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.
Best Answer
You can find an answer by fleshing out the part of your question that says "By the general theory of covering spaces, we know that". In particular, you might ask how the Galois connection between connected covers of a pointed manifold and subgroups of the fundamental group is explicitly constructed, and then analyze how the monodromy action matches the permutation representation.
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.