The Atiyah-Patodi-Singer signature formula for measured foliations

Given a compact manifold with boundary X endowed with a foliation F0 transverse to the boundary, and which admits a holonomy invariant transverse measure ., we define three types of signature for the pair (foliation, boundary foliation): the analytic signature, denoted by σΛan(X0;X0), is the leafwise L2-.-index of the signature operator on the extended manifold X obtained by attaching cylindrical ends to the boundary; the Hodge signature σ.; ΛHodge(X0, X0) is defined using the natural representation of the Borel groupoid R of X on the field of square integrable harmonic forms on the leaves; and the de Rham signature, σ.;dR(X0,X0) defined using the natural representation of the Borel groupoid R0 of X0 on the field of the L2-relative de Rham spaces of the leaves. We prove that these three signatures coincide σ.;an(X0, X0/ σHodge(X0; X0);dR.X0, X0) As a consequence of the index formula we proved in [Bull. Sci. Math. (2010), DOI 10.1016/ j.bulsci.2010.10.003], we finally obtain the Atiyah-Patodi-Singer signature formula for measured foliations: σ.ΛdR(X0, ZX0) = {L(T F0);Cλ)+1/2(Nλ(DF].