Math. Helv.'' , '''71''' (1996) pp. 70–97 {{MR|1371679}} {{ZBL|0859.57017}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> V.L. Ginzburg, "A smooth counterexample to the Hamiltonian Seifert conjecture in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130200/s13020053.png" />" ''Internat. Math. Res. Notices'' , '''13''' (1997) pp. 641–650 {{MR|1459629}} {{ZBL|0913.58022}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> J. Harrison, "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130200/s13020054.png" /> counterexamples to the Seifert conjecture" ''Topology'' , '''27''' (1988) pp. 249–278 {{MR|0963630}} {{ZBL|0669.57011}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> H. Hofer, "Pseudoholomorphic curves in symplectizations with applications to the Weinstein conjecture in dimension three" ''Invent. Math.'' , '''114''' (1993) pp. 515–563 {{MR|1244912}} {{ZBL|0797.58023}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> K. Kuperberg, "A smooth counterexample to the Seifert conjecture" ''Ann. of Math.'' , '''140''' (1994) pp. 723–732 {{MR|1307902}} {{ZBL|0856.57024}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> G. Kuperberg, K. Kuperberg, "Generalized counterexamples to the Seifert conjecture" ''Ann. of Math.'' , '''144''' (1996) pp. 239–268 {{MR|1418899}} {{MR|1394969}} {{ZBL|0856.57026}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> P.A. Schweitzer, "Counterexamples to the Seifert conjecture and opening closed leaves of foliations" ''Ann. of Math.'' , '''100''' (1974) pp. 386–400 {{MR|0356086}} {{ZBL|0295.57010}} </TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> H. Seifert, "Closed integral curves in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130200/s13020055.png" />-space and isotopic two-dimensional deformations" ''Proc. Amer. Math. Soc.'' , '''1''' (1950) pp. 287–302 {{MR|37508}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> F.W. Wilson, "On the minimal sets of non-singular vector fields" ''Ann. of Math.'' , '''84''' (1966) pp. 529–536 {{MR|0202155}} {{ZBL|0156.43803}} </TD></TR></table>