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

Ph.D Thesis, La Sapienza Rome