Equivariant sheaf cohomdlogy

In this paper we study Grothendieck's equivariant sheaf cohomology H(X,G;G) for non-discrete topological groups G and G-sheavesG on a G-Space X. For compact groups and locally compact, totally disconnected groups we obtain detailed results relating H(X,G;-) to H(X;-)^G and H(X/G;-). Furthermore we point out the connection between H(X,G;-) and Borel's equivariant cohomology H_G(X;-).     

Weights in the cohomology of toric varieties

We describe the weight filtration in the cohomology of toric varieties. We present a role of the Frobenius automorphism in an elementary way. We prove that equivariant intersection homology of an arbitrary toric variety is pure. We obtain results concerning Koszul duality: nonequivariant intersection cohomology is equal to the cohomology of the Koszul complexIH _T ^* (X)⊗H*(T). We also describe the weight filtration inIH ^*(X). Supported by KBN 2P03A 00218 grant. I thank, Institute of Mathematics, Polish Academy of Science for hospitality.     

Equivariant homology and cohomology of groups

We provide and study an equivariant theory of group (co)homology of a group G with coefficients in a Γ-equivariant G-module A, when a separate group Γ acts on G and A, generalizing the classical Eilenberg-MacLane (co)homology theory of groups. Relationship with equivariant cohomology of topological spaces is established and application to algebraic K-theory is given.     

Singuläre definition der äquivarianten Bredon homologie

Let G be a discrete group,o(G) the orbit category of G and M:o(G)a a covariant (contravariant) functor to abelian groups. We define a singular equivariant homology theory H_*(X;M) (resp. H^*(X;M)) which satisfies a dimension axiom, in the sense of Bredon (Lecture notes 34). It turns out, that all fundamental properties of these theories directly follow by naturality from the analogous theorems in the classical non equivariant case.     

Eine Spektralsequenz in der stetigen Kohomologietheorie topologischer Gruppen

Let H^*(G; M) be the continuous cohomology of a locally compact group G with coefficients in a topological RG-module M. If G operates without fixed points on a R-paracompact space X such that there is a slice through each point and X/G is R-paracompact, then there exists a spectral sequence converging to the equivariant cohomology H^*(X,G; M) of X with second term E₂ ^p.q?H^p(G; H^qX; M)) where the sheaf theoretical cohomology of X is suitable topologized. Several applications and a generalization to actions of G with non-empty fixed point sets are given.     

Algebraic models of change of groups functors in (co)free rational equivariant spectra

Free and cofree equivariant spectra are important classes of equivariant spectra which represent equivariant cohomology theories on free equivariant spaces. Greenlees-Shipley [24], [26] and Pol and the author [45] have given an algebraic model for rational (co)free equivariant spectra. In this paper, we extend this framework by proving that the Quillen functors of induction-restriction-coinduction between categories of (co)free rational equivariant spectra correspond to Quillen functors between the algebraic models in the case of connected compact Lie groups. This is achieved using new abstract techniques regarding correspondences of Quillen functors along Quillen equivalences, which we expect to be of use in other applications.     

Localization algebras and deformations of Koszul algebras

We show that the center of a flat graded deformation of a standard Koszul algebra A behaves in many ways like the torus-equivariant cohomology ring of an algebraic variety with finite fixed point set. In particular, the center of A acts by characters on the deformed standard modules, providing a “localization map”. We construct a universal graded deformation of A and show that the spectrum of its center is supported on a certain arrangement of hyperplanes which is orthogonal to the arrangement coming from the algebra Koszul dual to A. This is an algebraic version of a duality discovered by Goresky and MacPherson between the equivariant cohomology rings of partial flag varieties and Springer fibers; we recover and generalize their result by showing that the center of the universal deformation for the ring governing a block of parabolic category O{\mathcal{O}} for \mathfrakgl_n{\mathfrak{gl}_n} is isomorphic to the equivariant cohomology of a Spaltenstein variety. We also identify the center of the deformed version of the “category O{\mathcal{O}}” of a hyperplane arrangement (defined by the authors in a previous paper) with the equivariant cohomology of a hypertoric variety.     

Reconstructing the orbit type stratification of a torus action from its equivariant cohomology

We investigate what information on the orbit type stratification of a torus action on a compact space is contained in its rational equivariant cohomology algebra. Regarding the (labelled) poset structure of the stratification, we show that equivariant cohomology encodes the subposet of ramified elements. For equivariantly formal actions, we also examine what cohomological information of the stratification is encoded. In the smooth setting, we show that under certain conditions—which in particular hold for a compact orientable manifold with discrete fixed point set—the equivariant cohomologies of the strata are encoded in the equivariant cohomology of the manifold.

     

The dualizing spectrum of a topological group

To a topological group G, we assign a naive G-spectrum , called the dualizing spectrum of G. When the classifying space BG is finitely dominated, we show that detects Poincaré duality in the sense that BG is a Poincaré duality space if and only if is a homotopy finite spectrum. Secondly, we show that the dualizing spectrum behaves multiplicatively on certain topological group extensions. In proving these results we introduce a new tool: a norm map which is defined for any G and for any naive G-spectrum E. Applications of the dualizing spectrum come in two flavors: (i) applications in the theory of Poincaré duality spaces, and (ii) applications in the theory of group cohomology. On the Poincaré duality space side, we derive a homotopy theoretic solution to a problem posed by Wall which says that in a fibration sequence of fini the total space satisfies Poincaré duality if and only if the base and fiber do. The dualizing spectrum can also be used to give an entirely homotopy theoretic construction of the Spivak fibration of a finitely dominated Poincaré duality space. We also include a new proof of Browder's theorem that every finite H-space satisfies Poincaré duality. In connection with group cohomology, we show how to define a variant of Farrell-Tate cohomology for any topological or discrete group G, with coefficients in any naive equivariant cohomology theory E. When E is connective, and when G admits a subgroup H of finite index such that BH is finitely dominated, we show that this cohomology coincides with the ordinary cohomology of G with coefficients in E in degrees greater than the cohomological dimension of H. In an appendix, we identify the homotopy type of for certain kinds of groups. The class includes all compact Lie groups, torsion free arithmetic groups and Bieri-Eckmann duality groups. Received July 14, 1999 / Revised May 17, 2000 / Published online February 5, 2001     

On a conjecture of Bredon

Using the localization technique for equivariant cohomology theory we prove a conjecture of G.Bredon (s. [4], p.381) which states that under certain conditions (s. the theorem below) the cohomology with Z_p-coefficients of each component of the fixpoint set of a Z_p-space can be generated as an algebra by (at most) the same number of elements as the cohomology of the space itself.     

Bivariant K-theory via correspondences

We use correspondences to define a purely topological equivariant bivariant K-theory for spaces with a proper groupoid action. Our notion of correspondence differs slightly from that of Connes and Skandalis. Our construction uses no special features of equivariant K-theory. To highlight this, we construct bivariant extensions for arbitrary equivariant multiplicative cohomology theories.We formulate necessary and sufficient conditions for certain duality isomorphisms in the topological bivariant K-theory and verify these conditions in some cases, including smooth manifolds with a smooth cocompact action of a Lie group. One of these duality isomorphisms reduces bivariant K-theory to K-theory with support conditions. Since similar duality isomorphisms exist in Kasparov theory, the topological and analytic bivariant K-theories agree if there is such a duality isomorphism.     

Arc Spaces and Equivariant Cohomology

We present a new geometric interpretation of equivariant cohomology in which one replaces a smooth, complex G-variety X by its associated arc space J _∞ X, with its induced G-action. This not only allows us to obtain geometric classes in equivariant cohomology of arbitrarily high degree, but also provides more flexibility for equivariantly deforming classes and geometrically interpreting multiplication in the equivariant cohomology ring. Under appropriate hypotheses, we obtain explicit bijections between $ \mathbb{Z} $ -bases for the equivariant cohomology rings of smooth varieties related by an equivariant, proper birational map. We also show that self-intersection classes can be represented as classes of contact loci, under certain restrictions on singularities of subvarieties. We give several applications. Motivated by the relation between self-intersection and contact loci, we define higher-order equivariant multiplicities, generalizing the equivariant multiplicities of Brion and Rossmann; these are shown to be local singularity invariants, and computed in some cases. We also present geometric $ \mathbb{Z} $ -bases for the equivariant cohomology rings of a smooth toric variety (with respect to the dense torus) and a partial flag variety (with respect to the general linear group).     

Universal (Co) Homology Theories

It is shown that the homology and cohomology theories on separable C*–algebras given by nonstable E–theory are the universal such theories. By specializing to Abelian C*–algebras, we obtain a family of extraordinary Steenrod homology and cohomology theories on pointed compact metric spaces which are the universal such theories in the same way. For each of the extraordinary Steenrod (co)homology theories considered, we describe an –spectrum which represents the theory.     

(Co)Homology Theories for Oriented Algebras

We study three different (co)homology theories for a family of pullbacks of algebras that we call oriented. We obtain a Mayer Vietoris long exact sequence of Hochschild and cyclic homology and cohomology groups for these algebras. We give examples showing that our sequence for Hochschild cohomology groups is different from the known ones. In case the algebras are given by quiver and relations, and that the simplicial homology and cohomology groups are defined, we obtain a similar result in a slightly wider context. Finally, we also study the fundamental groups of the bound quivers involved in the pullbacks.     

REMARKS ON UNIVERSAL COEFFICIENT THEOREMS FOR GENERALIZED HOMOLOGY THEORIES

Abstract

New proofs of universal coefficient theorems for generalized homology theories (cf. ∮ 2, ∮ 3) including L. G. Brown's result, relating Brown-Douglas-Fillmore's Ext (X) with complex K-theory are presented. They are all based on a theorem asserting the existence of a chain functor for a generalized homology theory (cf. ∮ 1), which was originally designed for the construction of strong homology theories on strong shape categories.     

Equivariant–Bivariant Chern Character for Profinite Groups

For the action of a locally compact and totally disconnected group G on a pair of locally compact spaces X and Y we construct, by sheaf theoretic means, a new equivariant and bivariant cohomology theory. If we take for the first space Y an universal proper G-action then we obtain for the second space its delocalized equivariant homology. This is in exact formal analogy to the definition of equivariant K-homology by Baum, Connes, Higson starting from the bivariant equivariant Kasparov KK-theory. Under certain basic finiteness conditions on the first space Y we conjecture the existence of a Chern character from the equivariant Kasparov KK-theory of Y and X into our cohomology theory made two-periodic which becomes an isomorphism upon tensoring the KK-theory with the complex numbers. This conjecture is proved for profinite groups G. An essential role in our construction is played by a bivariant version of Segal localization which we establish for KK-theory.     

Bredon-style homology, cohomology and Riemann-Roch for algebraic stacks

One of the main obstacles for proving Riemann-Roch for algebraic stacks is the lack of cohomology and homology theories that are closer to the K-theory and G-theory of algebraic stacks than the traditional cohomology and homology theories for algebraic stacks. In this paper we study in detail a family of cohomology and homology theories which we call Bredon-style theories that are of this type and in the spirit of the classical Bredon cohomology and homology theories defined for the actions of compact topological groups on topological spaces. We establish Riemann-Roch theorems in this setting: it is shown elsewhere that such Riemann-Roch theorems provide a powerful tool for deriving formulae involving virtual fundamental classes associated to dg-stacks, for example, moduli stacks of stable curves provided with a virtual structure sheaf associated to a perfect obstruction theory. We conclude the present paper with a brief application of this nature.     

论局部乘积与POINCARE-ALEXANDER-LEFSCHETZ型对偶定理

<正> 在§1我们界说了局部乘积,它关联 Hausdorff 紧致空间 X 中闭子集X_0的同调以及 X_0在 X 中邻域的同调.在流形与有边流形上的 Poincaré-Alexander-Lefschetz 型对偶定理可以用这种局部乘积表示(§2).在§3,我们研讨了一类所谓摹流形状空间.局部的下调群与上调群的概念在 [3,233—263页;8]中曾不明显地使