Equivariant Homology Theories Research Articles

Injectivity results for coarse homology theories

We show injectivity results for assembly maps using equivariant coarse homology theories with transfers. Our method is based on the descent principle and applies to a large class of linear groups or, more general, groups with finite decomposition complexity.

Isomorphism conjectures with proper coefficients

Let G be a group and F a nonempty family of subgroups of G, closed under conjugation and under subgroups. Also let E be a functor from small Z-linear categories to spectra, and let A be a ring with a G-action. Under mild conditions on E and A one can define an equivariant homology theory HG(−,E(A)) of G-simplicial sets such that H⁎G(G/H,E(A))=E(A⋊H). The strong isomorphism conjecture for the quadruple (G,F,E,A) asserts that if X→Y is an equivariant map such that XH→YH is an equivalence for all H∈F, thenHG(X,E(A))→HG(Y,E(A)) is an equivalence. In this paper we introduce an algebraic notion of (G,F)-properness for G-rings, modeled on the analogous notion for G-C⁎-algebras, and show that the strong (G,F,E,P) isomorphism conjecture for (G,F)-proper P is true in several cases of interest in the algebraic K-theory context.

Equivariant pretheories and invariants of torsors

In the present paper we introduce and study the notion of an equivariant pretheory: basic examples include equivariant Chow groups, equivariant K-theory and equivariant algebraic cobordism. To extend this set of examples we define an equivariant (co)homology theory with coefficients in a Rost cycle module and provide a version of Merkurjev's (equivariant K-theory) spectral sequence for such a theory. As an application we generalize the theorem of Karpenko-Merkurjev on G-torsors and rational cycles; to every G-torsor E and a G-equivariant pretheory we associate a graded ring which serves as an invariant of E. In the case of Chow groups this ring encodes the information concerning the motivic J-invariant of E and in the case of Grothendieck's K_0 -- indexes of the respective Tits algebras.

The Baum–Connes conjecture via localisation of categories

We redefine the Baum–Connes assembly map using simplicial approximation in the equivariant Kasparov category. This new interpretation is ideal for studying functorial properties and gives analogues of the Baum–Connes assembly map for other equivariant homology theories. We extend many of the known techniques for proving the Baum–Connes conjecture to this more general setting.

The Baum-Connes Conjecture via Localization of Categories

We redefine the Baum-Connes assembly map using simplicial approximation in the equivariant Kasparov category. This interpretation is ideal for studying functorial properties and gives analogues of the Baum-Connes assembly map for all equivariant homology theories.

Commuting Homotopy Limits and Smash Products

In general the processes of taking a homotopy inverse limit of a diagram of spectra and smashing spectra with a fixed space do not commute. In this paper we investigate under what additional assumptions these two processes do commute. In fact we deal with an equivariant generalization which involves spectra and smash products over the orbit category of a discrete group. Such a situation naturally occurs if one studies the equivariant homology theory associated to topological cyclic homology. The main theorem of this paper will play a role in the generalization of the results obtained by Boekstedt, Hsiang and Madsen about the algebraic K-theory Novikov Conjecture to the assembly map for the family of virtually cyclic subgroups.

Chern characters for proper equivariant homology theories and applications to K- and L-theory

Article Chern characters for proper equivariant homology theories and applications to K- and L-theory was published on January 31, 2002 in the journal Journal für die reine und angewandte Mathematik (volume 2002, issue 543).

G-invariante Morse-funktionen

If f is a Morse function on a smooth manifold M there exists a homotopy equivalence from M to a CW complex X such that the critical points of f with index λ are in a one-one correspondence to the λ-cells of X. In the equivariant case, a similar result holds for a special type of invariant Morse functions. In this paper we prove the existence of such special invariant Morse functions on compact smooth G-manifolds. As a consequence, any compact smooth G-manifold is homotopy equivalent to a G-CW complex. Other applications deal with the Euler number of the fixed point set and Morse inequalities in equivariant homology theory.

Cyclic homology and equivariant homology

The purpose of this paper is to explore the relationship between the cyclic homology and cohomology theories of Connes [9-11], see also Loday and Quillen [20], and "IF equivariant homology and cohomology theories. Here II" is the circle group. The most general results involve the definitions of the cyclic homology of cyclic chain complexes and the notions of cyclic and cocyclic spaces so precise statements will be postponed until w 3. In this introduction we explain some of the formal similarities between the cyclic theory and the equivariant theory and give two examples where the general results apply. Let A be an associative algebra over a commutative ring K. Then one can form the cyclic homology HC.(A) and cohomology HC*(A) of A. These groups have periodicity operators

Equivariant Homology Theories on G-Complexes

Equivariant Homology Theories on G-Complexes

Equivariant homology theories on $G$-complexes

A definition is given for a cellular equivariant homology theory on G-complexes. The definition is shown to generalize to G-complexes with prescribed isotropy subgroups. A ring I is introduced to deal with the general definition. One obtains a universal coefficient theorem and studies the universal coefficients.

Singul�re definition der �quivarianten 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.

Singular Homology in Fibre Bundles

The object of this paper is to study the structure of fibre bundles ([7]) by means of their singular complexes ([2, 3]). An important tool for this purpose is the theory of semi-simplicial complexes ([6]), of which the singular complex is an example. The first two sections discuss the equivariant homology theory of a space with a topological group of operators, and the application of this theory to principal fibre bundles. This concept generalizes the notion of equivariant homology with a discrete group of operators ([4, 5]). In ? ?3-8 machinery is developed for the management of semi-simplicial complexes. Applications to fibre bundles are given in ? ?9-11. The remainder of the paper is devoted to proving the existence, for any group, of an almost universal principal bundle. This makes it possible (?12) to extend the notion of the characteristic ring of a fibre bundle ([1]) and to prove its uniqueness. In the discussion of semi-simplicial complexes, two changes from the notation of [6] should be signalized. If K is a semi-simplicial complex, Kq is used to denote the set of q-simplices in K. If ae K, the ith face of ais denoted by o-i (rather than a(i))