On learning about how to define smooth vector fields on a manifold $M$, I learned that one should first define a tangent bundle , $T(M)$, as $\cup T_p(M)$ together with a topology(smooth structure). And then, a smooth vector field $X$ would be a smooth map from $M \rightarrow T(M)$ s.t. $\pi \circ X=id_M$
However, it is obvious that the use of the auxilary manifold $T(M)$ should not be confined to merely offering a good def. of "smooth" vector fields. Well, at least you would not be using the global topology of $T(M)$. (Since the checking of "smoothness" is done locally)
I am inclined to believe that the global topology of $T(M)$ may play a more important role in determining the properties of vectorfields on $M$. So here are my two questions:
Can you give an example where the global topology of $T(M)$ is used to control certain properties(possibly about vec. fields) on $M$?
If the global topology on $T(M)$ is indeed important, how do we set about determining it? I have only seen trivial examples where for $M=S^1 or \mathbb{R}^n$, $T(M)=M\times \mathbb{R}^n$. But surely, there are examples where $T(M)\neq M\times \mathbb{R}^n$. And how do we determine the topology in that case?