Complex chains, their traces, and cycles

Let $\mathrm{\Omega }\subseteq \mathbb{C}$ be open. A complex chain in $\mathrm{\Omega }$ is a finite list

$\mathrm{\Gamma }=((m_0,{\gamma }_0),\dots ,(m_{r-1},{\gamma }_{r-1})),r\in \mathbb{N},$

each $m_k$ an integer (The integers as equivalence classes of pairs of naturals) and each ${\gamma }_k:[a_k,b_k]\to \mathbb{C}$ a complex contour (Rectifiable complex contours, reversal, concatenation, closedness, and orientation) whose trace lies in $\mathrm{\Omega }$. It is written ${\sum }_{k<r}{m}_k{\gamma }_k$. The list of length $r=0$ is the empty chain.

Its trace is the set

${\mathrm{\Gamma }}^{\ast }:=\bigcup \{{\gamma }_k^{\ast }:k<r\text{ and }{m}_k\mathrm{\ne }0\},$

a subset of $\mathrm{\Omega }$; a term with $m_k=0$ contributes nothing to it.

Its boundary is the function $\mathrm{\partial }\mathrm{\Gamma }:\mathbb{C}\to \mathbb{Z}$ given by

$\mathrm{\partial }\mathrm{\Gamma }(q):={\sum }_{\begin{array}{c}k<r\\ {\gamma }_k(b_k)=q\end{array}}{m}_k-{\sum }_{\begin{array}{c}k<r\\ {\gamma }_k(a_k)=q\end{array}}{m}_k,$

each sum being a finite sum over a subset of $\{k:k<r\}$ (A finite sum in a commutative monoid indexed by an arbitrary finite set, The cardinality $\mid A\mid$ of a finite set), and the subtraction is that of the commutative ring $\mathbb{Z}$ (The integers form a commutative ring). The chain $\mathrm{\Gamma }$ is a cycle when $\mathrm{\partial }\mathrm{\Gamma }(q)=0$ for every $q\in \mathbb{C}$.

Sum and negation. For chains ${\mathrm{\Gamma }}_1$ and ${\mathrm{\Gamma }}_2$ in $\mathrm{\Omega }$, the sum ${\mathrm{\Gamma }}_1+{\mathrm{\Gamma }}_2$ is the concatenated list, and $-\mathrm{\Gamma }$ is the list with every coefficient replaced by $-m_k$ and every contour unchanged. Write ${\mathrm{\Gamma }}_1-{\mathrm{\Gamma }}_2:={\mathrm{\Gamma }}_1+(-{\mathrm{\Gamma }}_2)$. The reversal ${\mathrm{\Gamma }}^{-}$ is the list with every coefficient unchanged and every contour replaced by its reversal; the additive inverse $-m_k$ is taken in $\mathbb{Z}$ (The integers form a commutative ring).

A chain is a list, and equality of chains is equality of lists. No free abelian group on the set of contours is introduced here, and no result on this page asserts that two differently presented chains are equal: every statement below is about a given list, and the operations above produce lists. This is a deliberate departure from the presentations that define a chain as a group element, and it is what removes the obligation to say when two chains coincide.

Which lists are cycles. The empty chain is a cycle, both boundary sums being empty and hence $0$ (A finite sum in a commutative monoid indexed by an arbitrary finite set). A list all of whose contours are closed is a cycle: for such a ${\gamma }_k$ the two endpoint values coincide, so $m_k$ enters the sum at $q={\gamma }_k(a_k)$ once positively and once negatively and cancels there, and enters neither sum at any other point. In particular a single closed contour, taken as the list of length $1$ with $m_0=1$, is a cycle. Terms with $m_k=0$ add $0$ to both sums and so never affect $\mathrm{\partial }\mathrm{\Gamma }$.

Cycles are more general than lists of closed contours. The condition is that the endpoints cancel after the coefficients are counted, not that each piece closes up: two contours with the same initial point and the same terminal point, carried with coefficients $+1$ and $-1$, form a cycle although neither is closed. The distinction is what the integral results below actually use: the vanishing of the boundary function is exactly the hypothesis under which the integral of a continuous derivative over $\mathrm{\Gamma }$ is zero.

Ambient set. A chain is a chain in $\mathrm{\Omega }$; the same list is a chain in every open set containing all the ${\gamma }_k^{\ast }$, and in particular in $\mathbb{C}$. When a nonempty connected $\mathrm{\Omega }$ is wanted it is called a complex domain (A complex domain is a nonempty connected open subset of $\mathbb{C}$). Finite sums of integers and of complex numbers are finite sums in their additive commutative monoids, as in A finite sum in a commutative monoid indexed by an arbitrary finite set.