@Thorgott The trivializations of $(\pi, f) : (\pi, f)^{-1}(X \times (a_i-\epsilon, a_i+\epsilon)) \to X \times (a_i-\epsilon, a_i+\epsilon)$ (for $i = 0, 1$) and $\pi : W[a_0, a_1] \to X$ are both obtained from the following fashion:
In the first case, choose local vector fields $\partial/\partial x_1, \cdots, \partial/\partial x_n$ and $\partial/\partial t$, lift them up by the submersivity property of $(\pi, f)$ and then flow along them to generate a trivializing chart. In the second case, use only $\partial/\partial x_1, \cdots, \partial/\partial x_n$, lift them up by the submersivity p…
