Conventions for chains, cycles and the homological adjective on this page

Four choices are made in the definitions above, and each is made for a reason that can be stated.

A chain is a finite list, not an element of a group. Complex chains, their traces, and cycles presents a chain as a list of pairs $(m_k,{\gamma }_k)$, with sum given by concatenation and negation by negating the coefficients. Presenting chains as elements of a free abelian group on the set of contours would require saying when two chains are equal, and every such identification would then have to be checked against the integral and the index. The list presentation avoids that obligation entirely: equality of chains is equality of lists, and no result above asserts that two differently presented chains coincide. What the results do assert is equality of the numbers ${\int }_{\mathrm{\Gamma }}fdz$ and $n(\mathrm{\Gamma },p)$, which is all any of them uses.

A cycle is a chain whose boundary function vanishes, and that is weaker than requiring every piece to be closed. In The integral of a continuous derivative over a cycle is zero, summing the endpoint increments of a primitive with the coefficients $m_k$ leaves the coefficient of $F(q)$ equal to the boundary value at $q$. The same boundary cancellation is also used in The index of a cycle about a point off its trace is an integer to make the logarithm increments sum to an integer index. Both arguments need the endpoint counts to cancel and nothing more, so imposing closedness on each ${\gamma }_k$ would strengthen the hypothesis without strengthening either conclusion. Two contours running between the same pair of distinct points, weighted $+1$ and $-1$, satisfy the condition and neither is closed.

The adjective is "homologically simply connected", written out every time. Homologically simply connected complex domains defines a condition on indices: every cycle in the domain is null-homologous in it (Null-homologous cycles and homologous cycles in an open set). Nothing above defines or uses a notion of simple connectivity phrased with loops or homotopies, and no statement above asserts a relation between the two. Keeping the qualifier is what makes that scope visible to a reader who arrives with the other notion in mind.

The winding number belongs to the parametrised contour, not to its trace. The winding number of a closed contour about a point off its trace is stated for a map $\gamma :[a,b]\to \mathbb{C}$, because the integral it is built from depends on that map: a complex contour is a rectifiable path together with its domain and its parametrisation (Rectifiable complex contours, reversal, concatenation, closedness, and orientation), and the trace is only the image set. Two closed contours can share a trace and have different indices at a point, since the parametrisation records how many times and in which direction the trace is traversed; the chain-level index of Integration over a complex chain and the index of a chain inherits the same dependence through its terms.