Consider the family of proper complex genus two curves with affine equation $y^2 = x(x-1)(x-a)(x-b)(x-c)$, defined over an open subset $U$ of $\mathbb{C}^3$. Here, $U$ consists of all triples $(a,b,c)$ such that the corresponding curve is smooth. 


Fix a point $x \in U$, and let $C$ denote the corresponding curve. The fundamental group $\pi\_1(U,x)$ acts on the set of isotopy classes of simple closed curves on $C$. Is this action transitive on the set of isotopy classes of non-separating simple closed curves? Is this action transitive on the set of isotopy classes of separating simple closed curves? (The answer in both cases is yes if $\pi\_1(U,x)$ is replaced by the mapping class group of $C$)



Simple closed curves on genus 2 surfaces
The action is not transitive on simple closed curves. For instance, some of the simple closed curves lift to the following unbranched, $\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$ cover, yet others do not: the projective smooth model of the affine curve $\text{Zero}(y^2-x(x-1)(x-a)(x-b)(x-c), 4xz^2-(z^2+1)^2) \subset \mathbb{C}^3$. Since your parameterization of the curves "fixes" the branch points over $x=0$, $x=1$, and $x=\infty$, you are also "fixing" the quotient fundamental group of the orbifold with underlying manifold $\mathbb{CP}^1$ (the "$x$-line") and $\mathbb{Z}/2\mathbb{Z}$-orbifold points at $x=0$, $x=1$, and $x=\infty$. This orbifold fundamental group is isomorphic to $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$. So the image of a free homotopy class in this quotient group is fixed by the monodromy action.


