Complex Analysis/cycle

Chain

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

Integration over a chain

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

${\displaystyle \int _{\Gamma }f(z)\,dz:=\sum _{i=1}^{k}n_{i}\int _{\gamma _{i}}f(z)\,dz}$

Definition: Cycle

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

Version 2: A cycle is a chain ${\displaystyle \Gamma$ :=\sum _{i=1}^{k}n_{i}\gamma _{i}} consisting of closed paths ${\displaystyle \gamma _{1},\ldots ,\gamma _{k}}$.

Connection Between Version 1 and Version 2

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

Trace of a path

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

Trace of a cycle/chain

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

Winding number

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

Interior points of a cycle

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

Exterior points of a cycle

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

zero-homologous cycle

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

Homologous cycles

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

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, ${\displaystyle \partial \colon C_{1}(X)\to \operatorname {Div} (X)}$, 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 ${\displaystyle \partial }$, 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.

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• Date: 12/17/2024