Consider the map $f$ sending the family of elliptic curves $M=\{(x,y,s): y^2=(x-1)(x^2-s): s\in \mathbb{C}\setminus \{0,1\}\}$ to $\mathbb{C}\setminus \{0,1\}$ via $(x,y,s)\mapsto s$.
This is a locally trivial fibration, hence induces a monodromy action $\rho$ of $\pi_1(\mathbb{C}\setminus \{0,1\}, p)$ on $H_1(E,\mathbb{Z})$ where $E=f^{-1}(p)$ and $p\ne 0,1$.
I am trying to understand the argument of Calson, Müller-Stach and Peters in their book Period Mappings and Period Domains, page 18 of how they deduce the monodromy action explicitly. In particular,
Let $a,b\in \pi_1(\mathbb{C}\setminus \{0,1\})$ be closed loops at $p$ around $0,1$, respectively, then $\pi_1(\mathbb{C}\setminus \{0,1\})=\langle a,b\rangle$.
One can determine a basis for $H_1(E,\mathbb{Z})$ as follows: consider $\mathbb{P}^1\setminus \{1,\pm \sqrt{p},\infty\}$, and draw two line segments $c_1,c_2$ connecting $\sqrt{p}$ with $-\sqrt{p}$ and $1$ with $\infty$. These are the branch cuts for our function $y=\sqrt{(x^2-p)(x-1)}$. Let $\delta$ be a closed loop going around $c_1$ and $\gamma$ be a closed loop passing through both $c_1$ and $c_2$. Their preimages under $f$ intersect with $E$ to form two loops around two holes of the torus $E$ (see page 6 of their book for more explanation).
My question is:
How to show that the monodromy action of $a$ on $H_1(E,\mathbb{Z})$ with respect to basis $\delta, \gamma$ is $\rho(a)(\delta)=\delta, \rho(a)(\gamma)=\delta+\gamma$?
What I gather so far: Start with the locally trivial fibration $f:M\to \mathbb{C}\setminus \{0,1\}$ and $a:[0,1]\to \mathbb{C}\setminus \{0,1\}$ a closed loop of $p$ around $0$, then $a^*f$ is a fiber bundle on $[0,1]$, hence is a trivial bundle. This means there exists a family of diffeomorphisms $f_t: f^{-1}(a(t))\to f^{-1}(p)$ such that $f_0$ is the identity map. The diffeomorphism $f_1$ then induces an isomorphism on the homology $H_1(E,\mathbb{Z})\to H_1(E,\mathbb{Z})$, which is our monodromy action. However, how to describe $f_1$ explicitly?
PS: It seems that this goes under the name of Picard–Lefschetz transformation, but I don't know what this is, and would love to hear an explanation of this using the above example. Or I would be very happy if someone could point out an easy place to read about this. Thank you!