  In this paper, we define a relative $L^2$-$\rho$-invariant for Dirac
operators on odd-dimensional spin manifolds with boundary and show that they
are invariants of the bordism classes of positive scalar curvature metrics
which are collared near the boundary. As an application, we show that if a
$4k+3$-dimensional spin manifold with boundary admits such a metric and if,
roughly speaking, there exists a torsion element in the difference of the
fundamental groups of the manifold and its boundary, then there are infinitely
many bordism classes of such psc metrics on the given manifold. This result in
turn implies that the moduli-space of psc metrics on such manifolds has
infinitely many path components. We also indicate how to define delocalised
$\eta$-invariants for odd-dimensional spin manifolds with boundary, which could
then be used to obtain similar results for $4k+1$-dimensional manifolds.
