Let $R$ be a map from the Riemann sphere to itself, upon which its restriction to an annulus $A$ is a covering map to another annulus $B$.
Suppose there are critical points in one of the complementary components of A. If I then "fill in" this complementary component to obtain a simply connected region, call U, doe the original covering map extend to a covering from U to another region?
If not, what conditions must hold for it to be so?
I hope this question makes sense.
For context, this problem stems off an original problem that I posted: Critical points of rational mappings of annuli Fatou Component
Any help would be greatly appreciated.