Reeb stability theorem

Remarks

The theorem is a global consequence of the local Reeb stability theorem. The local theorem says that a compact leaf with finite holonomy group has a neighbourhood made of leaves that are finite covers of it. Idea of the proof: since $\pi_1(L)$ is finite and its holonomy acts on a transverse interval preserving orientation, the holonomy is trivial, so nearby leaves are copies of $L$. The set of points on compact leaves diffeomorphic to $L$ is then open, and a limit argument shows it is closed, so it is all of $M$.

Example: if a closed $3$-manifold $M$ has a transversely orientable codimension one foliation with a leaf homeomorphic to $S^2$, the theorem shows that $M$ fibres over $S^1$ with fibre $S^2$. The hypothesis on the fundamental group is needed: the Reeb foliation of $S^3$ has a compact torus leaf, with infinite fundamental group, while all its other leaves are planes.

Going deeper

For a student

The theorem shows that spheres and other leaves with finite fundamental group are rigid inside codimension one foliations. It is a first step in the study of compact leaves, which leads to Novikov's compact leaf theorem. The key notion is holonomy, the return map of a transverse interval along loops in a leaf, see the holonomy groupoid of a foliation. A corollary is that $S^3$ carries no codimension one foliation with a leaf diffeomorphic to $S^2$, since $S^3$ does not fibre over the circle.