  Inspired by an analytic construction of Chang, Weinberger and Yu, we define
an assembly map in relative geometric $K$-homology. The properties of the
geometric assembly map are studied using a variety of index theoretic tools
(e.g., the localized index and higher Atiyah-Patodi-Singer index theory). As an
application we obtain a vanishing result in the context of manifolds with
boundary and positive scalar curvature; this result is also inspired and
connected to work of Chang, Weinberger and Yu. Furthermore, we use results of
Wahl to show that rational injectivity of the relative assembly map implies
homotopy invariance of the relative higher signatures of a manifold with
boundary.
