The Atiyah Patodi Singer index formula for measured foliations

Abstract

Let $X_0$ be a compact Riemannian manifold with boundary endowed with a oriented, measured even dimensional foliation with purely transverse boundary. Let $X$ be the manifold with cylinder attached and extended foliation. We prove that the $L^2$--measured index of a Dirac type operator is well defined and the following Atiyah Patodi Singer index formula is true $$ind_{L^2,\Lambda}(D^+) = <\widehat{A}(X,\nabla)Ch(E/S),C_\Lambda> + 1/2[\eta_\Lambda(D^{\mathcal{F}_\partial}) - h^+_\Lambda + h^-_\Lambda].$$ Here $\Lambda$ is a holonomy invariant transverse measure, $\eta_{\Lambda}(D^{\mathcal{F}_{\partial}})$ is the Ramachandran eta invariant \cite{Rama} of the leafwise boundary operator and the $\Lambda$--dimensions $h^\pm_\Lambda$ of the space of the limiting values of extended solutions is suitably defined using square integrable representations of the equivalence relation of the foliation with values on weighted Sobolev spaces on the leaves.