Non-trivial characteristic invariants of homogeneous foliated bundles

Characteristic Classes of Foliated Surface Bundles with Area-preserving Holonomy

Making use of the extended flux homomorphism defined in [13] on the group SympΣg of symplectomorphisms of a closed oriented surface Σg of genus g ≥ 2, we introduce new characteristic classes of foliated surface bundles with symplectic, equivalently area-preserving, total holonomy. These characteristic classes are stable with respect to g and we show that they are highly non-trivial. We also pro...

Index, Eta and Rho Invariants on Foliated Bundles

We study primary and secondary invariants of leafwise Dirac operators on foliated bundles. Given such an operator, we begin by considering the associated regular self-adjoint operator Dm on the maximal Connes-Skandalis Hilbert module and explain how the functional calculus of Dm encodes both the leafwise calculus and the monodromy calculus in the corresponding von Neumann algebras. When the fol...

The Index of Operators on Foliated Bundles

We compute the equivariant cohomology Chern character of the index of elliptic operators along the leaves of the foliation of a flat bundle. The proof is based on the study of certain algebras of pseudodifferential operators and uses techniques for analizing noncommutative algebras similar to those developed in Algebraic Topology, but in the framework of cyclic cohomology and noncommutative geo...

Locally trivial W∗-bundles

Cut-and-paste on Foliated Bundles

We discuss the behaviour of the signature index class of closed foliated bundles under the operation of cutting and pasting. Along the way we establish several index theoretic results: we define Atiyah-Patodi-Singer (≡ APS) index classes for Dirac-type operators on foliated bundles with boundary; we prove a relative index theorem for the difference of two APS-index classes associated to differe...