Coverings of stably parallelizable manifolds
We exhibit a six dimensional manifold with a line bundle on it which is not the pullback of a bundle on .
We study secondary obstructions to representing a line bundle as the pull-back of a line bundle on and we interpret them geometrically.
This note answers a question of V. V. Vershinin concerning the properties of Buchstaber's elements Θ2i+1(2) in the symplectic cobordism ring of the real projective plane. It is motivated by Roush's famous result that the restriction of these elements to the projective line is trivial, and by the relationship with obstructions to multiplication in symplectic cobordism with singularities.