The Atiyah Patodi Singer signature formula for measured foliations

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 $ind_{L^2,\Lambda}(D^{\mathcal{F},+}) = <\hat{A}(T\mathcal{F})Ch(E/S),C_\Lambda> + 1/2[\eta_\Lambda(D^{\mathcal{F}_\partial}) - h^+_{\Lambda} + h^-_{\Lambda}].$