The index of leafwise G-transversally elliptic operators on foliations

Abstract

We introduce and study the index morphism for G-invariant leafwise G-transversally elliptic operators on smooth closed foliated manifolds which are endowed with leafwise actions of the compact group G. We prove the usual axioms of excision, multiplicativity and induction for closed subgroups. In the case of free actions, we relate our index class with the Connes-Skandalis index class of the corresponding leafwise elliptic operator on the quotient foliation. Finally we prove the compatibility of our index morphism with the Gysin Thom isomorphism and reduce its computation to the case of tori actions. We also construct a topological candidate for an index theorem using the Kasparov Dirac element for euclidean G-representations.

References in corpus (6)

• Fractional analytic index
• Quantising proper actions on Spin${}^c$-manifolds
• Transverse noncommutative geometry of foliations
• The equivariant index theorem for transversally elliptic operators and the basic index theorem for Riemannian foliations
• Projective elliptic genera and elliptic pseudodifferential genera
• The index of G-transversally elliptic families I

Cited by in corpus (3)

• Fredholm conditions and index for restrictions of invariant pseudodifferential operators to isotypical components
• An equivariant orbifold index for proper actions
• The index of families of projective operators