Novikov's compact leaf theorem

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 Sergey Novikov in 1964. Earlier Charles Ehresmann had conjectured that every smooth codimension-one foliation on S³ had a compact leaf, which was true for all known examples; in particular, the Reeb foliation had a compact leaf that was 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 π 1 ( M 3 ) is finite,
2. the second homotopy group π 2 ( M 3 ) ≠ 0 ,
3. there exists a leaf L ∈ F such that the map π 1 ( L ) → π 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.