Complex Analysis/chain

Definition - Chain

Let $G\subseteq \mathbb {C}$ be a region, $n\in \mathbb {N}$ and let $\gamma _{i}\colon [a_{i},b_{i}]\to G$ be curves in $G$ and $n_{i}\in \mathbb {Z}$. Then the formal linear combination $\sum _{i=1}^{n}n_{i}\gamma _{i}$ is called a chain in $\mathbb {C}$. The set of all chains in $G$ that form an abelian group in a natural way is denoted by $C(G)$.

Definition - Trace of a chain

The trace of a chain $\Gamma$ is the union of the traces of the individual curves $\gamma _{i}$, i.e.

$\mathrm {Spur} (\Gamma ):=\bigcup _{i=1}^{n}\mathrm {Spur} (\gamma _{i})$

Cycle

A chain $\Gamma =\sum _{i=1}^{n}n_{i}\gamma _{i}\in C(G)$ with $\gamma _{i}\colon [a_{i},b_{i}]\to G$ is called a cycle if every point in $G$ appears the same number of times as the starting and ending point of curves in $G$, i.e. if

$\sum _{i=1}^{n}n_{i}|\{i:\gamma _{i}(a_{i})=z\}|=\sum _{i=1}^{n}n_{i}|\{i:\gamma _{i}(b_{i})=z\}|$

holds for every $z\in G$.

Inner and outer regions

Let $\Gamma$ be a cycle in $\mathbb {C}$, using theWinding number, we can consider a decomposition of $\mathbb {C}$ determined by $\Gamma$ into three parts, namely:

• The image of the trace of $\Gamma$
• The outer region, the points that are not traversed by $\Gamma$, i.e.

$A_{\Gamma }:=\{z\in \mathbb {C} \setminus \mathrm {Spur} (\Gamma ):n(\Gamma ,z)=0\}$

• The inner region are the points that are traversed by $\Gamma$, i.e.

$I_{\Gamma }:=\{z\in \mathbb {C} \setminus \mathrm {Spur} (\Gamma ):n(\Gamma ,z)\neq 0\}$

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