Rational computations of the topological K-theory of classifying spaces of discrete groups (original) (raw)

Topological K–(co)homology of classifying spaces of discrete groups

Algebraic & Geometric Topology, 2013

Let G be a discrete group. We give methods to compute for a generalized (co-)homology theory its values on the Borel construction EG × G X of a proper G-CW -complex X satisfying certain finiteness conditions. In particular we give formulas computing the topological K-(co)homology K * (BG) and K * (BG) up to finite abelian torsion groups. They apply for instance to arithmetic groups, word hyperbolic groups, mapping class groups and discrete cocompact subgroups of almost connected Lie groups. For finite groups G these formulas are sharp. The main new tools we use for the K-theory calculation are a Cocompletion Theorem and Equivariant Universal Coefficient Theorems which are of independent interest. In the case where G is a finite group these theorems reduce to well-known results of Greenlees and Bökstedt.

Topological K-theory for discrete groups and Index theory

2022

We give a complete solution, for discrete countable groups, to the problem of defining and computing a geometric pairing between the left hand side of the Baum-Connes assembly map, given in terms of geometric cycles associated to proper actions on manifolds, and cyclic periodic cohomology of the group algebra. Indeed, for any such group Γ (without any further assumptions on it) we construct an explicit morphism from the Left-Hand side of the Baum-Connes assembly map to the periodic cyclic homology of the group algebra. This morphism, called here the Chern-Baum-Connes assembly map, allows to give a proper and explicit formulation for a Chern-Connes pairing with the periodic cyclic cohomology of the group algebra. Several theorems are needed to formulate the Chern-Baum-Connes assembly map. In particular we establish a delocalised Riemann-Roch theorem, the wrong way functoriality for periodic delocalised cohomology for Γ-proper actions, the construction of a Chern morphism between the ...

The topological K-theory of certain crystallographic groups

Journal of Noncommutative Geometry, 2013

Let Γ be a semidirect product of the form Z n ⋊ρ Z/p where p is prime and the Z/p-action ρ on Z n is free away from the origin. We will compute the topological K-theory of the real and complex group C *-algebra of Γ and show that Γ satisfies the unstable Gromov-Lawson-Rosenberg Conjecture. On the way we will analyze the (co-)homology and the topological K-theory of the classifying spaces BΓ and BΓ. The latter is the quotient of the induced Z/p-action on the torus T n .

Survey on Classifying Spaces for Families of Subgroups

Progress in Mathematics

We define for a topological group G and a family of subgroups F two versions for the classifying space for the family F, the G-CW-version EF (G) and the numerable G-space version JF (G). They agree if G is discrete, or if G is a Lie group and each element in F compact, or if F is the family of compact subgroups. We discuss special geometric models for these spaces for the family of compact open groups in special cases such as almost connected groups G and word hyperbolic groups G. We deal with the question whether there are finite models, models of finite type, finite dimensional models. We also discuss the relevance of these spaces for the Baum-Connes Conjecture about the topological K-theory of the reduced group C *-algebra, for the Farrell-Jones Conjecture about the algebraic Kand L-theory of group rings, for Completion Theorems and for classifying spaces for equivariant vector bundles and for other situations.

Homotopy colimits of classifying spaces of abelian subgroups of a finite group

The classifying space BG of a topological group G can be filtered by a sequence of subspaces B(q, G), q ≥ 2, using the descending central series of free groups. If G is finite, describing them as homotopy colimits is convenient when applying homotopy theoretic methods. In this paper we introduce natural subspaces B(q, G) p ⊂ B(q, G) defined for a fixed prime p. Then B(q, G) is stably homotopy equivalent to a wedge of B(q, G) p as p runs over the primes dividing the order of G. Colimits of abelian groups play an important role in understanding the homotopy type of these spaces. Extraspecial 2-groups are key examples, for which these colimits turn out to be finite. We prove that for extraspecial 2-groups of order 2 2n+1 , n ≥ 2, B(2, G) does not have the homotopy type of a K(π, 1) space. For a finite group G, we compute the complex K-theory of B(2, G) modulo torsion.

Computations of K- and L-Theory of Cocompact Planar Groups

K-Theory, 2000

Using the isomorphism conjectures of Baum & Connes and Farrel & Jones, we compute the algebraic K-and L-theory and the topological K-theory of cocompact planar groups (= cocompact N.E.C-groups) and of groups G appearing in an extension 1 → Z n → G → π → 1 where π is a finite group and the conjugation π-action on Z n is free outside 0 ∈ Z n. These computations apply for instance to two-dimensional crystallographic groups and cocompact Fuchsian groups.

Equivariant K-homology and K-theory for some discrete planar affine groups

arXiv (Cornell University), 2022

We consider the semi-direct products G = Z 2 GL 2 (Z), Z 2 SL 2 (Z) and Z 2 Γ(2) (where Γ(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 EG 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 EG, 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 second and third groups, the computations in K 0 provide explicit generators that are matched by the Baum-Connes assembly map.

HOMOTOPY THEORY OF CLASSIFYING SPACES OF COMPACT LIE GROUPS

1994

The basic problem of homotopy theory is to classify spaces and maps between spaces, up to homotopy, by means of invariants like cohomology. In the last decade some striking progress has been made with this problem when the spaces involved are classifying spaces of compact Lie groups. For example, it has been shown, for G connected and simple, that if two self maps of BG agree in rational cohomology then they are homotopic. It has also been shown that if a space X has the same mod p cohomology, cup product, and Steenrod operations as a classifying space BG then (at least if p is odd and G is a classical group) X is actually homotopy equivalent to BG after mod p completion. Similar methods have also been used to obtain new results on Steenrod's problem of constructing spaces with a given polynomial cohomology ring. The aim of this paper is to describe these results and the methods used to prove them.