In this paper we consider a class of connected closed G-manifolds with a non-empty finite fixed point set, each M of which is totally non-homologous to zero in MG (or G-equivariantly formal), where G = Z2. With the help of the equivariant index, we give an explicit description of the equivariant cohomology of such a G-manifold in terms of algebra, so that we can obtain analytic descriptions of ...

For a projective variety endowed with a torus action, the equivariant cohomology is determined by the fixed points of codimension 1 subtori. Especially, when the fixed points of the torus are finite and fixed varieties under the action of codimension 1 subtori have dimension less than or equal to 2, equivariant cohomology can be described by discrete conditions on the pair of fixed points via G...

We suggest Novikov type inequalities in the situation of a compact Lie groups action assuming that the given closed 1-form is invariant and basic. Our inequalities use equivariant cohomology and an appropriate equivariant generalization of the Novikov numbers. We test and apply our inequalities in the case of a finite group. As an application we obtain Novikov type inequalities for a manifold w...

We introduce and study a new spectral sequence associated with a Poisson group action on a Poisson manifold and an equivariant momentum mapping. This spectral sequence is a Poisson analog of the Leray spectral sequence of a fibration. The spectral sequence converges to the Poisson cohomology of the manifold and has the E2-term equal to the tensor product of the cohomology of the Lie algebra and...

In this chapter we give a categorical definition of the integral cohomology ring of a stack. For quotient stacks [X/G] the categorical cohomology ring may be identified with the equivariant cohomology H∗ G(X). Identifying the stack cohomology ring with equivariant cohomology allows us to prove that the cohomology ring of a quotient Deligne-Mumford stack is rationally isomorphic to the cohomolog...

We describe recent work on positive descriptions of the structure constants of the cohomology of homogeneous spaces such as the Grassmannian, by degenerations and related methods. We give various extensions of these rules, some new and conjectural, to K-theory, equivariant cohomology, equivariant K-theory, and quantum cohomology.

We find presentations by generators and relations for the equivariant quantum cohomology of the Grassmannian. For these presentations, we also find determinantal formulae for the equivariant quantum Schubert classes. To prove this, we use the theory of factorial Schur functions and a characterization of the equivariant quantum cohomology ring.

For an algebra B with an action of a Hopf algebra H we establish the pairing between equivariant cyclic cohomology and equivariant K-theory for B. We then extend this formalism to compact quantum group actions and show that equivariant cyclic cohomology is a target space for the equivariant Chern character of equivariant summable Fredholm modules. We prove an analogue of Julg’s theorem relating...

A refined form of the ‘Folk Theorem’ that a smooth action by a compact Lie group can be (canonically) resolved, by iterated blow up, to have unique isotropy type was established in [1] in the context of manifolds with corners; the canonical construction induces fibrations on the boundary faces of the resolution resulting in an ‘equivariant resolution structure’. Here, equivariant K-theory and t...

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...