Equivariant K-theory and the Chern character for discrete groups

2010/10/29 by Efton Park, Park, Efton
Mathematics · #19K99 (Secondary) #19L47 (Primary) #47L65 #FOS: Mathematics #K-Theory and Homology (math.KT) #Operator Algebras (math.OA) #math.KT #math.OA #msc:19K99 #msc:19L47 #msc:47L65

paper · pdf · doi:10.48550/arxiv.1010.6271

arxiv created 2010/10/29 · arxiv updated 2010/11/01

Abstract

Let X be a compact Hausdorff space, let Γ be a discrete group that acts continuously on X from the right, define \widetildeX = \(x,γ) ∈ X × Γ: x⋅γ= x\, and let Γ act on \widetildeX via the formula (x,γ)⋅α= (x⋅α, α^-1γα). Results of P. Baum and A. Connes, along with facts about the Chern character, imply that Kⁱ_Γ(X) ⊗ ℂ ≅ Kⁱ(\widetildeX\slashΓ) ⊗ ℂ for i = 0, -1. In this note, we present an example where the groups Kⁱ_Γ(X) and Kⁱ(\widetildeX\slashΓ) are not isomorphic.

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