Novikov's compact leaf theorem
# 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 $S^3$

Theorem: A smooth codimension-one foliation of the 3-sphere $S^3$ 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 $S^3$ 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 $T^2.$

## Novikov's compact leaf theorem for any $M^3$

In 1965, Novikov proved the compact leaf theorem for any $M^3:$

Theorem: Let $M^3$ be a closed 3-manifold with a smooth codimension-one foliation F. Suppose any of the following conditions is satisfied:

* the fundamental group $\pi _{1}(M^{3})$ is finite,
* the second homotopy group $\pi _{2}(M^{3})\neq 0$,
* 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.

