A Segal conjecture for p-completed classifying spaces

Ragnarsson, Kári

doi:10.1016/j.aim.2007.04.006

Cited by 4 publications

(6 citation statements)

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“…Lewis-May-McClure proved this for B(G, K) = B(F all , K) in[15]. A minor adjustment generalizes their proof to any family of subgroups.This leads to the following description of stable maps between classifying spaces, which is Theorem D of the introduction, and generalizes[20, Theorem B].…”

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confidence: 67%

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Completion of G-spectra and stable maps between classifying spaces

Ragnarsson

2011

Advances in Mathematics

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We prove structural theorems for computing the completion of a G-spectrum at the augmentation ideal of the Burnside ring of a finite group G. First we show that a Gspectrum can be replaced by a spectrum obtained by allowing only isotropy groups of prime power order without changing the homotopy type of the completion. We then show that this completion can be computed as a homotopy colimit of completions of spectra obtained by further restricting isotropy to one prime at a time, and that these completions can be computed in terms of completion at a prime.As an application, we show that the spectrum of stable maps from BG to the classifying space of a compact Lie group K splits non-equivariantly as a wedge sum of p-completed suspension spectra of classifying spaces of certain subquotients of G × K. In particular this describes the dual of BG.

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confidence: 67%

“…Applying the functor π 0 , one independently recovers [20,Theorem B]. One can also focus on a single prime p to obtain a wedge sum description of F (BG ∧ p + , Σ ∞ BK + ) (see Corollary 6.5), and applying π 0 to that description one independently recovers [20, Theorem A].…”

Section: Introductionmentioning

confidence: 93%

Completion of G-spectra and stable maps between classifying spaces

Ragnarsson

2011

Advances in Mathematics

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“…Since this result is of independent interest, it will be presented separately in [24]. The proof given there is similar but more direct, with the added advantage that the double coset formula is preserved, and that the target can be the classifying space of a compact Lie group.…”

Section: Classifying Spectramentioning

confidence: 89%

Classifying spectra of saturated fusion systems

Ragnarsson

2006

Algebr. Geom. Topol.

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The assignment of classifying spectra to saturated fusion systems was suggested by Linckelmann and Webb and has been carried out by Broto, Levi and Oliver. A more rigid (but equivalent) construction of the classifying spectra is given in this paper. It is shown that the assignment is functorial for fusion-preserving homomorphisms in a way which extends the assignment of stable p -completed classifying spaces to finite groups, and admits a transfer theory analogous to that for finite groups. Furthermore the group of homotopy classes of maps between classifying spectra is described, and in particular it is shown that a fusion system can be reconstructed from its classifying spectrum regarded as an object under the stable classifying space of the underlying p -group. 55R35; 20D20, 55P42 IntroductionSaturated fusion systems were introduced by Puig in [20; 21] as a formalization of fusion systems of groups. To a finite group G with Sylow p -subgroup S one associates a fusion system F S .G/ over S . This is the category whose objects are the subgroups of S , and whose morphisms are the conjugations induced by elements in G . Puig axiomatized this construction, thus allowing abstract fusion systems without requiring the presence, or indeed existence, of an ambient group G . He also identified important properties enjoyed by those fusion systems that are induced by groups. Puig called fusion systems with these properties full Frobenius systems. These definitions were later simplified by Broto-Levi-Oliver, who introduced the term saturated fusion systems in [7] (see Definition 1.3 below). A further simplification has been obtained by .A useful tool for the study of saturated fusion systems would be a functor assigning a classifying space to each saturated fusion system. Exactly what a classifying space means in this context is made precise by the theory of p -local finite groups developed by Broto-Levi-Oliver in [7]. They define a p -local finite group as a triple .S; F; L/, where S is a finite p -group, F is a saturated fusion system over S , and L is a centric linking system associated to F , a category which contains just enough information to construct a classifying space jLjp for F .The motivating example for the definition of a p -local finite group comes from finite groups. In [6], Broto-Levi-Oliver give an algebraic construction for a centric linking system L c S .G/ associated to the fusion system F S .G/ of a finite group G , and show that jL c S .G/jp ' BGp .Given the classifying space jLjp , one can by [7] reconstruct the fusion system via the following homotopy-theoretic construction:where Ã P and Ã Q are the inclusions of the subgroups P and Q in S , and Â is the natural "inclusion" BS ! jLjp . This construction was first applied by to show that if the p -completed classifying spaces of two finite groups have the same homotopy types, then their fusion systems are isomorphic.The passage from saturated fusion systems to classifying spaces is more problematic. In general it is not known whether a saturated f...

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“…There is some overlap between this note and Ragnarsson's results in [8]. The authors would like to thank him for helpful discussions during a week long visit to Aberdeen in 2008.…”

Section: Introductionmentioning

confidence: 95%