Consider a polynomial $f
\in \mathbb C[x_1,\dots ,x_n]$. An atypical value of $f$ is a complex number about which $f:\mathbb C^n\to \mathbb C$ is not a *topological* fiber bundle. Writing $\mathrm{Atyp}(f)$ for the atypical values of $f$ we thus have a fiber bundle $$\mathbb C^n\setminus f^{-1}(\mathrm{Atyp}(f))\to \mathbb C\setminus \mathrm{Atyp}(f).$$

A book I'm reading now says the following.

The fundamental group $\pi_1(\mathbb C\setminus \mathrm{Atyp}(f))$ acts therefore on the generic fiber $f^{-1}(t_0)$ up to isotopy. The image of this action is called the geometric monodromy group. The pullback of $f$ along an admissible loop (image doesn't hit atypical values) is a fiber bundle which yields an automorphism of the fiber, called the

geometric monodromy.

I am confused by this. We have for any Hurewicz fibration $E\to B$ the transport functor $\pi_1(B)\to \mathsf{hTop}$ which lands in the homotopy category. This is the only meaning I can think of for "acts up to isotopy". On the other hand, this does not give an automorphism of the fiber in the topological category.

**Question.** Is the geometric monodromy defined as the *homotopy type* of the continuous map between fibers given by the homotopy lifting property? Is the geometric monodromy group(oid) given by the image $\pi_1(B)$ in $\mathsf{hTop}$?