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).(adsbygoogle = window.adsbygoogle || []).push({});

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 forholomorphic linebundles, 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 Ineedto calculate something, I'm more generally interested in how one would approach this problem.

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

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