Equivariant $K$-theory of central extensions and twisted equivariant $K$-theory: $SL_{3}\mathbb{Z}$ and $St_{3}{\mathbb{Z}}$

Abstract

We compare twisted equivariant K-theory of SL 3 Z with untwisted equivariant K-theory of a central extension St 3 Z. We compute all twisted equivariant K-theory groups of SL 3 Z, and compare them with previous work on the equivariant K-theory of BSt 3 Z by Tezuka and Yagita.Using a universal coefficient theorem by the authors, the computations explained here give the domain of Baum-Connes assembly maps landing on the topological K-theory of twisted group C * -algebras related to SL 3 Z, for which a version of KKtheoretic duality studied by Echterhoff, Emerson, and Kim is verified.The second author was financially supported under the project Aplicaciones de la K-teoría en teoría delíndice y las conjeturas de isomorfismo with ID

Proper actions and decompositions in equivariant K-theory

“…From discussion on page 59 in [BV16] we know that there is a unique element u 1 + u 2 ∈ H 3 (Sl 3 (Z); Z) restricting to nontrivial extensions of the three different copies of S 3 contained in Sl 3 (Z), on the other hand Lemma 9 (i) in [Sou78] implies that St 3 (Z) has the same property, then u 1 + u 2 represents the extension (4) . Then using Thm.…”

“…When the twisting α Q ′ on the right hand side term is defined in terms of cocycles in group cohomology (for details consult [Dwy08] or [BV16]). In this terminology, in [BV16] calculations of the equivariant twisted K-theory groups for E Sl 3 (Z) were obtained, then we have to determine the cocycle…”

Computations in $$C_{pq}$$ C pq -Bredon cohomology

“…• Equivariant cell decomposition for the action of Sl 3 (Z) on the homogeneous space Sl 3 (R)/SO 3 . There exists a triangulation of the quotient of an equivariant deformation retract of this homogeneous space described in [64], but also in [58], which is the main input for the computations of twisted equivariant K-Theory and K-homology in [9], [10].…”

“…We fix an orientation; namely, the ordering of the vertices O < Q < M < M < N < N < P induces an orientation in E and also in the quotient BSl 3 Z = E/ ≡. The cells coboundaries are detemined in section 5 of [9] and include restriction of representations and signs coming from the prescribed orientation chosen above.…”

A description of the assembly map for the Baum-Connes conjecture with coefficients

“…That description looks convenient to study the Baum-Connes conjecture in specific cases. For example by results in [15], [1] and [2] we have explicit computations of the equivariant K-homology groups of SL(3, Z), one can try to describe that elements in terms of operators appearing in this work ans compute the assembly map. We will explore that question in a future work.…”