Relative geometric assembly and mapping cones, Part I: The geometric model and applications

TL;DR

This paper introduces a geometric assembly map in relative K-homology, explores its properties using index theory, and applies it to manifolds with boundary, establishing new vanishing and homotopy invariance results.

Contribution

It defines a new geometric assembly map in relative K-homology and analyzes its properties, connecting index theory with geometric and topological invariants.

Findings

Established a vanishing result for manifolds with boundary and positive scalar curvature.

Proved that rational injectivity implies homotopy invariance of relative higher signatures.

Connected the geometric assembly map to index theoretic tools and prior work of Chang, Weinberger, Yu, and Wahl.

Abstract

Inspired by an analytic construction of Chang, Weinberger and Yu, we define an assembly map in relative geometric $K$-homology. The properties of the geometric assembly map are studied using a variety of index theoretic tools (e.g., the localized index and higher Atiyah-Patodi-Singer index theory). As an application we obtain a vanishing result in the context of manifolds with boundary and positive scalar curvature; this result is also inspired and connected to work of Chang, Weinberger and Yu. Furthermore, we use results of Wahl to show that rational injectivity of the relative assembly map implies homotopy invariance of the relative higher signatures of a manifold with boundary.