Homologically simply connected complex domains

A complex domain $\mathrm{\Omega }$ (A complex domain is a nonempty connected open subset of $\mathbb{C}$) is homologically simply connected when every complex chain which is a cycle and whose trace lies in $\mathrm{\Omega }$ (Complex chains, their traces, and cycles) is null-homologous in $\mathrm{\Omega }$ (Null-homologous cycles and homologous cycles in an open set); equivalently, when

$n(\mathrm{\Gamma },p)=0\text{for every cycle }\mathrm{\Gamma }\text{ in }\mathrm{\Omega }\text{ and every }p\in \mathbb{C}\setminus \mathrm{\Omega },$

with the index of Integration over a complex chain and the index of a chain.

The qualifier is part of the name and is kept in every use. The condition above is about indices, and it is the only notion of simple connectivity defined or used on this page: no notion involving loops, homotopies or a fundamental group is introduced here, and none is invoked in any proof below. Writing "homologically simply connected" everywhere is what keeps that scope visible to a reader who knows the other notions from elsewhere.

$\mathbb{C}$ itself is homologically simply connected, because $\mathbb{C}\setminus \mathbb{C}$ is empty and the condition is then vacuous. More generally the condition constrains a domain only through the points it omits.

Connectedness is part of the definition, since a complex domain is nonempty, open and connected. That is a convenience rather than a necessity for the index condition itself, and every result below that assumes homological simple connectivity therefore has a connected domain available.