1

If one study bundles (being locally trivial or not, that does not matter) in the category of topological spaces, the projection map simply needs to be surjective or ''onto''.

On the other hand, when one consider smooth manifolds, Wiki says that In the category of differentiable manifolds, fiber bundles arise naturally as submersions of one manifold to another. However, Jeffrey M. Lee's book Manifolds and differential geometry does not impose any condition on the projection except the condition of local triviality (because he is following Steenrod's approach).

Questions

1.- When dealing with fibre bundles (i.e. locally trivial bundles), are the submersion condition equivalent to the locally trivial condition?

2.- If I'm not interested in fibre bundles but only in bundles (following Husemoller approach) I must impose onto or submersion?

3.- Some simple example of an application that is surjective but not a submersion?

Thanks

PD: I think three questions are related and can be asked simultaneously.

Dog_69
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1 Answers1

1

The relationship between submersions and fibrations comes from Ehresmann's fibration theorem which asserts that a proper submersion between smooth manifolds is a fibration. Wikipedia is being somewhat imprecise; you need properness or else there are counterexamples.

I'm not sure what distinction you're making between fiber bundles and bundles.

The smooth function

$$f : \mathbb{R} \ni x \mapsto x^3 \in \mathbb{R}$$

is surjective (even bijective) but fails to be a submersion at $x = 0$.

Qiaochu Yuan
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  • Nice and simple counterxample for number 3. The distinction on 2 is that bundles hasn't got the locally trivial property. It would be the cocept defined by Husemoller in the category of topological spaces but now considering smooth manifolds. I think, 2 is equivalent to ask: What is an epimorphism in the category of smooth manifolds? Submersions? And for 1, I'm not sure to undesrtand your answer. Ehresmann's fibration theorems means that all bundle projections of fibre bundles $\pi:E\rightarrow M$ are proper maps? – Dog_69 Jan 02 '18 at 22:38
  • I don't know what there is to say about bundles that aren't locally trivial. Submersions are not the epimorphisms in the category of smooth manifolds. Ehresmann's theorem does not say that: it says that if you want to check that a map between smooth manifolds is a fibration, a sufficient (but not necessary) condition is that it be a proper submersion. – Qiaochu Yuan Jan 02 '18 at 22:48
  • Here is the definition of bundle according to Wiki: https://en.wikipedia.org/wiki/Bundle_(mathematics). I'm confussing because there, a bundle can be defined in the category theory as an epimorphism $\pi:E\rightarrow M$ between two objects $M$ and $E$ in such a category. So, I'm traying to find the equivalent concept in the category of smooth manifolds. And I'm also worried about why in fibre bundles the projection map must be a submersion. So I have asked if submersion condition for fibre bundles was related to locally trivial condition. I hope this comment explains my questions. – Dog_69 Jan 02 '18 at 23:03
  • Reading https://en.wikipedia.org/wiki/Fibered_manifold I think I have some answers. Firts: a bundle (the concept defined by Husemoller and Wikipedia but adapted to the category of smooth manifolds) doesn' need to be a submersion. When the bundle projection is a submersion, then the bundle is called a fibred manifold. Second: A fibre bundle is a fibred manifold (not a bundle) such that the locally trivial condition holds. – Dog_69 Jan 02 '18 at 23:29
  • To be honestly, the entry https://math.stackexchange.com/questions/2419907/monoepimorphisms-as-imsubmersions-in-the-category-of-manifolds would be added to my above comment. – Dog_69 Jan 02 '18 at 23:37