Bredon Homology Research Articles

Equivariant K-Homology and K-Theory for Some Discrete Planar Affine Groups

Abstract We consider the semi-direct products $G={\mathbb{Z}}^{2}\rtimes GL_{2}({\mathbb{Z}}), {\mathbb{Z}}^{2}\rtimes SL_{2}({\mathbb{Z}})$, and ${\mathbb{Z}}^{2}\rtimes \Gamma (2)$ (where $\Gamma (2)$ is the congruence subgroup of level 2). For each of them, we compute both sides of the Baum–Connes conjecture, namely the equivariant $K$-homology of the classifying space $\underline{E}G$ for proper actions on the left-hand side, and the analytical K-theory of the reduced group $C^{*}$-algebra on the right-hand side. The computation of the LHS is made possible by the existence of a 3-dimensional model for $\underline{E}G$, which allows to replace equivariant K-homology by Bredon homology. We pay due attention to the presence of torsion in $G$, leading to an extensive study of the wallpaper groups associated with finite subgroups. For the first and third groups, the computations in $K_{0}$ provide explicit generators that are matched by the Baum–Connes assembly map.

Bredon homology of Artin groups of dihedral type

For Artin groups of dihedral type, we compute the Bredon homology groups of the classifying space for the family of virtually cyclic subgroups with coefficients in the K-theory of a group ring.

On the algebras over equivariant little disks

We describe the structure present in algebras over the little disks operads for various representations of a finite group G, including those that are not necessarily universe or that do not contain trivial summands. We then spell out in more detail what happens for G=C2, describing the structure on algebras over the little disks operad for the sign representation. Here we can also describe the resulting structure in Bredon homology. Finally, we produce a stable splitting of coinduced spaces analogous to the stable splitting of the product, and we use this to determine the homology of the signed James construction.

Freeness and equivariant stable homotopy

We introduce a notion of freeness for R O $RO$ -graded equivariant generalized homology theories, considering spaces or spectra E $E$ such that the R $R$ -homology of E $E$ splits as a wedge of the R $R$ -homology of induced virtual representation spheres. The full subcategory of these spectra is closed under all of the basic equivariant operations, and this greatly simplifies computation. Many examples of spectra and homology theories are included along the way. We refine this to a collection of spectra analogous to the pure and isotropic spectra considered by Hill–Hopkins–Ravenel. For these spectra, the R O $RO$ -graded Bredon homology is extremely easy to compute, and if these spaces have additional structure, then this can also be easily determined. In particular, the homology of a space with this property naturally has the structure of a co-Tambara functor (and compatibly with any additional product structure). We work this out in the example of B U R $BU_{\mathbb {R}}$ and coinduced versions of this. We finish by describing a readily computable bar and twisted bar spectra sequence, giving Bredon homology for various E ∞ $E_{\infty }$ pushouts, and we apply this to describe the homology of B B U R $BBU_{\mathbb {R}}$ .

On the equivariant K– and KO–homology of some special linear groups

We compute the equivariant $KO$-homology of the classifying space for proper actions of $\textrm{SL}_3(\mathbb{Z})$ and $\textrm{GL}_3(\mathbb{Z})$. We also compute the Bredon homology and equivariant $K$-homology of the classifying spaces for proper actions of $\textrm{PSL}_2(\mathbb{Z}[\frac{1}{p}])$ and $\textrm{SL}_2(\mathbb{Z}[\frac{1}{p}])$ for each prime $p$. Finally, we prove the Unstable Gromov-Lawson-Rosenberg Conjecture for a large class of groups whose maximal finite subgroups are odd order and have periodic cohomology.

Hecke operators in Bredon (co)homology, K-(co)homology and Bianchi groups

In this paper, we provide a framework for the study of Hecke operators acting on the Bredon (co)homology of an arithmetic discrete group. Our main interest lies in the study of Hecke operators for Bianchi groups. Using the Baum–Connes conjecture, we can transfer computations in Bredon homology to obtain a Hecke action on the [Formula: see text]-theory of the reduced [Formula: see text]-algebra of the group. We show the power of this method giving explicit computations for the group [Formula: see text]. In order to carry out these computations we use an Atiyah–Segal type spectral sequence together with the Bredon homology of the classifying space for proper actions.

P-Toral Approximations Compute Bredon Homology

Abstract We study Bredon homology approximations for spaces with an action of a compact Lie group G. We show that if M is a coMackey functor satisfying mild p-locality conditions, then Bredon homology of a G-space X with coefficients in M is determined by fixed points of p-toral subgroups of G acting on X. As an application we prove a vanishing result for the Bredon homology of the complex ${{\mathcal{L}}}_{n}$ of direct-sum decompositions of ${{\mathbb{C}}}^{n}$.

Classifying spaces and Bredon (co)homology for transitive groupoids

We define the orbit category for transitive topological groupoids and their equivariant CW-complexes. By using these constructions we define equivariant Bredon homology and cohomology for actions of transitive topological groupoids. We show how these theories can be obtained by looking at the action of a single isotropy group on a fiber of the anchor map, extending equivariant results for compact group actions. We also show how this extension from a single isotropy group to the entire groupoid action can be applied to the structure of principal bundles and classifying spaces.

The Farrell–Tate and Bredon homology for [formula omitted] via cell subdivisions

We provide some new computations of Farrell–Tate and Bredon (co)homology for arithmetic groups. For calculations of Farrell–Tate or Bredon homology, one needs cell complexes where cell stabilizers fix their cells pointwise. We provide two algorithms computing an efficient subdivision of a complex to achieve this rigidity property. Applying these algorithms to available cell complexes for PSL4(Z) provides computations of Farrell–Tate cohomology for small primes as well as the Bredon homology for the classifying spaces of proper actions with coefficients in the complex representation ring.

Computing Bredon homology of groups

We describe the basic ingredients of a general computational framework for performing machine calculations in the cohomology of groups. This has been implemented in the GAP system for computational algebra and the paper is intended to aid those wishing to extend that implementation to their own needs.

Bredon homology of partition complexes

We prove that the Bredon homology or cohomology of the partition complex with fairly general coefficients is either trivial or computable in terms of constructions with the Steinberg module. The argument involves a theory of Bredon homology and cohomology approximation.

On the equivariant $K$-homology of PSL$_2$ of the imaginary quadratic integers

We establish formulae for the part due to torsion of the equivariant K-homology of all the Bianchi groups (PSL_2 of the imaginary quadratic integers), in terms of elementary number-theoretic quantities. To achieve this, we introduce a novel technique in the computation of Bredon homology: representation ring splitting, which allows us to adapt the recent technique of torsion subcomplex reduction from group homology to Bredon homology.

Brown’s criterion in Bredon homology

We translate Brown's criterion for homological finiteness properties to the setting of Bredon homology.

Intermediaries in Bredon (co)homology and classifying spaces

For certain contractible G-CW-complexes and F a family of subgroups of G, we construct a spectral sequence converging to the F-Bredon cohomology of G with E1-terms given by the F-Bredon cohomology of the stabilizer subgroups. As applications, we obtain several corollaries concerning the cohomological and geometric dimensions of the classifying space EFG. We also introduce, for any subgroup closed class of groups F, a hierarchically de ned class of groups and show that if a group G is in this class, then G has finite F ∩ G-Bredon (co)homological dimension if and only if G has jump F ∩ G-Bredon (co)homology.

On the classifying space for proper actions of groups with cyclic torsion

In this paper we introduce a common framework for describing the topological part of the Baum–Connes conjecture for a wide class of groups. We compute the Bredon homology for groups with aspherical presentation, one-relator quotients of products of locally indicable groups, extensions of ℤn by cyclic groups, and fuchsian groups. We take advantage of the torsion structure of these groups to use appropriate models of the universal space for proper actions which allow us, in turn, to extend some technology defined by Mislin in the case of one-relator groups.

Bredon homology and ramified covering G-maps

Let G be a finite group. The objective of this paper is twofold. First we prove that the cellular Bredon homology groups with coefficients in an arbitrary coefficient system M are isomorphic to the homotopy groups of certain topological abelian group. And second, we study ramified covering G-maps of simplicial sets and of simplicial complexes. As an application, we construct a transfer for them in Bredon homology, when M is a Mackey functor. We also show that the Bredon–Illman homology with coefficients in M satisfies the equivariant weak homotopy equivalence axiom in the category of G-spaces.

A new description of equivariant cohomology for totally disconnected groups

AbstractWe 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 well as a computation of equivariant periodic cyclic homology for a natural class of examples. In addition we discuss the relation between cosheaf homology and equivariant Bredon homology. Since the theory of Baum and Schneider generalizes cosheaf homology we finally see that all these approaches to equivariant cohomology for totally disconnected groups are closely related.

Bredon homology and equivariant [formula omitted]-homology of [formula omitted

We obtain the equivariant K -homology of the classifying space for proper actions E ¯ S L ( 3 , Z ) from the computation of its Bredon homology with respect to finite subgroups and coefficients in the representation ring. We also obtain the corresponding results for G L ( 3 , Z ) . Our calculations give therefore the topological side of the Baum–Connes conjecture for these groups.

Equivariant K -homology for some Coxeter groups

We obtain the equivariant K-homology of the classifying space EW for W, a right-angled or, more generally, an even Coxeter group. The key result is a formula for the relative Bredon homology of EW in terms of Coxeter cells. Our calculations amount to the K-theory of the reduced C*-algebra of W, via the Baum–Connes assembly map.

Equivariant Morse relations

For a finite group G, Costenoble and Waner defined a cellular (co-)homology theory for G-spaces X, which is graded on virtual representations of the equivariant fundamental groupoid …G(X). Using this homology, we associate an infinite (Morse) series with an equivariant Morse function f defined on a closed Riemannian G-manifold M. Wasserman has shown that when the critical locus of f is a disjoint union of orbits, M has a canonical decomposition into disc bundles. We show that if this decomposition ‘corresponds’ to a virtual representation ∞ of …G(M), then the Morse relations are satisfied by the ‘∞th homology groups’. For semi-free G-actions, we characterise the Morse functions which naturally give rise to such representations ∞ of …G(M). We also show that corresponding to any equivariant Morse function on a Z2-manifold, it is always possible to define virtual representations ∞ so that the Morse relation is satisfied by the ‘∞th homology groups’. In particular, the Morse relation is satisfied by Bredon homology.