Index, eta and rho-invariants on foliated bundles

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.

💡 Research Summary

The paper investigates both primary (index) and secondary (eta and rho) invariants associated with leafwise Dirac operators on foliated bundles. Starting from a leafwise Dirac operator (D) on a foliated bundle (\pi\colon E\to B) with foliation (\mathcal{F}), the authors lift (D) to the maximal Connes‑Skandalis Hilbert (C^)-module (\mathcal{E}_{\max}). This yields a regular self‑adjoint operator (D_m) that lives in the multiplier algebra of the maximal groupoid (C^)-algebra (\mathcal{A}{\max}). The functional calculus of (D_m) simultaneously encodes the leafwise calculus (through the von Neumann algebra (\mathcal{N}{\mathcal{F}}) generated by leafwise operators) and the monodromy calculus (through the von Neumann algebra (\mathcal{N}_{\Gamma}) generated by the holonomy groupoid).

When the foliation admits a holonomy‑invariant transverse measure (\mu), the paper shows that the canonical traces (\tau_{\mathcal{F}}) on (\mathcal{N}{\mathcal{F}}) and (\tau{\Gamma}) on (\mathcal{N}{\Gamma}) coincide after integration against (\mu). This compatibility permits the definition of a Fuglede‑Kadison determinant (\operatorname{Det}\mu) and a trace (\operatorname{Tr}_\mu) on the von Neumann algebras, which are then used to extend Atiyah’s index theorem for Galois coverings to the foliated setting. Concretely, for an elliptic leafwise operator (D) with appropriate boundary conditions, the (\mu)-index \