The Atiyah Patodi Singer signature formula for measured foliations

TL;DR

This paper extends the Atiyah-Patodi-Singer signature formula to measured foliations, providing a new index formula involving noncommutative geometry tools for manifolds with boundary and cylindrical ends.

Contribution

It proves a novel $L^2$-$ ext{Lambda}$ index formula for Dirac operators on measured foliations with boundary, generalizing classical index theorems using noncommutative geometry.

Findings

Derived an $L^2$-$ ext{Lambda}$ index formula for foliated manifolds with boundary.

Connected the index to eta invariants and foliation characteristic classes.

Extended classical index theory to the setting of measured foliations with cylindrical ends.

Abstract

Let $(X_0,\mathcal{F}_0) $ be a compact manifold with boundary endowed with a foliation $\mathcal{F}_0$ which is assumed to be measured and transverse to the boundary. We denote by $\Lambda$ a holonomy invariant transverse measure on $(X_0,\mathcal{F}_0) $ and by $\mathcal{R}_0$ the equivalence relation of the foliation. Let $(X,\mathcal{F})$ be the corresponding manifold with cylindrical end and extended foliation with equivalence relation $\mathcal{R}$. In the first part of this work we prove a formula for the $L^2$-$\Lambda$ index of a longitudinal Dirac-type operator $D^{\mathcal{F}}$ on $X$ in the spirit of Alain Connes' non commutative geometry $ind_{L^2,\Lambda}(D^{\mathcal{F},+}) = <\hat{A}(T\mathcal{F})Ch(E/S),C_\Lambda> + 1/2[\eta_\Lambda(D^{\mathcal{F}_\partial}) - h^+_{\Lambda} + h^-_{\Lambda}].$