  Let $A$ be a C*-algebra, $J \subset A$ a C*-subalgebra, and let $B$ be a
stable C*-algebra. Under modest assumptions we organize invertible
C*-extensions of $A$ by $B$ that are trivial when restricted onto $J$ to become
a group $Ext_J^{-1}(A,B)$, which can be computed by a six-term exact sequence
which generalizes the excision six-term exact sequence in the first variable of
$KK$-theory. Subsequently we investigate the relative K-homology which arises
from the group of relative extensions by specializing to abelian C*-algebras.
It turns out that this relative K-homology carries substantial information also
in the operator theoretic setting from which the BDF theory was developed and
we conclude the paper by extracting some of this information on approximation
of normal operators.
