Novikov's compact leaf theorem
In mathematics, Novikov's compact leaf theorem, named after Sergei Novikov, states that
Theorem: A smooth codimension-one foliation of the 3-sphere S3has a compact leaf. The leaf is a torus T2bounding 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.
In 1965, Novikov proved the compact leaf theorem for any M3:
Theorem: Let M3be a closed 3-manifold with a smooth codimension-one foliation F. Suppose any of the following conditions is satisfied:
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.
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