Novikov's compact leaf theorem (original) (raw)
From Wikipedia, the free encyclopedia
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
[edit]
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
[edit]
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 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.