Re: question about fiber bundles

From: Jannick (nospam_at_spam.com)
Date: 01/21/05 

Date: 20 Jan 2005 19:45:02 -0500

>>To be more precise: Let E,B be manifolds (as smooth as needed). Let
>>p:E --> B be a projection onto B. We know that p^(-1) {x} is
>>diffeomorphic to F, for all x in B. Does it imply that for any x in B
>>there exists an open set U containing x such that:
>>p^(-1) {U} is diffeomorphic to UxF, such that each p^(-1) {x} is
>>diffeomorphic {x} x F ?
>
>
> The projection map needs to be proper, i.e. it maps closed sets on
> closed sets. This might not be case in general.
>

Here is a counterexample for the assertion:

If p is a covering with strictly countable fibers, then you can
carefully cut off some closed set A of E such that p restricted to
E\minus A is still a local diffeomorphism with strictly countable
fibers. But there is no chance to locally construct a trivializing chart
with standard fiber IN.
Take the projection p:IN x IR --> IR as an example.