In Book of GEOMETRY, TOPOLOGY and PHYSICS of Mikio Nakahara 2nd Edition, p.353-354, he described the reconstruction of fibre
$$(E,\pi,M,F,G)$$
total space $E$, projection $\pi$, base $M$, fiber $F$, structure group $G$.
my own questions:
He said that the given data reconstruct a fibre bundle $E$ "uniquely." How do we show that it is uniquely constructed? Why not other possible fibre bundles?
How is the trivialization $\phi_i$ given by the above given data $(E,\pi,M,F,G)$? It is not obvious $$\phi_i: U_i \times F \to \pi^{-1}(U_i)$$ is determined by $(E,\pi,M,F,G)$...?
He said that: This procedure may be employed to construct a new fibre bundle from an old one. Isn't this claim kind of conflict with that the reconstruction of a fibre bundle $E$ "uniquely"? If not, what data requires to change among $(E,\pi,M,F,G)$ to construct a new fibre bundle from an old one? I don't think it is changing the trivialization $\phi_i$ because which is a gauge transformation.
To make sure everyone follow his statement, I attach Nakahara 9.2.2 below.
