Given a real oriented vector bundle E over the base space B of rank n, such that the Euler characteristic class in the n-th cohomology group of B vanishes, is it true that there exists a global nowhere-vanishing section of the bundle? 
Any idea where to find a proof or a counterexample?


Thanks!



Vanishing of Euler class
Hi Dima,


I think that the answer to your question is no: as it is pointed out in the book "Differential Forms in Algebraic Topology" of R. Bott and W. Tu, cohomological invariants are too coarse to ensure the existence of geometrical objects.


More precisely, Example 23.16 of the book of Bott and Tu shows that $S^4$ has a nontrivial rank 3 vector bundle with vanishing Euler class.


Best,


Matheus


