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Groups with
Spanier–Whitehead duality
Shintaro Nishikawa and Valerio Proietti |
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Vol. 5 (2020), No. 3, 465–500
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Abstract |
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Building on work by Kasparov, we study the notion of Spanier–Whitehead -duality for a discrete group. It is defined as duality in the -category between two -algebras which are naturally attached to the group, namely the reduced group -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
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Mathematical Subject Classification 2010
Primary: 46L85
Secondary: 46L80, 55P25
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Milestones
Received: 16 September 2019
Revised: 9 February 2020
Accepted: 24 February 2020
Published: 28 July 2020
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Authors |
| Shintaro Nishikawa | |
| Department of Mathematics Pennsylvania State University University Park, PA United States |
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| Valerio Proietti | |
| Department of Mathematics East China Normal University Shanghai China |
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