  Index maps taking values in the $K$-theory of a mapping cone are defined and
discussed. The resulting index theorem can be viewed in analogy with the
Freed-Melrose index theorem. The framework of geometric $K$-homology is used in
a fundamental way. In particular, an explicit isomorphism from a geometric
model for $K$-homology with coefficients in a mapping cone, $C_{\phi}$, to
$KK(C(X),C_{\phi})$ is constructed.
