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 S3

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 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, Reeb foliation had a compact leaf that was T2.

Novikov's compact leaf theorem for any M3

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 π1(M3) is finite,
  2. the second homotopy group \pi_2(M^3)\ne 0,
  3. 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.

References

  • S. Novikov. The topology of foliations//Trudy Moskov. Mat. Obshch, 1965, v. 14, p. 248–278.[1]
  • I. Tamura. Topology of foliations — AMS, v.97, 2006.
  • D. Sullivan, Cycles for the dynamical study of foliated manifolds and complex manifolds, Invent. Math., 36 (1976), p. 225–255. [2]
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