It seems that I have proven something wierd, namely that the Euler Characteristic of every vector bundle vanishes:
Every vector bundle has a non-trivial involution ($(p,u)\rightarrow (p,-v)$) and therefore is the boundary of another manifold. Using that bounding manifolds have Euler characteristic $0$ we get that every Vector bundle must also.
First of all, is the proof correct? And if it is, can anybody give any intuition behind it? While I have proved it , it seems very odd to me. For example which manifold does the Mobious Strip bound ?