We prove the following results for toric Deligne–Mumford stacks, under minimal compactness hypotheses: the Localization Theorem in equivariant K-theory; the equivariant Hirzebruch–Riemann–Roch theorem; the Fourier–Mukai transformation associated to a crepant toric wall-crossing gives an equivariant derived equivalence.

The category of rational O(2)-equivariant spectra splits as a product of cyclic and dihedral parts. Using the classification of rational G-equivariant spectra for finite groups G , we classify the dihedral part of rational O(2)-equivariant spectra in terms of an algebraic model.

In this paper a closing lemma for C1 nonsingular endomorphisms equivariant under free actions of finite-groups is proved. Hence a recurrent trajectory, as well as all of its symmetric conjugates, of a C1 nonsingular endomorphism equivariant under a free action of a finite group can be closed up simultaneously by an arbitrarily small C1 equivariant perturbation.

We give positive formulas for the restriction of a Schubert Class to a T -fixed point in the equivariant K-theory and equivariant cohomology of the Grassmannian. Our formulas rely on a result of Kodiyalam-Raghavan and Kreiman-Lakshmibai, which gives an equivariant Gröbner degeneration of a Schubert variety in the neighborhood of a T -fixed point of the Grassmannian.

We introduce the equivariant elliptic genus for open varieties and prove an equivariant version of the change of variable formula for blow-ups along complete intersections. In addition, we prove the equivariant elliptic genus analogue of the McKay correspondence for the ALE spaces.

We construct a canonical Thom isomorphism for virtual T-oriented T-equivariant spin bundles with vanishing Borel-equivariant second Chern class, which is natural under pull-back of vector bundles and exponential under Whitney sum. It extends in the rational case the non-equivariant sigma orientation of Hopkins, Strickland, and the author. The construction relates the sigma orientation to the re...

Let G be a locally compact group. We describe elements of KK(A,B) by equivariant homomorphisms, following Cuntz’s treatment in the non-equivariant case. This yields another proof for the universal property of KK: It is the universal split exact stable homotopy functor. To describe a Kasparov triple (E, φ, F ) for A,B by an equivariant homomorphism, we have to arrange for the Fredholm operator F...

The purpose of this paper is to prove the localization theorem for torus actions in equivariant intersection theory. Using the theorem we give another proof of the Bott residue formula for Chern numbers of bundles on smooth complete varieties. In addition, our techniques allow us to obtain residue formulas for bundles on a certain class of singular schemes which admit torus actions. This class ...

Spaces of maps into classifying spaces for equivariant crossed complexes, II: The general topological group case. Abstract The results of a previous paper [3] on the equivariant homotopy theory of crossed complexes are generalised from the case of a discrete group to general topological groups. The principal new ingredient necessary for this is an analysis of homotopy coherence theory for cross...

Extending work of Klyachko and Perling, we develop a combinatorial description of pure equivariant sheaves on an arbitrary nonsingular toric variety X. This combinatorial description can be used to construct very explicit moduli spaces of stable equivariant sheaves on X using Geometric Invariant Theory (analogous to techniques used in case of equivariant vector bundles by Payne and Perling). We...