Small Scale Index Theory, Scalar Curvature, and Gromov's Simplicial Norms

Abstract: 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 on Gromov's simplicial norm of the Poincar\'e dual of the $\hat{A}$-class for Riemannian manifolds satisfying a scalar curvature lower bound. This result can be considered as a topological finiteness theorem for manifolds with a lower scalar curvature bound.