Vol. 5, No. 3, 2020

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Groups with Spanier–Whitehead duality

Shintaro Nishikawa and Valerio Proietti

Vol. 5 (2020), No. 3, 465–500
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.

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Keywords
Spanier–Whitehead duality, Poincaré duality, Baum–Connes conjecture, direct splitting method, noncommutative topology
Mathematical Subject Classification 2010
Primary: 46L85
Secondary: 46L80, 55P25
Milestones
Received: 16 September 2019
Revised: 9 February 2020
Accepted: 24 February 2020
Published: 28 July 2020
Authors
Shintaro Nishikawa
Department of Mathematics
Pennsylvania State University
University Park, PA
United States
Valerio Proietti
Department of Mathematics
East China Normal University
Shanghai
China