Cycle (function theory)

Chain and cycle are mathematical objects that are particularly considered in function theory, but also occur as special cases in algebraic topology . The chain is a generalization of a curve and the cycle is a generalization of a closed curve . They are mainly used in function theory in the area of ​​integration.

In order to indicate that chain and cycle are special cases from the homology theory of algebraic topology, one also speaks of the 1-chain and the 1-cycle. In the algebraic topology itself, the term 1-cycle or p-cycle has prevailed instead of the term 1-cycle. In addition, it should be noted that the plural of the cycle is called the cycles , but the plural of the cycle is called the cycles .

Definitions

Chain

A chain on or on a Riemann surface is understood to be a formal, finite, integer linear combination ${\displaystyle X:=\mathbb {C} }$ ${\displaystyle X}$

${\displaystyle \Gamma :=\sum _{i=1}^{k}n_{i}\gamma _{i}\quad n_{i}\in \mathbb {Z} }$

of steady curves . The set of all chains that naturally form an Abelian group is noted with . ${\displaystyle \gamma _{i}\colon [0,1]\to X}$ ${\displaystyle X}$ ${\displaystyle C_{1}(X)}$

Integration via a one chain

Let be a closed complex (1,0) differential form , then the integral over the chain is through ${\displaystyle \omega }$ ${\displaystyle \Gamma }$

${\displaystyle \int _{\Gamma }\omega :=\sum _{i=1}^{k}n_{i}\int _{\gamma _{i}}\omega }$

Are defined. If the plane is complex, then the calculus of differential forms is not necessary. In this case , where is a differentiable function . The definition is then simplified to ${\displaystyle X}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle \omega =f(z)\mathrm {d} z}$ ${\displaystyle f\colon D\subset \mathbb {C} \to \mathbb {C} }$

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

cycle

A cycle is a chain in which each point occurs just as often as the start point as the end point of the curves , taking into account the multiplicity . ${\displaystyle a\in \mathbb {C} }$ ${\displaystyle n_{i}}$ ${\displaystyle \gamma _{i}}$

This definition can be reformulated with the help of the divisor group . Be ${\displaystyle \operatorname {Div} (X)}$

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

an illustration. For a curve you bet , if . Otherwise the divisor is the value +1 in , the value −1 in and otherwise the value 0. For a chain is by defined. The core ${\displaystyle c\colon [0,1]\to X}$ ${\displaystyle \partial c=0}$ ${\displaystyle c(0)=c(1)}$ ${\displaystyle \partial c}$ ${\displaystyle c(1)}$ ${\displaystyle c(0)}$ ${\displaystyle \Gamma }$ ${\displaystyle \partial }$ ${\displaystyle \textstyle \partial \Gamma :=\sum _{i=1}^{n}n_{i}\partial \gamma _{i}}$

${\displaystyle Z_{1}(X):=\operatorname {Kern} (\partial )}$

the figure is the group of cycles. ${\displaystyle \partial }$

Number of turns

The trace is the union of the images of the individual curves, i.e. H.

${\displaystyle \operatorname {Spur} \,\Gamma :=\bigcup _{i=1}^{N}\operatorname {im} \,\gamma _{i}}$.

If a subset, then a cycle is called in if and only if the track is in . ${\displaystyle D\subseteq \mathbb {C} }$ ${\displaystyle \Gamma }$ ${\displaystyle D}$ ${\displaystyle \operatorname {Spur} \,\Gamma \subseteq D}$ ${\displaystyle D}$

The revolution number is defined analogously to that of a closed curve, only using the integral defined above, i.e. H. for one writes ${\displaystyle z\not \in \operatorname {Spur} \,\Gamma }$

${\displaystyle \operatorname {ind} _{\Gamma }(z):={\frac {1}{2\pi \mathrm {i} }}\int _{\Gamma }{\frac {\mathrm {d} \zeta }{\zeta -z}}\in \mathbb {Z} }$.

The interior of a cycle are precisely those points for which the number of turns does not vanish:

${\displaystyle \operatorname {Int} \,\Gamma :=\{z\in \mathbb {C} \setminus \operatorname {Spur} \,\Gamma :\operatorname {ind} _{\Gamma }(z)\neq 0\}}$

Similarly, the exterior is exactly the set of points for which the number of turns vanishes:

${\displaystyle \operatorname {Ext} \,\Gamma :=\{z\in \mathbb {C} \setminus \operatorname {Spur} \,\Gamma :\operatorname {ind} _{\Gamma }(z)=0\}}$

A cycle is called zero homolog in if and only if the interior is completely in . This is exactly the case when the number of turns vanishes for all points from . ${\displaystyle D\subseteq \mathbb {C} }$ ${\displaystyle \operatorname {Int} \,\Gamma }$ ${\displaystyle D}$ ${\displaystyle \mathbb {C} \setminus D}$

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

Integral sentences

The chains and cycles are important in function theory because, as already mentioned, you can generalize the curve integral with them. In particular, the integral over a cycle can be understood as a generalization of the closed curve integral. The Cauchy integral theorem , the Cauchy integral formula and the residue theorem can be proved for cycles.

The Stokes' theorem can also be declared chains. Be a chain on which all curves are smooth and be a smooth function. Then the statement of Stokes' theorem reads ${\displaystyle \Gamma }$ ${\displaystyle X}$ ${\displaystyle \gamma _{i}}$ ${\displaystyle f\colon X\to \mathbb {R} }$

${\displaystyle \int _{\Gamma }\mathrm {d} f(z)\mathrm {d} z=\int _{\partial \Gamma }f(z)\mathrm {d} z}$,

where the operator from the section is one cycle and the derivative. The second integral must also be used as ${\displaystyle \partial }$ ${\displaystyle \mathrm {d} }$

${\displaystyle \int _{\partial \Gamma }f(z)\mathrm {d} z=\sum _{i=1}^{n}n_{i}f(z)|_{\gamma _{i}(0)}^{z\,=\gamma _{i}(1)}=\sum _{i=1}^{n}n_{i}\left(f(\gamma _{i}(1))-f(\gamma _{i}(0)\right)}$

be understood. If there is even a cycle whose curves are smooth, then Stokes' theorem simplifies to ${\displaystyle \Gamma }$

${\displaystyle \int _{\Gamma }\mathrm {d} f(z)\mathrm {d} z=0}$,

because then the sum is zero. ${\displaystyle \textstyle \sum _{i=1}^{n}n_{i}\left(f(\gamma _{i}(1))-f(\gamma _{i}(0)\right)}$

Classification in the homology theory

The terms chain and cycle are special cases of objects in the topology . In algebraic topology , one looks at complexes of p-chains and forms homology groups from them . These groups are invariants in the topology. A very important homology theory is that of the singular homology groups .

A chain, as it was defined here in the article, is a 1-chain of the singular complex , which is a certain chain complex. The operator defined in the section on the cycle is the first boundary operator of the singular complex and the group of divisors is therefore identical to the group of 0 chains. The group of cycles defined as the core of the boundary operator is a 1- cycle in the sense of the singular complex. ${\displaystyle \partial \colon C_{1}(X)\to \operatorname {Div} (X)}$ ${\displaystyle \partial }$

In addition to the core of the boundary operator, consider the image of this operator in algebraic topology and construct a corresponding homology group from these two sets. In the case of the singular complex, the singular homology is obtained . In this context, the previously defined terms homologous chain and zero homologous chain also have a more abstract meaning.

swell

• Wolfgang Fischer, Ingo Lieb: Function theory . 8th edition. Vieweg, Braunschweig 2003, ISBN 3-528-77247-6 .
• Otto Forster : Riemannsche surfaces , Springer 1977, English Lectures on Riemann surfaces , Graduate Texts in Mathematics, Springer-Verlag, 1991, ISBN 3-540-90617-7 , chapter 20

Individual evidence

1. ↑ Otto Forster : Riemannsche surfaces , Springer 1977, English Lectures on Riemann surfaces , Graduate Texts in Mathematics, Springer-Verlag, 1991, ISBN 3-540-90617-7 , chapter 20
2. ↑ Wolfgang Lück: Algebraic Topology: Homology and Manifolds . Vieweg, 2005.