Global Section of Fiber Bundles with Contractible Fiber

MagmaMorphic

Is it true in complete generality that every fiber bundle with contractible fiber have a global section? Or do some sort of restrictions on the bundle need to be made? I ran across a mention of this fact in Guillemin and Sternberg's "Supersymmetry..." and I'm not sure how to prove it.

zhentil

Do you know any obstruction theory? With that in hand, it's trivial. I'm not sure how far outside CW complexes obstruction theory extends, however.

If you don't know obstruction theory, here's a rough idea of how the argument works. Suppose we have a CW-structure on our base space, and assume inductively we've defined a section over the (k-1)-skeleton. We would like to extend the map over the k-skeleton. The potential problem with this is that we're attempting to extend a map defined over a sphere to the disk it bounds. The "obstruction" to being able to extend it is precisely that the map may be homotopically nontrivial. (think of the section as a map D->D x F, x -> (x,s(x))). This, of course, is not a problem at all if the fiber is contractible.

lavinia

I would think that over local trivializations of the bundle the contraction mapping would make a new section that is constant. I think you can piece these constant maps together over all local trivializations because the contraction can be done to any point in the fiber.

haven't checked this though.