Homology decompositions and groups inducing fusion systems

Libman, Assaf; Seeliger, Nora

doi:10.4310/hha.2012.v14.n2.a10

Cited by 7 publications

(22 citation statements)

References 13 publications

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“…Moreover we construct an explicit signalizer functor for the Robinson and our new group model. This is a more direct and most important complete solution to the seventh problem of Oliver's list [2] in the sense that it provides an explicit formula than the second author's previous work with Libman [18]. We believe that independently from our context signalizer functor constructions are of great interest to group theorists.…”

Section: Introductionmentioning

confidence: 79%

“…A signalizer functor in the sense of Aschbacher-Chermak on a group model G is an assignment P → θ(P ) for every F −centric subgroup P ≤ S such that θ(P ) is a complement of Z(P ) in C G (P ) and such that if gP g −1 ≤ Q for g ∈ G then θ(Q) ≤ gθ(P )g −1 . A signalizer functor gives rise to a centric linking system if it exists [18]. In [18] Libman and the author show that all previously discussed group models have a signalizer functor however without an explicit construction.…”

Section: Group Models For Fusion Systemsmentioning

confidence: 99%

“…A signalizer functor gives rise to a centric linking system if it exists [18]. In [18] Libman and the author show that all previously discussed group models have a signalizer functor however without an explicit construction. This is one of the main results of this article.…”

Section: Group Models For Fusion Systemsmentioning

confidence: 99%

“…For constrained fusion systems there exists finite group models [5]. We review all general constructions for group models for fusion systems known so far [14], [29], [18]. The finite group G = S ≀ Σ e(X) constructed by Park [26] is not a group model since S in the general case is not a Sylow p−subgroup of G, where e(X) is a certain characteristic biset associated with F .…”

Section: Group Models For Fusion Systemsmentioning

confidence: 99%

“…The first conceptual treatment of group models for fusion systems are due to Leary-Stancu [14] and Robinson [29]. These groups realize the linking system as well induced by a signalizer functor as proved in [18] (where the signalizer functor is not constructed explicitely) however both constructions fail to have the right cohomology in the general case [31,Proposition 4.2] and Proposition 6.9 this article. In this note we construct a new group model for saturated fusion systems involving amalgams to generalize and overcome the shortcomings of previous constructions.…”

Section: Introductionmentioning

confidence: 99%

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Signalizer functors, existence, and applications to the fundamental group

Seeliger

2011

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We solve the seventh problem of Oliver's list [M. Aschbacher, R. Kessar, B. Oliver, Fusion systems in algebra and topology, LMS Lecture Note Series: 31, Cambridge University Press, 2011] via an explicit signalizer functor construction in the sense of Aschbacher-Chermak for various group models. Moreover we prove the existence of centric linking systems via group models in certain cases which is the first problem and give applications to the fundamental group which is the eighth problem of the list respectively. We illustrate with many examples.

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Section: Introductionmentioning

confidence: 79%

Section: Group Models For Fusion Systemsmentioning

confidence: 99%

Section: Group Models For Fusion Systemsmentioning

confidence: 99%

Section: Group Models For Fusion Systemsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Signalizer functors, existence, and applications to the fundamental group

Seeliger

2011

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Cohomology of infinite groups realizing fusion systems

Gündoğan

Yalcin

2019

J. Homotopy Relat. Struct.

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Automorphisms of p-local compact groups

Gonzalez¹,

Levi²

2016

Journal of Algebra

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Abstract. Self equivalences of classifying spaces of p-local compact groups are well understood by means of the algebraic structure that gives rise to them, but explicit descriptions are lacking. In this paper we use Robinson's construction of an amalgam G, realising a given fusion system, to produce a split epimorphism from the outer automorphism group of G to the group of homotopy classes of self homotopy equivalences of the classifying space of the corresponding p-local compact group.A p-local compact group is an algebraic object which is modelled on the p-local homotopy theory of classifying spaces of compact Lie groups and p-compact groups. These objects were constructed by Broto, Levi and Oliver in [BLO2,BLO3], and have been studied in an increasing number of papers since. Many aspects of the theory are still wide open, and among such aspects, a reasonable notion of maps between p-local compact groups remains quite elusive. Self equivalences of p-local compact groups are relatively well understood in terms of certain automorphisms of the underlying Sylow subgroup, but computing these groups of automorphisms is generally out of reach. This paper offers a way of regarding such automorphisms as being induced, in the appropriate sense, by automorphisms of certain discrete groups associated to that, in the finite case, were introduced by G. Robinson in [R]. The construction will be generalised here to the compact analog. We will refer to such groups as Robinson groups Before stating our results, we briefly recall the terminology, with a more detailed discussion in the next section. A p-local compact group is a triple G = (S, F , L), where S is a discrete p-toral group (to be defined later), and F and L are categories whose objects are certain subgroups of S. The classifying space of G, which we denote by BG is the p-completed nerve |L| ∧ p . For any space X, let Out(X) denote the group of homotopy classes of self homotopy equivalences of X. We are now ready to state our main result.Theorem A. Let G = (S, F , L) be a p-local compact group. Then there exist a group G, with a map µ : BG → BG, and a a split epimorphismFurthermore, the following statements hold:denote the subgroup of automorphisms which restrict to the identity on S, and let Out(G, 1 S ) denote its quotient by the subgroup of inner automorphisms induced by conjugation by elements of C G (S). Then Ker(ω) ∼ = Out(G, 1 S ) if p = 2, and for p = 2 there is a short exact sequence,Z/Z 0 → 1.

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Homology decompositions and groups inducing fusion systems

Cited by 7 publications

References 13 publications

Signalizer functors, existence, and applications to the fundamental group

Signalizer functors, existence, and applications to the fundamental group

Cohomology of infinite groups realizing fusion systems

Automorphisms of p-local compact groups

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