I am interested in the topology of locally trivial fiber bundles over $S^{1}$ but have not been able to find a resource covering this case at a level I, as a non expert in topological matters, am able to understand.
In particular, given a locally trivial fiber bundle $\pi: E \rightarrow S^{1}$ it seems reasonable to me that the topology of the total space $E$ should be fully specified by the gluing of the space on the boundary. I.e. the total space can be given as $[0,1] \times F$ where $(0,f)$ and $(1, \phi(f))$ are identified. Here $F = \pi^{-1}(x \in E)$ is the standard fiber and $\phi: F \rightarrow F$ is a self-homoemorphism. The topology of the total space should then only depend on the conjugacy class of $\phi$ in the mapping class group of $F$.
My question is the following
- Is the above correct in general, and if so is there some reference that covers this simple case I can have a look at?
- In the above I made reference to the topology of the total space. Up to what level (homeomorphism/homotopy etc) does the mapping class group element $\phi$ belongs to specify the global topology?