  This paper contributes to the classification of positive scalar curvature
metrics up to bordism and up to concordance. Let $M$ be a closed spin manifold
of dimension $\ge 5$ which admits a metric with positive scalar curvature. We
give lower bounds on the rank of the group of psc metrics over $M$ up to
bordism in terms of the corank of the canonical map $KO_*(M)\to
KO_*(B\pi_1(M))$, provided the rational analytic Novikov conjecture is true for
$\pi_1(M)$.
