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 Sergei Novikov in 1964. Earlier, Charles Ehresmann had conjectured that every smooth codimension-one foliation on S3 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 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:
the fundamental group π 1 ( M 3 ) {\displaystyle \pi _{1}(M^{3})} is finite, the second homotopy group π 2 ( M 3 ) ≠ 0 {\displaystyle \pi _{2}(M^{3})\neq 0} , 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.[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]