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In this paper, we define a relative $L^2$-$\rho$-invariant for Diracoperators on odd-dimensional spin manifolds with boundary and show that theyare invariants of the bordism classes of positive scalar curvature metricswhich 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 thefundamental groups of the manifold and its boundary, then there are infinitelymany bordism classes of such psc metrics on the given manifold. This result inturn implies that the moduli-space of psc metrics on such manifolds hasinfinitely many path components. We also indicate how to define delocalised$\eta$-invariants for odd-dimensional spin manifolds with boundary, which couldthen be used to obtain similar results for $4k+1$-dimensional manifolds.

Relative torsion and bordism classes of positive scalar curvature metrics on manifolds with boundary