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How to prove a vector bundle is non-trivial using transition functions

  1. Apr 5, 2014 #1
    My knowledge on this topic is a bit sketchy. I realize that there is a whole branch of math out there devoted to checking whether a certain vector bundle is trivial or not, and I partly know some stuff about characteristic classes that seems to do just that (the cases of the Euler classes and Chern classes are well-known to me).

    But what if we define a vector bundle in terms of its transition functions? Is there a natural framework to check whether it's non-trivial? I know a bit of Cech cohomology, and it seems that the cocycle condition satisfied by vector bundle makes this piece of math quite relevant. I know of it being used for holomorphic line bundles, but I don't know of a more general approach where one calculates some related Cech cohomology for a general vector bundle given in terms of its transition functions.

    Can anyone point me into the right direction? It's not even that I need to calculate something, I'm more generally interested in how one would approach this problem.
  2. jcsd
  3. Apr 6, 2014 #2
    First of all, non-trivial characteristic classes are a sufficient, but not necessary condition for non-triviality.

    You're right about Cech cohomology. The result is that bundle isomorphism classes correspond to Cech cohomology classes of the base space. I'm a bit foggy on the details myself. I think you want to work with the sheaf of smooth functions with values in GL(n). Then, you'd want to try to show that if the Cech cocycle defined by your transition functions is null-cohomologous, your bundle is trivial. There are some technicalities because GL(n) is non-Abelian, so you have to figure out how to make sense of that.

    Last edited: Apr 6, 2014
  4. Apr 14, 2014 #3
    You could look at the bundle of frames of the vector bundle. This is a ##GL(V)##-principal bundle and this bundle is trivial if and only if it admits a global nowhere-vanishing smooth (unit) section. You get the transition functions directly from the vector bundle. Correspondingly, ##V## is trivial if and only if the frame bundle is trivial. If it is not trivial, you could go around in three overlapping charts and see whether you can produce a contradiction.
    I hope this helps.
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