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

TL;DR

The paper develops a quantitative bridge between lower bounds on scalar curvature and topological complexity by introducing a small-scale index framework. Through epsilon-homology and Connes-Chern characters, it connects Dirac-operator indices to the Gromov simplicial norm of the Poincaré dual of the Â-class, yielding explicit bounds that depend on the scalar curvature bound, injectivity radius, and volume data. The methodology blends Connes’ noncommutative geometry, quantitative K-theory, Bismut’s local index theory, and Getzler rescaling to produce both local index densities and global norm bounds, including a $C^0$-stable version for aspherical manifolds. This approach generalizes Lichnerowicz vanishing phenomena and offers a scalar-curvature analogue of Cheeger finiteness, providing a robust analytic toolkit for controlling topological invariants via geometric constraints. The work also develops a comprehensive analytic framework (weighted Sobolev spaces, resolvent estimates, and holomorphic functional calculus) to justify the asymptotics and kernel computations that underpin the main results.

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 Poincaré 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.