  We study the topology of the space of positive scalar curvature metrics on
high dimensional spheres and other spin manifolds. Our main result provides
elements of infinite order in higher homotopy and homology groups of these
spaces, which, in contrast to previous approaches, are of infinite order and
survive in the (observer) moduli space of such metrics.
  Along the way we construct smooth fiber bundles over spheres whose total
spaces have non-vanishing A-hat-genera, thus establishing the
non-multiplicativity of the A-hat-genus in fibre bundles with simply connected
base.
