Index, eta and rho-invariants on foliated bundles

TL;DR

This paper explores primary and secondary invariants of leafwise Dirac operators on foliated bundles, extending classical index theorems and introducing a foliated rho-invariant with stability and homotopy invariance properties.

Contribution

It extends Atiyah's index theorem and defines a foliated rho-invariant, demonstrating its stability and homotopy invariance under Baum-Connes assumptions.

Findings

Extended Atiyah's index theorem to foliations.

Defined a foliated rho-invariant for signature operators.

Proved homotopy invariance of the foliated rho-invariant.

Abstract

We study primary and secondary invariants of leafwise Dirac operators on foliated bundles. Given such an operator, we begin by considering the associated regular self-adjoint operator $D_m$ on the maximal Connes-Skandalis Hilbert module and explain how the functional calculus of $D_m$ encodes both the leafwise calculus and the monodromy calculus in the corresponding von Neumann algebras. When the foliation is endowed with a holonomy invariant transverse measure, we explain the compatibility of various traces and determinants. We extend Atiyah's index theorem on Galois coverings to these foliations. We define a foliated rho-invariant and investigate its stability properties for the signature operator. Finally, we establish the foliated homotopy invariance of such a signature rho-invariant under a Baum-Connes assumption, thus extending to the foliated context results proved by Neumann, Mathai, Weinberger and Keswani on Galois coverings.