Classifying vector bundles Given a manifold $M$, is there a way of classifying up to isomorphism all possible vector bundles over $M$ of a given rank? Some other questions on this site deal with specific cases, which all seem very difficult. Are there cases where it is completely known, and what conditions must we place on $M$ to be able to say anything? What makes the other cases so difficult?
Related to that, suppose we are given some space, for example the Möbius band, which is a line bundle over $S^1$. Is there a way to classify all spaces which have the Möbius band as a vector bundle, for example? 
 A: The usual way of classifying $n$-dimensional vector bundles is to look at the classifying space, which is the infinite Grassmannian $Gr(n,\infty)$. There is a ``tautological" bundle over $Gr(n,\infty)$, and every $n$-dimensional vector bundle over a manifold $M$ is a pull-back of the tautological bundle by some map $M\to Gr(n,\infty)$. This pull-back is invariant under homotopy so you really are studying homotopy classes of maps $M\to Gr(n,\infty)$. That is the bundles are in 1-1 correspondence with the set $[M,Gr(n,\infty)]$ and there are lots of tools available, such as characteristic classes, for studying this set.
For your question about the Möbius band, it is a 2-manifold, and the dimension of the base plus the fiber must be $2$. The only nontrivial possibility is that it is a line bundle over a $1$-manifold, and the only connected $1$-manifold is the circle. In general a vector bundle is homotopy equivalent to its base space, so at least the homotopy type of the manifold $M$ is uniquely determined by the bundle. There are examples of non-homeomorphic manifolds that become homeomorphic after crossing with $\mathbb R$, so you can have two different $n$-dimensional vector bundles have homeomorphic total space.
