All
I am looking at the following theorem in the book of Bredon and my question is how does it work for manifolds instead of spheres? The proof is understood, but I am wondering if it is enough to just said that it also works for manifolds? Since taking at each point of given manifold a sphere, we establish orientations locally which are all consistent, so global orientation for manifold is chosen. So we have this theorem for such each sphere, which means it should hold for the whole manifold, as that orientations coincide. Could you please advise if my understanding is correct or give a hint how should I proceed for manifolds?
