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Novikov's compact leaf theorem

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Result about foliation of compact 3-manifolds

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

Theorem: A smooth codimension-one foliation of the 3-sphere S has a compact leaf. The leaf is a torus T 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 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.

Novikov's compact leaf theorem for any M

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

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

  1. the fundamental group π 1 ( M 3 ) {\displaystyle \pi _{1}(M^{3})} is finite,
  2. the second homotopy group π 2 ( M 3 ) 0 {\displaystyle \pi _{2}(M^{3})\neq 0} ,
  3. there exists a leaf L F {\displaystyle L\in F} such that the map π 1 ( L ) π 1 ( M 3 ) {\displaystyle \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.
  • 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.
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