# Foliated bundles and suspensions

Hi,

Is there any criteria, except for the existence of a flat connection, for a foliated bundle $E$ to be a suspension ( a foliated flat bundle)? For example, the Kronecker foliation on the torus is a suspension $\mathbb{R}\times_{\mathbb{Z}} \mathbb{S}^1$ , i.e. its of the form $M\times_\Gamma F$, where $\Gamma$ acts freely and transitively on the manifold $M$ and there is a free action $\rho: \Gamma \rightarrow Diff(F)$.

Note: The Kronecker foliation is induced by the vector field , $a \frac{\partial}{\partial x} + b \frac{\partial}{\partial y}$ on $\mathbb{R}^2$ with $a,b$ constants. The action of $\mathbb{Z}$ on $\mathbb{R}\times \mathbb{S}^1$ is given by:

$(r, \exp{iz}).m= (r+m, \exp{(iz+m\alpha)})$ for some $\alpha \in \mathbb{R}$.

Thanks

-
Indrava, could you please explain what is a foliated bundle, or foliated flat bundle? I don't think these are completely standard math expression (at least I don't know what they mean). – Dmitri Jan 24 '10 at 0:36
I'm confused by your terminology. If F -> E -> B is a bundle of manifolds with dim F = n, then the bundle is flat (i.e. arises from a map pi_1(B) -> Diff(F) by your construction) iff E admits a codimension n foliation transverse to the fibers. Proof: monodromy. Can you give an example of a "foliated bundle" (in your terminology) which is not a "foliated flat bundle"? – Tom Church Jan 24 '10 at 0:43
Tom, your iff is not exactly right. If F is not compact then leaves can scape to infinity preventing the lift of paths from the base to them. – Jorge Vitório Pereira Jan 24 '10 at 0:56
Thanks for catching that! Actually for F noncompact the definition of "foliation transverse to the fibers" is often taken to require that each leaf cover the base (e.g. "Geometric Theory of Foliations", p.91), but that's not at all obvious from the terminology, so it's a good point to emphasize. – Tom Church Jan 24 '10 at 1:14
Hi! Thanks a lot for your replies! @Dmitri: I'm sorry I didn't know that these terms are not widely used, the definition of a foliated bundle can be found in the link given by jvp or at foliations.org/surveys/FoliationLectNotes_Milnor.pdf. The term "foliated flat bundle" is mentioned in the Survey article by Kordyukov here: arxiv.org/PS_cache/math/pdf/0504/0504095v2.pdf @Tom: Sorry for the confusion. – Indrava Roy Jan 24 '10 at 1:46

I am interpreting the terminology as follows:

• foliated bundle = foliation on the total space of the bundle
• foliated flat bundle = suspension

If the fibers of the bundle are compact then the only thing you need is transversality between leaves and fibers.

If the fibers are not compact then besides transversality you also need to prevent the leaves from scaping to infinity. To do it, you can ask the natural projections from leaves to the base of your bundle to be covering maps.

See Chapter V of Geometric Theory of Foliations.

-
Thank you for the answer! Also thanks for the link, I didn't know about this book, seems great! However, I have a related question (which might be a naive one): How would u know that a given foliated manifold can not be expressed as a foliated flat bundle? – Indrava Roy Jan 24 '10 at 1:52
Yes, thats the right interpretation. Sorry again for the confusion. – Indrava Roy Jan 24 '10 at 1:54