Content

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   Introduction
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   Part I: Foliated Atiyah's theorem and the Baum-Connes map
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   Part II: Stability properties of the foliated -invariant
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   Part III: Hilbert-Poincaré complexes for foliations
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   Part IV: Application to homotopy invariance
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  Background on foliations and Operator algebras
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   Foliated Charts and Foliated Atlases
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   Holonomy
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   Groupoids associated to a foliation
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   Noncommutative integration theory on foliations
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   Tangential measures
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   Transverse measures
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   Integrating a tangential measure against a holonomy invariant measure
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  Operator algebras on foliations
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   The convolution algebra on a groupoid
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   Representations of Bc
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   C*-algebra of a foliation with coefficients in a vector bundle
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   Von Neumann Algebras for foliations
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   Traces on foliations
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  Foliated Atiyah's theorem
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  Pseudodifferential operators on Groupoids
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   Longitudinal Pseudodifferential operators on Foliations and its monodromy groupoid
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   Almost local pseudodifferential operators on foliations
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  Measured Index of Dirac Operators
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   Statement of foliated Atiyah's theorem
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   Construction of the parametrix
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   Atiyah-Bott formula for the measured index
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  K-theoretic Index of Dirac operators
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   Hilbert C*-modules on foliations
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   Dirac Operators on Hilbert C*-modules
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   Remarks on the Baum-Connes map
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  Functional calculus of Dirac operators
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   Functional calculus of normal regular operators on Hilbert C*-modules
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   Functional calculus for the operator Dm
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  Stability properties of foliated -invariants
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   The foliated and invariants
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   -invariant as a determinant
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   Metric independence of the -invariant
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   Leafwise diffeomorphism invariance of the -invariant
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  Hilbert-Poincaré complexes on foliations
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   Hilbert modules associated with leafwise maps
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   Alternative description of GXX'(f)
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   Relation with Connes-Skandalis module
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    Hilbert-Poincaré complexes for foliations
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   Homotopy equivalence of HP-complexes on foliations
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   Regularization and extension to adjointable operators on Hilbert-modules
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   Poincaré lemma for Hilbert modules
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  Applications: Extending Keswani's proof for foliations
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   von Neumann algebras associated with a leafwise homotopy equivalence
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   Traces
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   Operators on Hilbert modules
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  Determinants and the Large time path
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   Determinants of paths
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   The Large time path
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   The determinant of the large time path
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   Remarks on the Small Time Path and homotopy invariance
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  Signatures and homotopy equivalences of Hilbert-Poincaré complexes
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   Unbounded HP-complexes
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   Signatures of HP-complexes
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   Homotopy invariance of the signature
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   Morita equivalence of HP-complexes