# Monodromy correspondence

Lately I've been studying monodromy and covering maps (in particular ramified covering mapos of Riemann surfaces), and I came across something I didn't fully understand. Let $V$ be a connected real manifold, and let $\rho:\pi_1(V,q)\to S_d$ be a group homomorphism with a transitive image. Let $H=\{a\in\pi_1(V,q):\rho(a)(1)=1\}$. It is easy to prove that $H$ is a subgroup of $\pi_1(V,q)$ of index $d$. By the general theory of covering spaces, we know that there is a covering space $F:U\to V$ of degree $d$ such that $\pi_1(U,x)\cong H$.

Now, we know that $\pi_1(V,q)$ acts on $F^{-1}(q)$ by taking a class of curves $[\gamma]$ and sending a point $x$ in the fiber of $q$ to the endpoint of the lifting of $\gamma$ to $U$ with initial point $x$. This action induces a homomorphism $\pi_1(V,q)\to S_d$ (with transitive image).

Now the question: Supposedly, given a homomorphism $\rho:\pi_1(V,q)\to S_d$ with transitive image, if we take the covering $F:U\to V$ associated to the group $H$ mentioned above, why is it that the homomorphism described by the action of $\pi_1(V,q)$ on the fibers of $q$ the same (or maybe with conjugate images) as the homomorphism $\rho$?

I read this statement in Algebraic Curves and Riemann Surfaces by Rick Miranda, and can't figure out why the homomorphisms should be the same.

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Given a subgroup $H$ of the $\pi_1(V,q)$, you define the points of the corresponding pointed cover $(U,p)$ as equivalence classes of paths starting from the basepoint, under relations of homotopy equivalence and equivalence under the action of $H$ by pre-composition of loops (and then give it a suitable topology, but that isn't important here). The covering map is then described by expanding the equivalence relation to allow loops from all of $\pi_1(V,q)$.
Pre-composition of loops therefore induces an action of $\pi_1(V,q)$ on the orbit of $p$ that identifies the orbit with the right cosets of $H$ in $\pi_1(V,q)$. The monodromy action as you describe it is given by post-composition of loops, and it identifies the orbit of $p$ with the left cosets of $H$ in $\pi_1(V,q)$. The inversion map on a group naturally identifies right cosets of any subgroup with left cosets.
Now, consider the initial setup with $\rho: \pi_1(V,q) \to S_d$, and $H$ as the stabilizer of the element $1$. Since we assume the action is transitive, the elements of a $d$ element permutation representation are naturally identified with left cosets of $H$ under the left multiplication action.