Novikov's compact leaf theorem

Theorem: A smooth codimension-one foliation of the 3-sphere S³ has a compact leaf. The leaf is a torus T² bounding a solid torus with the Reeb foliation.

The theorem was proved by Sergei Novikov in 1964. Earlier, Charles Ehresmann had conjectured that every smooth codimension-one foliation on S³ had a compact leaf, which was known to be true for all known examples; in particular, the Reeb foliation has a compact leaf that is T².

In 1965, Novikov proved the compact leaf theorem for any M³:

Theorem: Let M³ be a closed 3-manifold with a smooth codimension-one foliation F. Suppose any of the following conditions is satisfied:

1. the fundamental group ${\displaystyle \pi _{1}(M^{3})}$ is finite,
2. the second homotopy group ${\displaystyle \pi _{2}(M^{3})\neq 0}$,
3. there exists a leaf ${\displaystyle L\in F}$ such that the map ${\displaystyle \pi _{1}(L)\to \pi _{1}(M^{3})}$ induced by inclusion has a non-trivial kernel.

Then F has a compact leaf of genus g ≤ 1.

In terms of covering spaces:

A codimension-one foliation of a compact 3-manifold whose universal covering space is not contractible must have a compact leaf.

• S. Novikov. The topology of foliations//Trudy Moskov. Mat. Obshch, 1965, v. 14, p. 248–278.
• I. Tamura. Topology of foliations — AMS, v.97, 2006.
• D. Sullivan, Cycles for the dynamical study of foliated manifolds and complex manifolds, Invent. Math., 36 (1976), p. 225–255.