Small scale index theory, scalar curvature, and Gromov's simplicial norms

In this article, we study the topological complexity of manifolds with a lower scalar curvature bound. We introduce a small scale index theorem to establish an upper bound for Gromov's simplicial norm of the Poincar\'e dual of the A-hat class for manifolds with spin universal covering, in terms of a scalar curvature lower bound, volume upper bound, and injectivity radius lower bound of the universal covering. This result can be viewed both as a generalization of Lichnerowicz vanishing theorem and as a scalar curvature analogue to Cheeger finiteness theorem.