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

Authors

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)

Abstract

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.

Keywords

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

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Published 31 May 2016