Is There a Theorem for Fibration Over R for Non-Compact Manifolds?

[holy_toaster]

Hi! I have a question that maybe somebody can answer, I hope...

There is a theorem that holds for compact manifolds M. It says that M fibres over S^1 if and only if there is a closed, non-singular one-form on M. (Meaning M is the total space of a fibre-bundle p : M -> S^1)

Now my question is if there is a similar theorem for non-compact manifolds, to fibre over the reals R ? The proof from the compact case can not be applied to the non-compact case because it relies on any closed, non-singular one-form being non-exact on a compact manifold.

Any ideas if such a result exists or where I could find it?

 

[zhentil]

Nope, not true. Take any function f:M->R with isolated critical points. Delete the critical points and call the new manifold M'. It certainly needn't fiber.

 

[zhentil]

If you're looking for a far more general statement of sufficient conditions, look up Ehresmann's Theorem.

 

[holy_toaster]

Aha. Thank you. I think that theorem is what I was looking for.

 

[mathwonk]

note the key hypothesis of properness of the map, which is the relative version of compactness. Indeed the generality is almost illusory, since under the proper submersion hypothesis it seems the inverse image of any closed ball in the target is a compact submanifold with boundary of the source.