my-server
← Wiki

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<sup>3</sup>

Theorem: A smooth codimension-one foliation of the 3-sphere S<sup>3</sup> has a compact leaf. The leaf is a torus T<sup>2</sup> 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<sup>3</sup> 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&nbsp;T<sup>2</sup>.

Novikov's compact leaf theorem for any M<sup>3</sup>

In 1965, Novikov proved the compact leaf theorem for any&nbsp;M<sup>3</sup>:

Theorem: Let M<sup>3</sup> be a closed 3-manifold with a smooth codimension-one foliation&nbsp;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&nbsp;≤&nbsp;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.

References