Homotopy Classification of Self-Maps of BG via G-actions

Jackowski, Stefan; McClure, James E.; Oliver, Bob

doi:10.2307/2946568

Cited by 88 publications

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“…Then (10). Also, by (11), q( x) = q( y), so ε S ( x) and ε S ( y) are liftings of the same morphism ε S,S (q( x)). Thus ε S ( x) = ε S ( y) by Lemma 5.2(a).…”

Section: Lemma 53 For Each P Q ∈ Ob( T ) and Each X ∈ N Smentioning

confidence: 99%

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Extensions of linking systems with p-group kernel

Oliver

Ventura

2007

Math. Ann.

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Abstract. We study extensions of p-local nite groups where the kernel is a pgroup. In particular, we construct examples of saturated fusion systems F which do not come from nite groups, but which have normal p-subgroups A C F such that F=A is the fusion system of a nite group. One of the tools used to do this is the concept of a \transporter system", which is modelled on the transporter category of a nite group, and is more general than a linking system.vet G e nite groupD with ylow pEsugroup S P yl p @GAF he fusion system of G @t pA is the tegory p S @GA whose ojets re the sugroups of GD nd where wor p S @GA @P; QA is the set of monomorphisms from P to Q indued y onjugtion y elements of GF he transporter system of G t p is the tegory S @GA with the sme ojets s p S @GAD nd with morphism sets wor S @GA @P; QA a N G @P; QAX the set of elements of G whih onjugte P into QF e sugroup P S is lled pEentri in G ifG @P A of order prime to pY nd the centric linking system of G is the tegory v c S @GA whose ojets re the sugroups of S whih re pEentri in GD nd where wor v c S @GA @P; QA a N G @P; QA=C H G @P AF ell of these denitions re repeted in more detil t the eginning of etion IF sn severl ppersD suh s fvyI nd yPD the fusion nd linking systems of G re shown to ply entrl role in desriing homotopy theoreti properties of the pEompleted lssifying spe BG p F estrt fusion nd linking systems hve lso een dened nd studiedD nd re shown in fvyP to hve mny of the sme properties s the fusion nd linking systems of nite groupsF e p-local nite group is dened to e triple @S; p; vAD where S is nite pEgroupD p is saturated fusion system over S @henitions IFP nd IFQAD nd v is centric linking system ssoited to p @henition IFTAF xorml nd entrl pEsugroups of fusion systems nd linking systems re lso dened @henition IFRAF gertin types of extensions of pElol nite groupsD nd in prtiulr entrl extenE sionsD were studied in fgqvyPF yne hope ws tht extensions ould provide new wy to onstrut exoti exmplesF fut in the se of entrl extensionsD this ws shown to e impossileF fy fgqvyPD heorem TFIQ nd gorollry TFIRD p=A; e v=AAF yne prolem when doing this is tht the fusion system p=A ontins too little informtionX p nnot e desried s n extension of p=A y A in ny senseF

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“…Then (10). Also, by (11), q( x) = q( y), so ε S ( x) and ε S ( y) are liftings of the same morphism ε S,S (q( x)). Thus ε S ( x) = ε S ( y) by Lemma 5.2(a).…”

Section: Lemma 53 For Each P Q ∈ Ob( T ) and Each X ∈ N Smentioning

confidence: 99%

“…can now be combined with results in[11] about the functors Λ * (−; −), to obtain the following corollary.…”

mentioning

confidence: 99%

Extensions of linking systems with p-group kernel

Oliver

Ventura

2007

Math. Ann.

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“…Then [y] imply y ∈ D, then D satisfies the hypothesis of the proposition. We may consider a chain of such subcategories ∅ = D 0 ⊂ · · · ⊂ D n = C in which each subcategory has one more isomorphism class of objects than the previous one, so that n is the number of isomorphism classes in C, and from this we obtain a stratification of kC of length n. For each kC-module M there is a corresponding filtration in which each factor is zero except on one isomorphism class of objects, where it is the value of M. This is exactly the filtration which has been used in the calculation of higher limits in [15] and [22].…”

Section: Some Ei Category Algebras Are Standardly Stratifiedmentioning

confidence: 98%

“…We do this in Section 2. It turns out that representations of the orbit category of a finite group with respect to a family of subgroups are always standardly stratified, and the stratification is related to the procedure used to compute the derived functors of the limit functor in [15] and [22].…”

Section: Introductionmentioning

confidence: 99%

Standard stratifications of EI categories and Alperin's weight conjecture

Webb

2008

Journal of Algebra

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We characterize the finite EI categories whose representations are standardly stratified with respect to the natural preorder on the simple representations. The orbit category of a finite group with respect to any set of subgroups is always such a category. Taking the subgroups to be the p-subgroups of the group, we reformulate Alperin's weight conjecture in terms of the standard and proper costandard representations of the orbit category. We do this using the properties of the Ringel dual construction and a theorem of Dlab, which have elsewhere been described for standardly stratified algebras where there is a partial order on the simple modules. We indicate that these results hold in the generality of an algebra whose simple modules are preordered, rather than partially ordered.

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“…The main result concerning this category is that the following map Proof. Using the mod p decomposition of unitary groups via p-stubborn subgroups [16] we obtain a decomposition of the mapping space as a homotopy inverse limit,…”

Section: Non-modular Homotopy Fixed Points On Generalized Grassmanniansmentioning

confidence: 99%

Quillen Grassmannians as non-modular homotopy fixed points

Castellana

2004

Math. Z.

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The aim of this paper is to compute the homotopy fixed points of the homotopy action described by unstable Adams operations on the classifying spaces of unitary groups. The same technique can be applied to compute homotopy fixed points of the action of certain automorphisms on p-compact groups called generalized Grassmannians. We use the description of Quillen Grassmannians to describe the set of homotopy representations of elementary abelian p-groups into them. (2000): 55R35, 55R40, 20D20 Mathematics Subject Classification

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Homotopy Classification of Self-Maps of BG via G-actions

Cited by 88 publications

References 0 publications

Extensions of linking systems with p-group kernel

Extensions of linking systems with p-group kernel

Standard stratifications of EI categories and Alperin's weight conjecture

Quillen Grassmannians as non-modular homotopy fixed points

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