Global Section of Fiber Bundles with Contractible Fiber

In summary, the conversation discusses the existence of a global section for a fiber bundle with contractible fiber. The mention of this fact is found in Guillemin and Sternberg's "Supersymmetry..." and can be proven using obstruction theory. The argument involves extending a map over a sphere to the disk it bounds, and the "obstruction" is that the map may be homotopically nontrivial. However, this is not a problem if the fiber is contractible. It is suggested that over local trivializations, the contraction mapping can be used to create a new constant map, which can be pieced together to form a global section. The validity of this approach has not been confirmed.
  • #1
MagmaMorphic
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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.
 
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  • #2
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.
 
  • #3
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.
 

FAQ: Global Section of Fiber Bundles with Contractible Fiber

1. What is a global section of a fiber bundle with a contractible fiber?

A global section of a fiber bundle with a contractible fiber is a continuous map that assigns a unique point in the fiber to every point in the base space. This means that the entire fiber is covered by the global section, and the fiber itself is contractible, meaning it can be continuously deformed to a single point.

2. Why is a contractible fiber important in fiber bundles?

A contractible fiber is important in fiber bundles because it allows for a global section to exist. This global section can then be used to study the properties of the bundle and its base space, as well as to define important concepts such as connections and parallel transport.

3. How is a global section related to the concept of a cross section?

A global section is a type of cross section, but not all cross sections are global sections. A cross section is simply a continuous map that assigns a point in the fiber to every point in the base space, while a global section is a special type of cross section that covers the entire fiber and has the additional property of being a continuous deformation of the identity map on the base space.

4. Can a global section always be defined for a fiber bundle with a contractible fiber?

No, a global section cannot always be defined for a fiber bundle with a contractible fiber. This is because the existence of a global section depends on the topology of both the base space and the fiber, and some topological spaces do not allow for global sections to be defined.

5. How does the global section affect the topology of the fiber bundle?

The global section has a significant impact on the topology of the fiber bundle. It allows for the construction of local trivializations, which can then be used to define a topological structure on the bundle. This topological structure is important for studying the properties of the bundle and its relationship to the base space.

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