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.
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:
| 3) | |
| \pi | |
| 1(M |
| 3)\ne | |
| \pi | |
| 2(M |
0
L\inF
\pi1(L)\to\pi
| 3) | |
| 1(M |
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.