The action of the mapping class group on metrics of positive scalar curvature

We present a rigidity theorem for the action of the mapping class group π0$_{ π0}$(Diff(M)) on the space R$^{+}$ (M) of metrics of positive scalar curvature for high dimensional manifolds M. This result is applicable to a great number of cases, for example to simply connected 6-manifolds and high dimensional spheres. Our proof is fairly direct, using results from parametrised Morse theory, the 2-index theorem and computations on certain metrics on the sphere. We also give a non-triviality criterion and a classification of the action for simply connected 7-dimensional Spin-manifolds.