Interior Kasparov product for $\varrho$-classes on Riemannian foliated bundles

TL;DR

This paper extends the construction of lower shriek maps to adiabatic deformation groupoid C*-algebras and proves a Kasparov product formula for $ ho$-classes in Riemannian foliated bundles, advancing index theory in foliation geometry.

Contribution

It introduces an asymptotic morphism for adiabatic deformation groupoid C*-algebras and establishes a Kasparov product formula for $ ho$-classes in Riemannian foliated bundles.

Findings

Constructed an asymptotic morphism for adiabatic deformation groupoid C*-algebras.

Proved an interior Kasparov product formula for $ ho$-classes.

Extended lower shriek maps to the setting of adiabatic deformation groupoids.

Abstract

Let $\iota\colon \mathcal{F}_0\to\mathcal{F}_1 $ be a suitably oriented inclusion of foliations over a manifold $M$, then we extend the construction of the lower shriek maps given by Hilsum and Skandalis to adiabatic deformation groupoid C*-algebras: we construct an asymptotic morphism $(\iota_{ad}^{[0,1)})_!\in E_n\left(C^*(G_{ad}^{[0,1)}), C^*(H_{ad}^{[0,1)})\right)$, where $G$ and $H$ are the monodromy groupoids associated with $\mathcal{F}_0$ and $\mathcal{F}_1$ respectively. Furthermore, we prove an interior Kasparov product formula for foliated $\varrho$-classes associated with longitudinal metrics of positive scalar curvature in the case of Riemannian foliated bundles.