Realizing the analytic surgery group of Higson and Roe geometrically, Part III: Higher invariants

TL;DR

This paper establishes an isomorphism between geometric and analytic models of the Higson-Roe surgery group, introduces a delocalized Chern character, and connects positive scalar curvature invariants with the geometric model.

Contribution

It constructs a geometric isomorphism for the Higson-Roe surgery group, introduces a delocalized Chern character, and relates scalar curvature invariants to the geometric model.

Findings

Established an isomorphism between geometric and analytic surgery groups.

Constructed a delocalized Chern character for the geometric model.

Linked positive scalar curvature invariants to the geometric model.

Abstract

We construct an isomorphism between the geometric model and Higson-Roe's analytic surgery group, reconciling the constructions in the previous papers in the series on "Realizing the analytic surgery group of Higson and Roe geometrically" with their analytic counterparts. Following work of Lott and Wahl, we construct a Chern character on the geometric model for the surgery group; it is a "delocalized Chern character", from which Lott's higher delocalized $\rho$-invariants can be retrieved. Following work of Piazza and Schick, we construct a geometric map from Stolz' positive scalar curvature sequence to the geometric model of Higson-Roe's analytic surgery exact sequence.