We consider twisted equivariant K-theory for actions of a compact Lie group G on a space X where all the isotropy subgroups are connected and of maximal rank. We show that the associated rational spectral sequenceà la Segal has a simple E 2 -term expressible as invariants under the Weyl group of G. Namely, if T is a maximal torus of G, they are invariants of the π 1 (X T )-equivariant Bredon cohomology of the universal cover of X T with suitable coefficients. In the case of the inertia stack ΛY this term can be expressed using the cohomology of Y T and algebraic invariants associated to the Lie group and the twisting. A number of calculations are provided. In particular, we recover the rational Verlinde algebra when Y = { * }.
In this paper we obtain a description of the Grothendieck group of complex vector bundles over the classifying space of a p‐local finite group (S,F,L) in terms of representation rings of subgroups of S. We also prove a stable elements formula for generalized cohomological invariants of p‐local finite groups, which is used to show the existence of unitary embeddings of p‐local finite groups. Finally, we show that the augmentation C∗(|L|p∧;double-struckFpfalse)→double-struckFp is Gorenstein in the sense of Dwyer–Greenlees–Iyengar and obtain some consequences about the cohomology ring of false|scriptLfalse|p∧.
Abstract. In this paper we construct faithful representations of saturated fusion systems over discrete p-toral groups and use them to find conditions that guarantee the existence of unitary embeddings of p-local compact groups. These conditions hold for the Clark-Ewing and Aguadé-Zabrodsky p-compact groups as well as some exotic 3-local compact groups. We also show the existence of unitary embeddings of finite loop spaces.
IntroductionIn the theory of compact Lie groups, the existence of a faithful unitary representation for every compact Lie group is a consequence of the Peter-Weyl theorem. This paper is concerned with the existence of analogous representations for several objects in the literature which are considered to be homotopical counterparts of compact Lie groups.In 1994, W.G. Dwyer and C.W. Wilkerson [17] introduced p-compact groups. They are loop spaces which satisfy some finiteness properties at a particular prime p. For example, if G is a compact Lie group such that its group of connected components is a finite p-group, then its p-completion G , where a bijective correspondence between connected p-compact groups and reflection data over the p-adic integers was established.Many ideas from the theory of compact Lie groups have a homotopical analogue for p-compact groups. Faithful unitary representations correspond to homotopy monomorphisms at the prime p from the classifying space BX of a pcompact group into BU (n) ∧ p for some n. A homotopy monomorphism at p is map g such that the homotopy fiber F of g ∧ p is BZ/p-null, that is, the evaluation map Map(BZ/p, F ) → F is a homotopy equivalence. For simplicity, we will call such maps BX → BU (n) In this article we deal with the same question for the combinatorial structures called p-local compact groups, which encode the p-local information of some spaces at a prime p. They were introduced in [9] to model p-completed classifying spaces of compact Lie groups, p-compact groups, as well as linear torsion groups, and they 2010 Mathematics Subject Classification. 55R35, (primary), 20D20, 20C20 (secondary).
In this paper we define complex equivariant K-theory for actions of Lie
groupoids using finite-dimensional vector bundles. For a Bredon-compatible Lie
groupoid, this defines a periodic cohomology theory on the category of finite
equivariant CW-complexes. We also establish an analogue of the completion
theorem of Atiyah and Segal. Some examples are discussed.Comment: 26 pages, v3 Correction to some lemmas and definitions in section 4.
Added a page at the end of section 3. Other minor correction
In this paper we provide characterizations of p-nilpotency for fusion systems and p-local finite groups that are inspired by known result for finite groups. In particular, we generalize criteria by Atiyah, Brunetti, Frobenius, Quillen, Stammbach and Tate.1991 Mathematics Subject Classification. Primary 55R35, 20D15, Secondary 20D20, 20C20, 20N99.
We show that the twisted K-theory of the classifying space of a plocal finite group is isomorphic to the completion of the Grothendieck group of twisted representations of the fusion system with respect to the augmentation ideal of the representation ring of the fusion system. We use this result to compute the K-theory of the Ruiz-Viruel exotic 7-local finite groups.2010 Mathematics Subject Classification. 55R35 (primary), 19A22, 19L50, 20D20 (secondary).
In this paper we define twisted equivariant K -theory for actions of Lie groupoids. For a Bredon-compatible Lie groupoid G, we show that this defines a periodic cohomology theory on the category of finite G-CW-complexes with G-stable projective bundles by comparing with a suitable representable cohomology theory. A classification of these bundles is shown. We also obtain a completion theorem and apply these results to proper actions of groups.
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