Interior Kasparov products for ϱ-classes on Riemannian foliated bundles

Abstract

Let ι:F0→F1 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 (ιad[0,1))!∈En(C⁎(Gad[0,1)),C⁎(Gad[0,1))), where G and H are the monodromy groupoids associated with F0 and F1 respectively. Furthermore, we prove an interior Kasparov product formula for foliated ϱ-classes associated with longitudinal metrics of positive scalar curvature in the case of Riemannian foliated bundles.