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2020 Groups with Spanier–Whitehead duality
Shintaro Nishikawa, Valerio Proietti
Ann. K-Theory 5(3): 465-500 (2020). DOI: 10.2140/akt.2020.5.465

Abstract

Building on work by Kasparov, we study the notion of Spanier–Whitehead K-duality for a discrete group. It is defined as duality in the KK-category between two C-algebras which are naturally attached to the group, namely the reduced group C-algebra and the crossed product for the group action on the universal example for proper actions. We compare this notion to the Baum–Connes conjecture by constructing duality classes based on two methods: the standard “gamma element” technique, and the more recent approach via cycles with property gamma. As a result of our analysis, we prove Spanier–Whitehead duality for a large class of groups, including Bieberbach’s space groups, groups acting on trees, and lattices in Lorentz groups.

Citation

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Shintaro Nishikawa. Valerio Proietti. "Groups with Spanier–Whitehead duality." Ann. K-Theory 5 (3) 465 - 500, 2020. https://doi.org/10.2140/akt.2020.5.465

Information

Received: 16 September 2019; Revised: 9 February 2020; Accepted: 24 February 2020; Published: 2020
First available in Project Euclid: 11 August 2020

zbMATH: 07237239
MathSciNet: MR4132744
Digital Object Identifier: 10.2140/akt.2020.5.465

Subjects:
Primary: 46L85
Secondary: 46L80 , 55P25

Keywords: Baum–Connes conjecture , direct splitting method , noncommutative topology , Poincaré duality , Spanier–Whitehead duality

Rights: Copyright © 2020 Mathematical Sciences Publishers

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Vol.5 • No. 3 • 2020
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