  We prove that for many degrees in a stable range the homotopy groups of the
moduli space of metrics of positive scalar curvature on S^n and on other
manifolds are non-trivial. This is achieved by further developing and then
applying a family version of the surgery construction of Gromov-Lawson to an
exotic smooth families of spheres due to Hatcher.
  As described, this works for all manifolds of suitable dimension and for the
quotient of the space of metrics of positive scalar curvature by the (free)
action of the subgroup of diffeomorphisms which fix a point and its tangent
space.
  We also construct special manifolds where the quotient of the space of
metrcis of positive scalar curvature by the full diffeomorphism group has
non-trivial higher homotopy groups.
