The purpose of this paper is to develop an equivariant intersection theory for actions of linear algebraic groups on algebraic schemes. The theory is based on our construction of equivariant Chow groups. They are algebraic analogues of equivariant cohomology groups which have all the functorial properties of ordinary Chow groups. In addition, they enjoy many of the properties of equivariant coh...

We first introduce an invariant index for G-equivariant elliptic differential operators on a locally compact manifold M admitting a proper cocompact action of a locally compact group G. It generalizes the Kawasaki index for orbifolds to the case of proper cocompact actions. Our invariant index is used to show that an analog of the Guillemin-Sternberg geometric quantization conjecture holds if M...

For an algebra B coming with an action of a Hopf algebra H and a twist automorphism, we introduce equivariant twisted cyclic cohomology. In the case when the twist is implemented by a modular element in H we establish the pairing between even equivariant cyclic cohomology and equivariant K-theory for B. We then extend this formalism to compact quantum group actions and show that our cyclic coho...

We derive a formula for the η-invariants of equivariant Dirac operators on quotients of compact Lie groups, and for their infinitesimally equivariant extensions. As an example, we give some computations for spheres. Quotients M = G/H of compact Lie groups provide many important examples of Riemannian manifolds with non-negative sectional curvature. The primary characteristic classes and numbers...

We consider smooth actions of totally disconnected groups on simplicial complexes and compare different equivariant cohomology groups associated to such actions. Our main result is that the bivariant equivariant cohomology theory introduced by Baum and Schneider can be described using equivariant periodic cyclic homology. This provides a new approach to the construction of Baum and Schneider as...

We define and study equivariant analytic and local cyclic homology for smooth actions of totally disconnected groups on bornological algebras. Our approach contains equivariant entire cyclic cohomology in the sense of Klimek, Kondracki and Lesniewski as a special case and provides an equivariant extension of the local cyclic theory developped by Puschnigg. As a main result we construct a multip...

1 Equivariant Cohomology 1 1.1 Group cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Equivariant cohomology of topological spaces . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Equivariant vector bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.4 Equivariant pushforward . . . . . . . . . . . . . . . . . . . . . . . ....

This work is divided in two cases. In the first case, we consider a spin manifold M as set of fixed points an S1-action on X, and second case loop space M. For each build vector bundle, connection bundle endomorphisms. These objects are used to global operators which define analytical index case. equal topological equivariant Atiyah–Singer index, expression where Witten genus appears.