What can be said about the leaves of a regular foliation?

Question

I was wondering about the following.

Let $M$ be a (smooth, closed, connected and oriented) manifold endowed with a regular foliation (i.e. such that all the leaves are smooth submanifolds of the same dimension and such that their tangent bundles form a smooth distribution in $TM$). Also assume all leafs are closed (thanks to @MikeMiller for pointing this out). What can be said about the leaves of the foliation? That is, how "different" can they be? Can they change topologically? If not, what additional structure do we need to define unique isomorphisms between leaves?

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I think I can prove the following (i.e. I have an idea about the proof, but haven't checked the details. I can post my idea if you want.)

Let $F$ be a fixed leaf. Then there exists an open neighborhood $U$ of $F$ in $M$, obtained as an union of leaves, such that every leaf in $U$ is a covering of $F$.

Answer 1

First, even if all the leaves are embedded submanifolds, nearby leaves can be drastically different. For instance, the following is a counterexample to your covering space conjecture: let $\Sigma_{g,1}$ be the surface of genus $g$ with a small open ball removed. Let $M = \Sigma_{g,1} \times S^1$, and foliate it with leaves which, away from the boundary, are $\Sigma_{g,1} \times \{t\}$, but near the boundary the leaves spiral forever towards the torus as in the Reeb foliation. This is a special case of what Gabai calls generalized Reeb components here (on page 612). Then the leaves are diffeomorphic to the interior of $\Sigma_{g,1}$, except the boundary leaf, which is a torus $T^2$. Of course you could glue things like this together to get a torus which on one side has a $\Sigma_{g,1}$ as a leaf and on the other side, a $\Sigma_{h,1}$.

The germ of a foliated neighborhood of a compact leaf $L$ is completely determined by the "germinal holonomy" of the leaf $L$; for a precise statement, see proposition 2.3.9 in Candel-Conlon, Foliations I. This leads to a few points.

First, the Reeb stability theorem: if $\pi_1(L)$ is finite, then a neighborhood of $L$ is foliated by leaves that cover $L$, and if the foliation. This follows from the previous point, given that you can conjugate a germ of a diffeomorphism of finite order to a linear one. (In particular, I did not assume that all the leaves near $L$ are compact: that comes for free.) For codimension 1 foliations, Thurston later generalized this to the condition that $H^1(L;\Bbb R) = 0$ (provided the foliation has trivial normal bundle). The actual statement is much stronger; see his article here.

Next, suppose you have a compact leaf with finite fundamental group. Someone (I think Reeb?) proved that the set of compact leaves with finite fundamental group is closed in the manifold $M$; and hence the union of these is open (by Reeb stability) and closed, so every leaf is compact with finite fundamental group. You can use this to prove eg that the only (transversely oriented, closed) 3-manifold foliation with an $S^2$ leaf is $S^2 \times S^1$.

Now, a sort of converse to the spirit of your question. Suppose $M$ is foliated by compact leaves. You can pick a small foliated neighborhood of any one of these and invoke 2.3.9 as before. In the case of $S^1$, here's a nice description: because all of the leaves must be $S^1$, in particular in our foliated neighborhood, the germinal diffeomorphism given by holonomy must be finite order (lest there be a $\Bbb R$ showing up in the same neighborhood). Elements of finite order in the group of germinal diffeomorphisms are conjugate to linear ones, and hence the foliation has a local form: it locally looks like the foliation of $S^1 \times D^n$ where $S^1 \times \{0\}$ is a leaf, and as you go around the $S^1$ factor, the foliation rotates. (Note that this in particular the local model of a Seifert space.) This is precisely what it would mean for a foliation by $S^1$s to be an $S^1$-bundle over an orbifold, so all foliations by circles are bundles over an orbifold.

I expect this to be true in much further generality: I expect that any foliation by compact leaves is a bundle over an orbifold (where here I'm allowed to change the covering type of the fiber). If you believe that there are only countably many fiber bundles over smooth manifolds such that the base and leaf are both compact, then hopefully you'll believe me when I say the same is true in the orbifold setting: hence there should only be countably many foliations by compact leaves. So such things are very rigid constructs, and there are not at all many of them!

Answer 2

The leaves can change topologically, you can have compact and non compact leaves as in the suspension. Let $M$ be a compact manifold and $\hat M$ its universal cover Let $h:\pi_1(M)\rightarrow Diff(F)$ consider the manifold $N$ which is the quotient of of $\hat M\times F$ by the diagonal action of $\pi_1(M)$. The foliation $\hat M\times y$ of $\hat M\times F$ is preserved by $h$ and defines on $N$ a foliation ${\cal F}$. Suppose that $F$ is not compact and $h$ has a unique fixed point $x_0$ and the stabilizer of every element different of $x_0$ is $\{1\}$. Let $p:\hat M\times F\rightarrow N$ be the projection, the unique compact leaf of ${\cal F}$ is the leaf through $p(x_0)$. An example of this is a flat vector bundle $p:V\rightarrow M$, only the zero section is a compact leaf of the horizontal foliation. You can eventually construct examples for which the orbit of every element by $h$ is finite. In this case, all the leaves are compact and are not necessarily diffeomorphic if you have a fixed point by the $h$ and if there are points which are not fixed by $h$. This can be achieved if $\pi_1(M)$ is finite.

This situation is quite the general situation. Haefliger (I believe in his thesis) has described the neighborhood of the leaf (compact) of every foliation by using the holonomy representation. Take a compact leaf ${\cal F}_x$ of the foliation ${\cal F}$ through $x$ on a compact manifold. Let $T$ be a local transveral of the foliation, you can describe a neighborhood $U$ of ${\cal F}_x$ which saturated as the suspention of $ \hat {\cal F}_x\times T$ by the holonomy representation.

The fact that the leaves are isomorphic is related to the fact that the holonomy of every leaf is trivial. I believed that Epstein has conjectured if all the leaves of a foliation on a manifold are compact and without holonomy, then they are diffeomorphic.

Answer 3

If `regular foliation' is defined in terms of how the leaves look, what relation, if any, is there to the foliation of a constraint surface in a Poisson manifold when the constraints are first class (Dirac) and have 0 as a common regular value?