Here is a question which I asked myself (and couldn't answer) while reading "The topology of spaces of rational functions" by G. Segal.

Let $X$ be a smooth complete complex curve (=a compact Riemann surface) of genus $g$ and let $Rat(X,d)$ be the space of all regular (=holomorphic) maps from $X$ to $\mathbf{P}^1(\mathbf{C})$ of degree $d$. In this question I'm interested in the fundamental group of the open subset $U(X,d)$ of $Rat(X,d)$ formed by all $f$ such that all critical points of $f$ are simple and all critical values are distinct. (A critical point is a point at which the derivative of $f$ vanishes; a critical value is the image of a critical point.) To be more specific, let's say I'd like to

find a "nice" system of generators of $\pi_1(U(X,d))$;

to describe, for each of these generators, its image under the map induced by the map $G$ from $U(X,d)$ to the configuration space $B(\mathbf{P}^1(\mathbf{C}),k)$ of unordered subsets of $\mathbf{P}^1(\mathbf{C})$ of cardinality $k=2(d+g-1)$ that takes $f$ to its branch divisor (i.e. the divisor of the critical points).

Here are some remarks that may be useful (or may not):

First, here is how one can think of the fundamental group of $Rat(X,d)$. By associating to every function its divisor of poles we get a map $F$ from $Rat(X,d)$ to the $d$-th symmetric power $S^d(X)$ of $X$.

Assume $d> 2g-2$. By the Riemann-Roch theorem, for any degree $d$ divisor $D$ the linear space ${\cal{L}}(D)=H^0(X,{\cal{O}}(D))$ (which is formed by all rational functions $f$ such that for any $x\in X$ the order of the pole of $f$ at $x$ is at most the multiplicity of $x$ in $D$) is $d-g+1$. So $F$ is surjective and a fiber of $F$ is $\mathbf{C}^{d-g+1}$ minus some number of hyperplanes (these are given by the condition that order the pole of $f$ at a point $x$ of $D$ is less then the multiplicity of $x$ in $D$).

The map $F$ is probably not a fibration. However, the fundamental group of $Rat(X,d)$ is spanned by the loops in a general fiber of $F$ going around one of the hyperplanes, and lifts of the loops in $S^d(X)$ (these are all of the form "one of the points moves along a loop in $X$ and the other stand still").

Second, recall that the Jacobian $J(X)$ of $X$ is defined as follows. Integration along cycles gives an injective map $H_1(X,\mathbf{Z})\to\mathbf{C}^g=Hom(H^0(X,\Omega_X),\mathbf{C})$ and the Jacobian of $X$ is the quotient. Moreover, once we have chosen a base point $x$ in $X$, we get a natural map $j:X\to J(X)$ defined as follows: for any $x'\in X$ take a path $\gamma$ from $x$ to $x'$ and set $j(x')$ to be the image in $J(X)$ of the "integration along $\gamma$ function". This is well defined map that can be extended by $\mathbf{Z}$-linearity to $S^d(X)$.

Abel's theorem says that two disjoint effective divisors are the divisors of the zeros and the poles of a rational function if and only if their images under $j$ coincide. This may be useful in this problem, but I don't see how.