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.
Categories: math.DG math.KT