# Complex Analysis/cycle
## Introduction

Chain and cycle are mathematical objects studied in Complex Analysis but also appear as special cases in Algebraic topology. A chain generalizes a curve, and a cycle generalizes a closed curve. They are primarily used in integration in complex analysis.

## Definitions
### Chain

A chain on a set $G\subset \mathbb {C}$ is defined as a finite integer linear combination of paths $\gamma _{1},\ldots ,\gamma _{k}$: $\Gamma :=\sum _{i=1}^{k}n_{i}\gamma _{i}\quad n_{i}\in \mathbb {Z}$. $\gamma _{1},\ldots ,\gamma _{k}$ are generally continuous Curve in $G$.

### Integration over a chain

Let $f:G\to \mathbb {C}$ be integrable, and let $\Gamma$ be a chain of piecewise continuously differentiable paths (paths of integration) $\gamma _{1},\ldots ,\gamma _{k}$ in $G\subset \mathbb {C}$. The integral over the chain $\Gamma$ is defined by:
    $\int _{\Gamma }f(z)\,dz:=\sum _{i=1}^{k}n_{i}\int _{\gamma _{i}}f(z)\,dz$

### Definition: Cycle

Version 1: A cycle is a chain $\Gamma :=\sum _{i=1}^{k}n_{i}\gamma _{i}$, where every point $a\in \mathbb {C}$ appears as the starting point as many times as it appears as the endpoint of the curves $\gamma _{i}$, taking multiplicities $n_{i}$ into account.
Version 2: A cycle is a chain $\Gamma :=\sum _{i=1}^{k}n_{i}\gamma _{i}$ consisting of closed paths $\gamma _{1},\ldots ,\gamma _{k}$.

### Connection Between Version 1 and Version 2

Version 2 is essential for complex analysis. Based on the properties of Version 1, any cycle $\Gamma :=\sum _{i=1}^{k}n_{i}\gamma _{i}$ can be transformed into a chain ${\hat {\Gamma }}:=\sum _{i=1}^{m}{\hat {n}}_{i}{\hat {\gamma }}_{i}$ of closed paths ${\hat {\gamma }}_{1},\ldots ,{\hat {\gamma }}_{m}$. If the paths $\gamma _{1},\ldots ,\gamma _{k}$ are piecewise continuously differentiable, then the closed paths ${\hat {\gamma }}1,\ldots ,{\hat {\gamma }}m$ are also continuously differentiable. For all holomorphic functions $f:G\to \mathbb {C}$, it holds that: $\int \Gamma f(z),dz=\int {\hat {\Gamma }}f(z),dz$.

### Trace of a path

The trace of a path $\gamma :[a,b]\to G$ is defined as: $\operatorname {Trace} (\gamma _{i}):=\operatorname {Image} (\gamma ):={\gamma (t),|,t\in [a,b]}$.

### Trace of a cycle/chain

The trace of a chain $\Gamma$ is the union of the Image (mathematics) of its individual curves, i.e.: $\operatorname {Trace} (\Gamma ):=\bigcup _{i=1}^{N}\operatorname {Image} (\gamma _{i})$. If $\operatorname {Trace} (\Gamma )\subset \mathbb {C}$ is a subset of $G\subset \mathbb {C}$, then $\Gamma$ is called a cycle in $G$ if and only if the trace $\operatorname {Trace} (\Gamma )\subseteq G$ lies in $G$.

### Winding number

The Winding number is defined analogously to that of a closed curve but uses the integral defined above. For $z\not \in \operatorname {Trace} (\Gamma )$, it is given by: $n(\Gamma ,z):={\frac {1}{2\pi \mathrm {i} }}\int _{\Gamma }{\frac {\mathrm {d} \zeta }{\zeta -z}}\in \mathbb {Z}$.

### Interior points of a cycle

The interior of a cycle consists of all points for which the winding number is non-zero: $\operatorname {Int} (\Gamma ):={z\in \mathbb {C} \setminus \operatorname {Trace} (\Gamma ):n(\Gamma ,z)\neq 0}$.

### Exterior points of a cycle

Analogously, the exterior is the set of points for which the winding number is zero: $\operatorname {Ext} (\Gamma ):={z\in \mathbb {C} \setminus \operatorname {Trace} (\Gamma ):n(\Gamma ,z)=0}$.

### zero-homologous cycle

A cycle is called null-homologous for a set $G\subseteq \mathbb {C}$ if and only if the interior $\operatorname {Int} (\Gamma )$ lies entirely within $G$. This is equivalent to the winding number vanishing for all points in $\mathbb {C} \setminus G$.

### Homologous cycles

Two cycles $\Gamma _{1}$, $\Gamma _{2}$ are called homologous in $G\subseteq \mathbb {C}$ if and only if their formal difference $\Gamma _{1}-\Gamma _{2}$ is null-homologous in $G$.

## Integral Theorems

Chains and cycles are important in complex analysis because, as mentioned, they generalize curve integrals. In particular, the integral over a cycle generalizes the closed curve integral. The Cauchy's integral theorem, the Cauchy's integral formula, and the Residue theorem can be proven for cycles.

## Relation to Homology Theory

To indicate that chains and cycles are special cases of objects in Homology (mathematics) of algebraic topology, they are sometimes referred to as 1-chains and 1-cycles.. In algebraic topology, the term 1-cycle or p-cycle is commonly used instead of cycle..

### Embedding in Homology Theory

The terms chain and cycle are special cases of Mathematical object in topology. In Algebraic topology, one considers complexes of p-chains and constructs Homology (mathematics) from them. These groups are Invariant (mathematics) in topology. A very important Homology (mathematics) is that of Singular homology.

### 1-Chain of the Singular Complex

A chain, as defined here, is a 1-chain of the Singular homology, which is a specific chain complex. The operator defined in the section on cycles, $\partial \colon C_{1}(X)\to \operatorname {Div} (X)$, is the first boundary operator of the singular complex. The group of divisors is therefore identical to the group of 0-chains. The group of cycles, defined as the kernel of the boundary operator $\partial$, is a 1-Chain complex in the sense of the singular complex.

### Algebraic Topology

In algebraic topology, one considers both the kernel of the boundary operator and the image of this operator, constructing a corresponding homology group from these two sets. In the case of the singular complex, one obtains Singular homology. In this context, the previously defined terms homologous chain and null-homologous chain take on a more abstract meaning.