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.

arXiv comment33 pages. In the second version typos have been corrected

References in corpus (10)

• VB-groupoids and representation theory of Lie groupoids
• Rho-classes, index theory and Stolz' positive scalar curvature sequence
• Singular spaces, groupoids and metrics of positive scalar curvature
• Positive scalar curvature and product formulas for secondary index invariants
• The surgery exact sequence, K-theory and the signature operator
• On the deformation groupoid of the inhomogeneous pseudo-differential Calculus
• A Baum-Connes conjecture for singular foliations
• Functoriality for higher rho invariants of elliptic operators
• The Higson-Roe sequence for étale groupoids II. The universal sequence for equivariant families
• Secondary cup and cap products in coarse geometry