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 curves 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 images 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.^[1]. In algebraic topology, the term 1-cycle or p-cycle is commonly used instead of cycle.^[2]. Additionally, note that the plural of cycle is "cycles," while the plural of Zykel is "Zykel" in German.

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.

References

Subject classification: this is a literature resource.

Otto Forster: Riemann surfaces, Springer 1977; English edition: Lectures on Riemann surfaces, Graduate Texts in Mathematics, Springer-Verlag, 1991, ISBN 3-540-90617-7, Chapter 20

Notes

↑ Otto Forster: Riemann surfaces, Springer 1977; English edition: Lectures on Riemann surfaces, Graduate Texts in Mathematics, Springer-Verlag, 1991, ISBN 3-540-90617-7, Chapter 20

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