  For a group $G$ (of type $F$) acting properly on a coarse Poincar\'{e}
duality space $X$, Kapovich-Kleiner introduced a coarse version of Alexander
duality between $G$ and its complement in $X$. More precisely, the cohomology
of $G$ with group ring coefficients is dual to a certain \v{C}ech homology
group of the family of increasing neighborhoods of a $G$-orbit in $X$. This
duality applies more generally to coarse embeddings of certain contractible
simplicial complexes into coarse $PD(n)$ spaces. In this paper we introduce a
relative version of this \v{C}ech homology that satisfies the
Eilenberg-Steenrod Exactness Axiom, and we prove a relative version of coarse
Alexander duality.
  As an application we provide a detailed proof of the following result, first
stated by Kapovich-Kleiner. Given a $2$-complex formed by gluing $k$ halfplanes
along their boundary lines and a coarse embedding into a contractible
$3$-manifold, the complement consists of $k$ deep components that are arranged
cyclically in a pattern called a Jordan cycle. We use the Jordan cycle as an
invariant in proving the existence of a $3$-manifold group that is virtually
Kleinian but not itself Kleinian.
