Inverse image of a Jordan curve

Question

If there exists a rational map $R$ from the extended plane $\hat{\mathbb{C}}$ to itself, and a Jordan curve $J$ on the plane, such that $R$ has no critical value on the curve, can we say that the inverse image of this Jordan curve $R^{-1}(J)$ is a finite union of disjoint Jordan curves ?

Answer 1

The answer is yes. Let $X$ be this full preimage, from your condition, for every $x\in X$ there exists a neighborhood $U$ which is mapped on a neighborhood $V$ of $f(x)$. By compactness, there is a covering of $X$ by such neighborhoods, and this implies that $X$ is a compact $1$-dimensional manifold embedded to the sphere. So it is a finite union of disjoint Jordan curves.