Noncommutative Geometry and Quantum Physics
Vietri sul Mare, August 31 - September 5, 2009
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Speaker: Paolo Antonini, Regensburg
Title: The Atiyah Patodi Singer signature formula for measured foliations
Abstract: Let (X0, F0) be a compact manifold with boundary endowed with a foliation F0 which is assumed to be measured and transverse to the boundary. We denote by Λ a holonomy invariant transverse measure on (X0, F0) and by R0 the equivalence relation of the foliation. Let (X, F) be the corresponding manifold with cylindrical end and extended foliation with equivalence relation R.
In the first part we describe a formula for the L2-Λ index of a longitudinal Dirac-type operator DF on X in the spirit of Alain Connes' non commutative geometry.
In the second part we specialize ourselves to the signature operator. We define three types of signature for the pair (foliation, boundary foliation): the analytic signature, denoted σΛ,an (X, ∂X0) is the L2-Λ-index of the signature operator on the cylinder; the Hodge signature σΛ,Hodge(X0, F0) is defined using the natural representation of R on the field of square integrable harmonic forms on the leaves and the de Rham signature, σΛ,dR (X, ∂X0), defined using the natural representation of R0 on the field of relative de Rham spaces of the leaves. We prove that these three signatures coincide

σΛ,an (X, ∂X0) = σΛ,Hodge(X0, F0) = σΛ,dR (X, ∂X0).

As a consequence of these equalities and of the index formula we finally obtain the main result of this work, the Atiyah-Patodi-Singer signature formula for measured foliations:

σΛ,dR (X, ∂X0) = <L(TF0), CΛ> + 1/2 [ηΛ(D F∂ )].
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