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

TL;DR

This paper introduces a small scale index theorem that links scalar curvature bounds to topological invariants like Gromov's simplicial norm, extending classical results and providing new geometric-topological constraints.

Contribution

It presents a novel small scale index theorem that bounds Gromov's simplicial norm using scalar curvature, volume, and injectivity radius bounds for manifolds with spin universal covers.

Findings

Bound Gromov's simplicial norm via scalar curvature and geometric bounds

Generalizes Lichnerowicz vanishing theorem

Provides scalar curvature analogue to Cheeger finiteness theorem

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 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.