on paths homotopic to a point

Question

Let $U\subset \mathbb{C}$ be a domain and $\Gamma:[0,1]\to U$ be a closed Jordan path. Denote $G$ the interior of $\Gamma$, i.e. $G$ is a bounded set such that $\Gamma=\partial G$. We assume that for given path $\Gamma$, there exists such $G$.

Is it true that the path $\Gamma$ is homotopic to a point if and only if the interior of $\Gamma$ does not contain any point of $\mathbb{C}\setminus U$ ?

If yes, is there a rigorous proof of this statement ? If no, can you give a counterexample ?

You can add any further restrictions if needed.

Thank you for any suggestions.

Answer 1

One direction is harder than the other.

1. [The hard direction.] First of all, it appears that you are taking for granted the Jordan separation theorem, i.e. that for every Jordan curve $J\subset {\mathbb C}$ there exists a bounded open subset $G\subset {\mathbb C}$, called the interior of $J$, such that $\partial G=J$. This is already nontrivial.

I will assume that you are also familiar with the Riemann mapping theorem which implies that such $G$ admits a conformal mapping to the unit disk, $f: \Delta\to G$.

In this setting, Caratheodory proved that $f$ extends to a homeomorphism of closures $F: \bar{\Delta}\to \bar{G}=G\cup J$. You can find a short self-contained proof in this Wikipedia article.

Remark. Notice that Caratheodory's theorem also implies that $\gamma$ is a uniform limit of rectifiable simple loops $\gamma_n: S^1\to G$. I will use this fact in Part 2.

From this, it follows that if $G\subset U$ then $J$ (regarded as a parameterized closed curve $\gamma: S^1\to U$) is null-homotopic in $U$.

2. For the converse: Suppose that $G$ contains a point $a\notin U$. Then (see the remark above) there exists a Jordan curve $J_n\subset G\cap U$ such that $a$ belongs to $G_n$, the interior of $J_n$. Therefore, we obtain a nonzero integral $$ \int_{J_n} \frac{dz}{z-a}\ne 0 $$ of the holomorphic function $\frac{1}{z-a}$ in $U$. It follows that $J_n$ is not null-homotopic in $U$. You can find a proof in any reasonably advanced complex analysis textbook (if a book contains a proof of the Riemann Mapping Theorem, it will contain a proof of this fact as well).

I claim that for sufficiently large $n$, $J_n$ is homotopic to $J$ in $U$. This follows from:

Lemma. Let $U$ be an open subset of ${\mathbb C}$ and we have a sequence $\gamma_n$ of (not necessarily simple) loops in $U$ converging uniformly to a loop $\gamma$ in $U$. Then for all sufficiently large $n$, the loops $\gamma, \gamma_n: S^1\to U$ are homotopic.

Proof. For each $z\in S^1$ define the line segment $$ L(z,t)= t \gamma_n(z) +(1-t)\gamma(z), 0\le t\le 1. $$ Then for all large $n$, the image of the continuous map $L: S^1\times [0,1]\to {\mathbb C}$ is contained in $U$ and, hence, defines the required homotopy. qed

Lastly, you use the elementary fact that if $\gamma_n$ is not null-homotopic to $U$ and $\gamma$ is homotopic to $\gamma$ in $U$ then $\gamma$ is also not null-homotopic in $U$. (Homotopy between loops is an equivalence relation.)