Novikov's Compact Leaf Theorem

{{Short description, 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''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 $\pi_2(M^3)\ne 0$,''
# ''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.''

References

* ''S. Novikov''. The topology of foliations//Trudy Moskov. Mat. Obshch, 1965, v. 14, p. 248–27

* ''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


Foliations
Theorems in topology