I am reading Arnol'd's book "Singularities of Differentiable Maps, Volume 2", and I'm trying to understand geometric monodromy, as he defines it on page 10. I have simplified the assumptions a bit since this is the case I am interested in.
Suppose $E^n$ is a complex manifold and $f: E \to \mathbb{D}$ is a holomorphic map, where $\mathbb{D} \subset \mathbb{C}$ is the unit disk. Assume moreover that $f$
- is proper;
- is a smooth submersion when restricted to $\partial E$;
- has finitely many critical points in $\mathring{E}$, and critical values in $\mathring{\mathbb{D}}$.
For a loop $\gamma: [0,1] \to \mathbb{D}$ avoiding critical values, he defines the monodromy $h_\gamma$ as follows: letting $\gamma(0) = z_0$, obtain a family of maps $\Gamma_t: E_{z_0} \to E_{\gamma(t)}$ by the homotopy lifting property, and set $$ h_\gamma = \Gamma_1: E_{z_0} \to E_{z_0}. $$ He claims that this family $\Gamma_t$ can be chosen "consistent with the direct product structure" on $f\vert_{\partial E}$, and hence that $\Gamma_1$ is the identity on $\partial E_{z_0}$.
My question: what sense of compatibility is he referring to, and how does this imply that the monodromy is trivial near $\partial E_{z_0}$?
I know that since we assume $f\vert_{\partial E}$ to be a smooth submersion over $\mathbb{D}$, we have that $f: \partial E \to \mathbb{D}$ is isomorphic to the trivial bundle $\partial E \times \mathbb{D} \to \mathbb{D}$. However, I do not understand what compatibility with this product structure means, can we use the $\Gamma_t$ to trivialise the restricted bundle?
As for the maps $\Gamma_t$, one could choose a connection on $E$ and take $\Gamma_t$ to be the induced parallel transport maps. Is this perhaps related to a specific trivialization of $f\vert_{\partial E}$?