The completion theorem in K-theory for proper actions of a discrete group

Lück, Wolfgang; Oliver, Bob

doi:10.1016/s0040-9383(99)00077-4

Cited by 64 publications

(125 citation statements)

References 17 publications

(14 reference statements)

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“…For infinite discrete groups satisfying appropriate finiteness conditions, Adem [3] and Lück [34] have studied the relationship between the K -theory of the classifying space B and the representation rings of the finite subgroups of . Lück and Oliver [35] considered the case of an infinite discrete group acting properly, ie with finite stabilizers, on a space X . They showed that the -equivariant K -theory of X , completed appropriately, agrees with the topological K -theory of the homotopy orbit space E X .…”

Section: Introductionmentioning

confidence: 99%

Yang–Mills theory over surfaces and the Atiyah–Segal theorem

Ramras

2008

Algebr. Geom. Topol.

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In this paper we explain how Morse theory for the Yang-Mills functional can be used to prove an analogue, for surface groups, of the Atiyah-Segal theorem. Classically, the Atiyah-Segal theorem relates the representation ring R(Γ) of a compact Lie group Γ to the complex K -theory of the classifying space BΓ. For infinite discrete groups, it is necessary to take into account deformations of representations, and with this in mind we replace the representation ring by Carlsson's deformation K -theory spectrum K def (Γ) (the homotopy-theoretical analogue of R(Γ)). Our main theorem provides an isomorphism in homotopyfor all compact, aspherical surfaces Σ and all * > 0. Combining this result with work of Tyler Lawson, we obtain homotopy theoretical information about the stable moduli space of flat unitary connections over surfaces.

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Section: Introductionmentioning

confidence: 99%

Yang–Mills theory over surfaces and the Atiyah–Segal theorem

Ramras

2008

Algebr. Geom. Topol.

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“…We mention that these spaces E(G, F) and in particular EG play an important role in the formulation of the Baum-Connes Conjecture [3, Conjecture 3.15 on p. 254], the Isomorphism Conjecture in algebraic K-and L-theory of Farrell and Jones [5], the generalization of the completion theorem of Atiyah and Segal for finite groups to infinite discrete groups [12] and in the construction of classifying spaces for equivariant bundles [4, Section I.8 and I.9]. More information about models for EG can be found for instance in [3].…”

Section: Introductionmentioning

confidence: 99%

On the universal space for group actions with compact isotropy

Meintrup¹

2000

Geometry and Topology: Aarhus

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Let G be a locally compact topological group and EG its universal space for the family of compact subgroups. We give criteria for this space to be G-homotopy equivalent to a d-dimensional G-CW -complex, a finite G-CW -complex or a G-CW -complex of finite type. Essentially we reduce these questions to discrete groups, and to the homological algebra of the orbit category of discrete groups with respect to certain families of subgroups.

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“…Theorem 11 (W. Lück and B. Oliver [35]). If Γ is a (countable) discrete group and X is a proper Γ -compact Γ -space, then…”

Section: The Reduced Crossed-productmentioning

confidence: 99%