Is a fiber bundle specified by its fiber and its base space?

Question

I want to model certain physical concept using fiber bundles, because I believe it is the most suitable language therefor. I know what both the base space and the fiber in this situation are. Do these determine the bundle uniquely (like when a structure group is involved)? I mean, is it necessary to specify a total space? Can it somehow (topologically/geometrically speaking) be described as the disjoint union of fibers?

I must add that the base space and the fibers are smooth manifolds, deprived a priori of any further structure.

Thanks in advance for your attention.

Answer 1

I don't think so. e.g. The Mobius strip for instance is a fiber bundle with base $S^1$ and fiber $[0,1]$, but it is certainly not equivalent to $S^1\times [0,1]$.

Answer 2

No. Over a (say) connected, compact manifold $X$, for example, the Stiefel-Whitney class $w_1$ gives a bijection from real line bundles over $X$ to $H^1(X, \mathbb{Z}_2)$, and the Chern class $c_1$ gives a bijection from complex line bundles over $X$ to $H^2(X, \mathbb{Z})$.