I'm reading this post by Charles Siegel on Monodromy Representations and there is a construction in example a not unterstand.
We look at the family $y^2z=x(x-z)(x-\lambda z)$ of projective elliptic curves parametrized by $\lambda$. We’re ignoring degenerations for the moment so this lives over $\mathbb{P}^1\setminus\{0,1,\infty\}$. This is also visible as $\mathbb{C}^\times\setminus\{1\}$.
Since topologically this space is homotopy equivalent to two-loop space $S^1 \vee S^1$, we have $\pi_1(\mathbb{C}^\times\setminus\{1\})= \mathbb{Z} * \mathbb{Z}$ we’ve got two nontrivial loops, one around zero, the other around one, and there are no relations, this is the free group of rank $2$:
We can envision all of this as living in the plane with four points marked as ${0,1,\infty,\lambda}$, and a cut from ${0}$ to ${\lambda}$ and a cut from ${1}$ to ${\infty}$. Then the loops are what happens if we loop ${\lambda}$ around things. We get one loop from rotating the cut between ${0}$ and ${\lambda}$ all the way around. The other one is a bit trickier to describe directly, but we can describe it in terms like this.
Later we consider the action of the two loops from $\pi_1(\mathbb{P}^1 \setminus \{0,1,\infty\})$ on the homology group $H_1$ of the elliptic curve. The generators of $H_1$ are:
We’ll focus on the first one. Look at ${H_1}$ of the elliptic curve. It has two generators, call them ${\delta}$ and ${\gamma}$, where ${\delta}$ is a loop around the ${0\lambda}$ cut, and ${\gamma}$ is a loop through the two cuts. This is a standard homology basis, and we’ll look at the action of our element of $\pi_1(\mathbb{P}^1\setminus\{0,1,\infty\})$ on the homology:
There are some aspects I not understand in this construction. Firstly, what does the author mean when he wrote 'The loops are what happens if we loop $\lambda$ around things. We get one loop from rotating the cut between ${0}$ and ${\lambda}$ all the way around.' Which loops do we obtain by this operation rotating the cut between ${0}$ and ${\lambda}$? I not understand which objects we obtain doing it. The only involved loops here are the two generators of $\pi_1(\mathbb{P}^1\setminus\{0,1,\infty\})$ called $\gamma_1$ and $\gamma_2$ and the two generators of $H_1$ of the elliptic curve $\delta$ and $\gamma$ in second picture, which as far as I understand the second picture represents one "sheet" of the elliptic curve. I would like do understand which loops the author has in mind?
Is the which the author has in 'We get one loop from rotating the cut between $0$ and $λ$ all the way around' just literally this one:
Secondly why the induced action by $\gamma_2 \in \pi_1(\mathbb{P}^1\setminus\{0,1,\infty\})$ on $H_1$ correspond precisely to the rotations of the cut between ${0}$ and ${\lambda}$ discussed in first part? Can somebody explain the geometry of this action?