Reeb sphere theorem

Morse foliation

A singularity of a foliation F is of Morse type if in its small neighborhood all leaves of the foliation are levels of a Morse function, being the singularity a critical point of the function. The singularity is a center if it is a local extremum of the function; otherwise, the singularity is a saddle.

The number of centers c and the number of saddles s , specifically c − s, is tightly connected with the manifold topology.

We denote ind p = min(k, n − k), the index of a singularity p , where k is the index of the corresponding critical point of a Morse function. In particular, a center has index 0, index of a saddle is at least 1.

A Morse foliation F on a manifold M is a singular transversely oriented codimension one foliation of class C² with isolated singularities such that:

Reeb sphere theorem

This is the case c > s = 0, the case without saddles.

Theorem: Let M n be a closed oriented connected manifold of dimension n ≥ 2 . Assume that M n admits a C 1 -transversely oriented codimension one foliation F with a non empty set of singularities all of them centers. Then the singular set of F consists of two points and M n is homeomorphic to the sphere S n .

It is a consequence of the Reeb stability theorem.

Generalization

More general case is c > s ≥ 0.

In 1978, E. Wagneur generalized the Reeb sphere theorem to Morse foliations with saddles. He showed that the number of centers cannot be too much as compared with the number of saddles, notably, c ≤ s + 2 . So there are exactly two cases when c > s :

He obtained a description of the manifold admitting a foliation with singularities that satisfy (1).

Theorem: Let M n be a compact connected manifold admitting a Morse foliation F with c centers and s saddles. Then c ≤ s + 2 . In case c = s + 2 ,

Finally, in 2008, C. Camacho and B. Scardua considered the case (2), c = s + 1 . Interestingly, this is possible in a small number of low dimensions.

Theorem: Let M n be a compact connected manifold and F a Morse foliation on M . If s = c + 1 , then