  This is a continuation of our previous work with Botvinnik on the
nontriviality of the secondary index invariant on spaces of metrics of positive
scalar curvature, in which we take the fundamental group of the manifolds into
account. We show that the secondary index invariant associated to the vanishing
of the Rosenberg index can be highly nontrivial, for positive scalar curvature
Spin manifolds with torsionfree fundamental groups which satisfy the
Baum--Connes conjecture. For example, we produce a compact Spin 6-manifold such
that its space of positive scalar curvature metrics has each rational homotopy
group infinite dimensional.
  At a more technical level, we introduce the notion of "stable metrics" and
prove a basic existence theorem for them, which generalises the Gromov--Lawson
surgery technique, and we also give a method for rounding corners of manifold
with positive scalar curvature metrics.
