Novikov's compact leaf theorem

In mathematics, Novikov's compact leaf theorem, named after Sergei Novikov, states that


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

Novikov's compact leaf theorem for S3
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.

Novikov's compact leaf theorem for any M3
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 $$\pi_1(M^3)$$ is finite,
 * 2) the second homotopy group $$\pi_2(M^3)\ne 0$$,
 * 3) there exists a leaf $$L\in F$$ such that the map $$\pi_1(L)\to\pi_1(M^3)$$ 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.