Monodromy Representation of Elliptic Curve

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

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:

enter image description here

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:

enter image description here

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:

enter image description here

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?


Source : Link , Question Author : Isak the XI , Answer Author : Community

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