TITLE: Principal bundles on 3-manifolds
QUESTION [8 upvotes]: If G is a simply connected Lie Group then why is every G-bundle over an orientable 3-manifold trivial? (Why is orientability important?)

REPLY [2 votes]: Are you sure that orientability is necessary? The result is proved in lemma 4.1.1.  here and I do not see where orientability is used.