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 $\mathsf{hTop}$?



Definition of geometric monodromy
As you say, a Hurewicz fibration defines a map from $\pi\_1(B)$ to the automorphisms of the fiber in the homotopy category.


But a topological fiber bundle has more structure than a Hurewicz fibration. In fact, a topological fiber bundle defines a map from $\pi\_1(B)$ to the mapping class group of the fiber (i.e. the component group of the group of topological automorphisms of the fiber). This is what "acts up to isotopy" means. 


To see this, maybe one way is to observe that the set of pairs of a point in $B$ and an isomorphism between the fiber over that point and $F$ is an an $\operatorname{Aut}(F)$-torsor, and modding out by the identity component gives a $\pi\_0(\operatorname{Aut}(F))$-torsor, which gives a homomorphism $\pi\_1(B) \to \pi\_0(\operatorname{Aut}(F))$.


