Positive scalar curvature and product formulas for secondary index invariants

TL;DR

This paper develops new secondary invariants for positive scalar curvature metrics on spin manifolds, establishes product formulas, and provides new proofs of index theorems using localization algebras and K-theory.

Contribution

It introduces partial secondary invariants, product formulas, and a new conceptual proof of the secondary partitioned manifold index theorem, advancing the understanding of scalar curvature and index theory.

Findings

Established product formulas for higher rho-invariants.

Proved a partitioned manifold index theorem for higher relative index.

Constructed examples of metrics with positive scalar curvature distinguished by invariants.

Abstract

We introduce partial secondary invariants associated to complete Riemannian metrics which have uniformly positive scalar curvature outside a prescribed subset on a spin manifold. These can be used to distinguish such Riemannian metrics up to concordance relative to the prescribed subset. We exhibit a general external product formula for partial secondary invariants, from which we deduce product formulas for the higher rho-invariant of a metric with uniformly positive scalar curvature as well as for the higher relative index of two metrics with uniformly positive scalar curvature. Our methods yield a new conceptual proof of the secondary partitioned manifold index theorem and a refined version of the delocalized APS-index theorem of Piazza-Schick for the spinor Dirac operator in all dimensions. We establish a partitioned manifold index theorem for the higher relative index. We also show that secondary invariants are stable with respect to direct products with aspherical manifolds that have fundamental groups of finite asymptotic dimension. Moreover, we construct examples of complete metrics with uniformly positive scalar curvature on non-compact spin manifolds which can be distinguished up to concordance relative to subsets which are coarsely negligible in a certain sense. A technical novelty in this paper is that we use Yu's localization algebras in combination with the description of K-theory for graded C*-algebras due to Trout. This formalism allows direct definitions of all the invariants we consider in terms of the functional calculus of the Dirac operator and enables us to give concise proofs of the product formulas.