A connected spherical complement forces every cycle in the domain to be null-homologous

Since ${\mathrm{\Gamma }}^{\ast }\subseteq \mathrm{\Omega }$, the index $n(\mathrm{\Gamma },p)$ is defined for every $p\in \mathbb{C}\setminus \mathrm{\Omega }$. By [L1], the function $p\mapsto n(\mathrm{\Gamma },p)$ is locally constant on $\mathbb{C}\setminus {\mathrm{\Gamma }}^{\ast }$, and there is $R>0$ with $n(\mathrm{\Gamma },p)=0$ whenever $\mathrm{\mid }p\mathrm{\mid }>R$. Thus the subset [given, L1, construct] $E=\{\mathrm{\infty }\}\cup \{p\in \mathbb{C}\setminus \mathrm{\Omega }:n(\mathrm{\Gamma },p)=0\}$ contains $\mathrm{\infty }$ together with a punctured neighborhood of $\mathrm{\infty }$ in the sphere.

The set $E$ is open in $\widehat{\mathbb{C}}\setminus \mathrm{\Omega }$ by the local constancy from [L1], and its complement in $\widehat{\mathbb{C}}\setminus \mathrm{\Omega }$ is open for the same reason. Since $\widehat{\mathbb{C}}\setminus \mathrm{\Omega }$ is connected and $E$ is nonempty by step 1.1, it follows that [step 1.1, L1, algebra] $E=\widehat{\mathbb{C}}\setminus \mathrm{\Omega }\mathrm{.}$ Therefore $n(\mathrm{\Gamma },p)=0$ for every $p\in \mathbb{C}\setminus \mathrm{\Omega }$.

By [L2], the vanishing from step 2.1 is exactly the statement that $\mathrm{\Gamma }$ is null-homologous in $\mathrm{\Omega }$.