Covering complexity, scalar curvature, and quantitative $K$-theory

TL;DR

This paper links the covering complexity of Riemannian spin manifolds to scalar curvature bounds using operator $K$-theory and a vanishing theorem, advancing understanding of geometric and topological invariants.

Contribution

It introduces a novel relationship between covering complexity and scalar curvature via quantitative operator $K$-theory and Lipschitz topological $K$-theory, building on previous vanishing results.

Findings

Established a link between covering complexity and scalar curvature bounds.

Utilized pairing between operator $K$-theory and Lipschitz topological $K$-theory.

Extended vanishing theorems for the quantitative higher index.

Abstract

We establish a relationship between a certain notion of covering complexity of a Riemannian spin manifold and positive lower bounds on its scalar curvature. This makes use of a pairing between quantitative operator $K$-theory and Lipschitz topological $K$-theory, combined with an earlier vanishing theorem for the quantitative higher index.