Homology, Homotopy and Applications

Volume 18 (2016)

Number 1

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

Pages: 49 – 70

DOI: https://dx.doi.org/10.4310/HHA.2016.v18.n1.a4


Noé Bárcenas (Centro de Ciencias Matemáticas, Universidad Nacional Autónoma de México, Morelia, Michoacán, México)

Mario Velásquez (Departamento de Matemáticas, Pontificia Universidad Javeriana, Bogotá D.C., Colombia)


We compare twisted equivariant $K$-theory of $SL_{3}{\mathbb{Z}}$ with untwisted equivariant $K$-theory of a central extension $St_3{\mathbb{Z}}$. We compute all twisted equivariant $K$-theory groups of $SL_{3}{\mathbb{Z}}$, and compare them with previous work on the equivariant $K$-theory of $BSt_3{\mathbb{Z}}$ 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 $SL_{3}{\mathbb{Z}}$, for which a version of $KK$-theoretic duality studied by Echterhoff, Emerson, and Kim is verified.


twisted equivariant $K$-theory, Bredon cohomology, Baum–Connes conjecture with coefficients, twisted group $C^*$-algebra, $KK$-theoretic duality

2010 Mathematics Subject Classification

19K33, 19L47, 19L64

Published 31 May 2016