A connected plane domain that is not homologically simply connected

Given: The annulus $A=\{z\in \mathbb{C}:\frac{1}{2}<\mathrm{\mid }z\mathrm{\mid }<2\}$ and the contour $C(t)=\exp (it)$ on $[0,2\pi ]$, taken as the chain with the single term $(1,C)$.

A complex domain is homologically simply connected when every cycle with trace in it is null-homologous in it (Homologically simply connected complex domains).

A cycle $\mathrm{\Gamma }$ with trace in an open $\mathrm{\Omega }$ is null-homologous in $\mathrm{\Omega }$ when $n(\mathrm{\Gamma },q)=0$ for every $q\in \mathbb{C}\setminus \mathrm{\Omega }$ (Null-homologous cycles and homologous cycles in an open set).

For $a\in \mathbb{C}$, $r>0$ and $k\in \mathbb{Z}$, the contour $a+r\exp (ikt)$ on $[0,2\pi ]$ is a closed complex contour with index $k$ for $\mathrm{\mid }z-a\mathrm{\mid }<r$ and $0$ for $\mathrm{\mid }z-a\mathrm{\mid }>r$, and with trace $\{\mathrm{\mid }z-a\mathrm{\mid }=r\}$ when $k\mathrm{\ne }0$ (A circle traversed $k$ times has winding number $k$ inside and $0$ outside).

For a positively oriented circle $a+r\exp (it)$ with $r>0$, $(2\pi i{)}^{-1}{\int }_{\gamma }dz\mathrm{/}(z-a)=1$ (The normalized integral around a positively oriented circle centred at a is 1).

A complex domain is a nonempty, connected, open subset of $\mathbb{C}$ (A complex domain is a nonempty connected open subset of $\mathbb{C}$).

For $c\in \mathbb{C}$ and $R\ge 0$ the set $\{z:\mathrm{\mid }z-c\mathrm{\mid }>R\}$ is path-connected (The exterior of a closed disc in the plane is path-connected).

A single closed contour with coefficient $1$ is a cycle whose trace is that contour's trace (Complex chains, their traces, and cycles), and its index is the winding number of that contour (Integration over a complex chain and the index of a chain).

For $n\ge 2$ the unit sphere $S^{n-1}\subseteq {\mathbb{R}}^n$ is path-connected (For $n\ge 2$, the sphere $S^{n-1}$ is path-connected and connected); the map $x\mapsto x\mathrm{/}\parallel x{\parallel }_2$ into $S^{n-1}$ is continuous (Radial normalisation $x\mapsto x\mathrm{/}\parallel x{\parallel }_2$ is continuous on ${\mathbb{R}}^n\setminus \{0\}$); and $S^{n-1}=\{x:\parallel x{\parallel }_2=1\}$ (Euclidean spheres and closed balls as subspaces of ${\mathbb{R}}^n$).

A subset is path-connected when any two of its points are joined by a continuous map from $[0,1]$ with image inside it (Paths, path-connected spaces and path components), a path-connected subset is connected (Every path-connected space is connected, and every path component lies inside a component), and a function continuous on each member of a finite closed cover is continuous (Continuity may be checked on any open cover, and on any finite closed cover; composites of continuous maps are continuous).

$\mathrm{\mid }zw\mathrm{\mid }=\mathrm{\mid }z\mathrm{\mid }\mathrm{\mid }w\mathrm{\mid }$ (Conjugation is an involutive real-field automorphism, $z\bar{z}=\mathrm{\mid }z{\mathrm{\mid }}^2$, and modulus is definite, multiplicative, and subadditive).

A set is open exactly when each of its points admits a ball inside it, and $B(x,r)=\{y:d(x,y)<r\}$ (The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement, Open ball, closed ball and sphere in a metric space).