Novikov's compact leaf theorem


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

Novikov's compact leaf theorem for ''S''3

Theorem: A smooth codimension-one foliation of the 3-sphere ''S3 has a compact leaf. The leaf is a torus T''2 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 ''M''3

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 is finite,
  2. the second homotopy group,
  3. there exists a leaf such that the map 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.''