Equivariant index and Chern character
نویسندگان
چکیده
These maps correspond respectively to the families index theorem of Atiyah and Singer for fibrewise pseudodifferential operators for a smooth fibration of a compact manifold M −→ Y, to the equivariant index for invariant and transversally elliptic operators for the smooth action of a compact Lie group on a compact manifold Z and to the pseudodifferential extension of the numerical index map of Atiyah, Patodi and Singer for a compact manifold with boundary Z. The Kgroups forming the domains are respectively the topological (compactly supported) K-theory of the fibrewise contangent bundle of the fibration, the G-equivariant Ktheory of the (generally singular) space of fibre conormals of the group action and the compactly supported K-theory of the cotangent bundle of the manifold with boundary with the fibration of the boundary smashed to its base, the conormal line. The targets for the first two maps are the K-theory of the base and the ring of virtual representations of the group. In this talk I had planned to try to present approaches to the analytic definitions of these maps, and the corresponding theorems – identifying them with topological push-forward maps – which I hoped would indicate how one can freely combine them to give an equivariant, families index of Atiyah-Patodi-Singer type. Unfortunately due to limitiations of time I will not be able to discuss the boundary case (in joint work with Frédéric Rochon) so will be content here with discussing an approach to a families equivariant index theorem. I will also discuss the closely related problem of deriving a Chern character formula, for the image of the index in an appropriate cohomology. First consider the families index theorem. Here is a diagram of the construction of the analytic index which corresponds to push-forward under the map π : T ∗(M/Y ) −→ Y which has fibres diffeomorphic to T ∗Z, where Z is the model fibre of the fibration M −→ Y :
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