In complex plane, if $C$ is a closed curve that is homotopic to a point, and $C$ is the boundary of a domain $E$, is $E$ simply connected?
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I think you need to assume that $C$ is a simple closed curve. Otherwise, take a long skinny ellipse (a "sausage") and bend it until the two ends just touch, like your thumb touching your forefinger when you use them to make a circle. There's clearly a curve that bounds the sausage (even after the ends touch at a single point), and this curve is clearly contractible within the sausage shape, but the sausage shape (assuming it includes the boundary) is not contractible -- it has $\pi_1 = \mathbb Z$.
Even with the simple-closed-curve assumption I'm not entirely convinced the statement is true, but I suspect that it is.
I have to rewrite the whole what I said:
I think you need some conditions on $C$ and $E$ to avoid pathology and to say something useful. The conditions that come to my mind include:
1) $E$ is path-connected, (since $E$ is a domain, it is open and connected but I don't think they are strong enough to prevent pathological phenomena to take place.)
2) the pair ($E, C$) is reasonably tame, for example every point of $C$ has a neighbourhood in $E \cup C$ which is homeomorphic to $(-1, 1) \times [0, 1)$ where (0, 0) corresponds to the point,
3) the homotopy of $C$ to a point occurs inside $E$.
Of course you may need more.