Extending Trivializations and Structure Groups

In summary, the conversation discusses the extension of trivializations over the 0-skeleton to the 1-skeleton in a line bundle with a connected structure group. The understanding is that the pullback bundle is trivial for every k-cell, but the connectedness of the structure group allows for the extension of a given trivialization. There is also mention of a canonical trivialization over the cells and a question about examples of trivializations that do not extend beyond the 0-skeleton. The discussion also touches on the idea of extending maps from the interior of a cell to its boundary.
  • #1
Bacle
662
1
Hi, Everyone:

LetB: E-->X be a line bundle, with scructure group G and X has a CW -decomposition.

I am trying to understand why/how, if the structure group G of B is connected,
then any trivialization over the 0-skeleton of X can be extended to a trivialization
of the 1-skeleton.

I understand that for every k-cell f:D^k --.X (D^k is the k-disk) , the
pullback bundle is trivial (by contractibility of D^k), but I don't see how/why
the connectedness of G alllows us to extend a given trivialization from the
0-skeleton to the 1-skeleton.

There is also a mention of a canonical trivialization over the cells. Anyone
know what that is.?

Thanks.
 
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  • #2
I wonder if someone knows examples of trivializations (i.e., global sections)
that do not extend beyond the 0-skeleton, maybe the 1-skeleton. It seems to
come down to extending maps from the interior of a cell to its boundary, maybe
with retractions.
Am I on the right track.?
 

1. What is the purpose of extending trivializations and structure groups?

The purpose of extending trivializations and structure groups is to generalize the concept of trivializations and structure groups to more complex mathematical objects, such as fiber bundles and principal bundles. This allows for a deeper understanding of the underlying structures and symmetries in these objects.

2. How is the extension of trivializations and structure groups done?

The extension of trivializations and structure groups is done through the use of transition functions. These functions describe how local trivializations on different patches of a bundle are related to each other, and allow for a consistent global trivialization to be constructed.

3. What are the benefits of extending trivializations and structure groups?

The benefits of extending trivializations and structure groups include a more comprehensive understanding of the geometric and topological properties of fiber and principal bundles, as well as the ability to define and study more complex mathematical structures.

4. How does extending trivializations and structure groups relate to differential geometry?

Extending trivializations and structure groups is an important concept in differential geometry, as it allows for the study of smooth and continuous structures on manifolds. By extending trivializations and structure groups, we can define smooth bundles and principal bundles, which are essential in many areas of mathematics and physics.

5. Are there any applications of extending trivializations and structure groups?

Yes, there are many applications of extending trivializations and structure groups in various fields of mathematics and physics. For example, in gauge theory, extending structure groups allows for the study of non-abelian gauge fields and their associated principal bundles. In addition, the theory of characteristic classes, which has applications in topology and algebraic geometry, relies heavily on the concept of extending trivializations and structure groups.

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