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

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In summary, the conversation discusses a theorem that states for compact manifolds, there is a closed, non-singular one-form on the manifold if and only if it fibres over S^1. The question is posed if a similar theorem exists for non-compact manifolds to fibre over the reals. It is mentioned that the proof for the compact case cannot be applied to the non-compact case due to the reliance on the non-exactness of any closed, non-singular one-form on a compact manifold. The conversation then delves into the topic of Ehresmann's Theorem and its relevance to the question. It is noted that the key hypothesis for this theorem is properness of the map, which is
  • #1
holy_toaster
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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?
 
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  • #2
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.
 
  • #3
If you're looking for a far more general statement of sufficient conditions, look up Ehresmann's Theorem.
 
  • #4
Aha. Thank you. I think that theorem is what I was looking for.
 
  • #5
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.
 
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FAQ: Is There a Theorem for Fibration Over R for Non-Compact Manifolds?

What is the "Theorem for fibration over R"?

The "Theorem for fibration over R" is a mathematical theorem that states that if a function is defined over a closed interval on the real number line, and it satisfies certain conditions, then it can be extended to a function over the entire real number line.

What are the conditions that need to be satisfied for the "Theorem for fibration over R" to hold?

The conditions are that the function must be continuous and have a finite number of discontinuities, and the behavior of the function at the discontinuities must be well-behaved. This means that the function must approach the same limit from both sides at each discontinuity.

How is the "Theorem for fibration over R" useful in mathematics?

This theorem is useful because it allows for the simplification and generalization of many mathematical problems. It can also be used to prove other theorems or solve equations, particularly in the fields of analysis and topology.

Can the "Theorem for fibration over R" be applied to functions with multiple variables?

Yes, this theorem can be extended to functions with multiple variables. However, the conditions for the theorem to hold become more complex and may require additional mathematical tools.

Are there any limitations to the "Theorem for fibration over R"?

Yes, there are some limitations to this theorem. It may not be applicable to all types of functions, and the conditions for the theorem to hold may be difficult to verify in some cases. Additionally, it can only be applied to functions defined on a closed interval on the real number line.

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