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

Abstract

Let $\iota :{\mathcal{F}}_0\to {\mathcal{F}}_1$ 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 $({\iota }_{ad}^{[0,1)}{)}_{!}\in E_n\left(C^{\ast }(G_{ad}^{[0,1)}),C^{\ast }(H_{ad}^{[0,1)})\right)$, 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.

33 pages. In the second version typos have been corrected

References in corpus (2)

• Functoriality for higher rho invariants of elliptic operators
• Secondary cup and cap products in coarse geometry