I am reading Bott-Tu "Differential forms in Algebraic Topology" and trying to understand proof of Leray-Hirsch. During the process I realized that I do not understand basic definitions.
Let $M$ be smooth $n$-dimensional manifold of finite type (admits fintie good cover). Let $\pi:E\to M$ be a fiber bundle over $M$ with fiber $M$. My question is: does this imply that I can get the "bundle structure" from homomorphisms that would go $$\phi_\alpha:E|_{U_\alpha}\xrightarrow{\simeq}U_\alpha\times F$$ with $\{U_\alpha\}_{\alpha=1}^k$ being the given good cover?
I would say that the answer is no, because it is just a cover, not a chart, doesn't come with diffeomorphisms onto $\mathbb{R}^n$.
I feel that I think about something very wrong, so if someone can help me realize what it would be really great. Here are my thoughts:
What does the good cover give me? Let's say $M$ can be covered by a good cover $\{U,V\}$. I know that it means that the intersection $U\cap V$ is contractible, i.e. diffeomorphic to $\mathbb{R}^n$. How do I connect it with the fiber bundle? Does it actually induce a smooth structure for $M$? Just by inclusions? But this would imply that all elements of a good cover have to be diffeomorphic to $\mathbb{R}^n$?