Duality pairings with the analytic structure group

TL;DR

This paper introduces a new duality pairing in the analytic structure group and K-theory that enables extraction of numerical invariants like rho-invariants from positive scalar curvature metrics.

Contribution

It constructs a novel slant product linking the analytic structure group with K-theory, extending previous work and enabling the extraction of numerical invariants.

Findings

Constructed a new slant product on the analytic structure group.

Established a duality pairing that yields numerical invariants.

Extended the framework for analyzing positive scalar curvature metrics.

Abstract

We construct a slant product $\mathrm{S}^{G\times H}_p(X\times Y)\otimes \mathrm{K}_{-q}(\bar{\mathfrak{c}}^{\mathrm{red}} Y\rtimes H)\to \mathrm{K}_{p-q}(\mathrm{C}^\ast_G X)$ on the analytic structure group of Higson and Roe and the K-theory of the stable Higson compactification taking values in the (equivariant) Roe algebra. This complements the slant products constructed in earlier work of Engel and the authors ( arXiv:1909.03777 [math.KT] ). The distinguishing feature of our new slant product is that it specializes to a duality pairing $\mathrm{S}^H_p(Y) \otimes \mathrm{K}_{-p}(\bar{\mathfrak{c}}^{\mathrm{red}} (Y)\rtimes H)\to \mathbb{Z}$ which can be used to extract numerical invariants out of elements in the analytic structure group such as rho-invariants associated to positive scalar curvature metrics.