are there some general method in judging if a fible bundle is trivial?
At least,for vector bundles,there is a well-developed theory,that is Charicteristic Classes.the triviality of vector bundles is equivalent to the vanishment of its characteristic classes.
for principal bundles,the triviality is equivalent to the exsience of a cross section.
(any good perspective on this assertion?besides,how to tell if there is a cross section)
for general fiber bundle (E,F,B,G),(here,E is total space,F the fiber,B the base space,G the structure group),we can construct its associate principal bundle (E',G,B,G),i.e.to replace the fiber F with the topological group G.there is a theorem that a fiber bundle is trivial iff its associate principal bundle is trivial.
hence the problem is reduced to find a cross section of principal bundle.
I want to know if there is some other methods that are more usable?
Thank you!
Best Answer
I will not really give an answer, but note that Turaev has a recent preprint on the archive (also this reply is too long for a comment!)
Abstract We study the existence problem and the enumeration problem for sections of Serre fibrations over compact orientable surfaces. When the fundamental group of the fiber is finite, a complete solution is given in terms of 2-dimensional cohomology classes associated with certain irreducible representations of this group. The proofs are based on Topological Quantum Field Theory.. Comment: 38 pages
Publication details Download http://arxiv.org/abs/0904.2692
This discusses the existence of sections for fibre bundles over surfaces. As Paul mentions above, the answer is 'obstruction theory' in general.