On the 2-compact group DI(4)

Notbohm, Dietrich

doi:10.1515/crll.2003.010

Cited by 12 publications

(14 citation statements)

References 26 publications

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“…By construction, the restriction of Bψ q to the maximal torus of BSol(q ∞ ) is the map induced by x → x q , and hence this is an "Adams map" as defined by Notbohm [18]. In fact, by [18, Theorem 3.5], there is an Adams map from BDI(4) to itself, unique up to homotopy, of degree any 2-adic unit.…”

Section: Bdi(4)mentioning

confidence: 99%

Construction of 2–local finite groups of a type studied by Solomon and Benson

2002

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A p-local finite group is an algebraic structure with a classifying space which has many of the properties of p-completed classifying spaces of finite groups. In this paper, we construct a family of 2-local finite groups, which are exotic in the following sense: they are based on certain fusion systems over the Sylow 2-subgroup of Spin 7 (q) (q an odd prime power) shown by Solomon not to occur as the 2-fusion in any actual finite group. Thus, the resulting classifying spaces are not homotopy equivalent to the 2-completed classifying space of any finite group. As predicted by Benson, these classifying spaces are also very closely related to the Dwyer-Wilkerson space BDI(4). AMS Classification numbers 918 Ran Levi and Bob OliverAs one step in the classification of finite simple groups, Ron Solomon [22] considered the problem of classifying all finite simple groups whose Sylow 2-subgroups are isomorphic to those of the Conway group Co 3 . The end result of his paper was that Co 3 is the only such group. In the process of proving this, he needed to consider groups G in which all involutions are conjugate, and such that for any involution x ∈ G, there are subgroups K H C G (x) such that K and C G (x)/H have odd order and H/K ∼ = Spin 7 (q) for some odd prime power q . Solomon showed that such a group G does not exist. The proof of this statement was also interesting, in the sense that the 2-local structure of the group in question appeared to be internally consistent, and it was only by analyzing its interaction with the p-local structure (where p is the prime of which q is a power) that he found a contradiction.In a later paper [3], Dave Benson, inspired by Solomon's work, constructed certain spaces which can be thought of as the 2-completed classifying spaces which the groups studied by Solomon would have if they existed. He started with the spaces BDI(4) constructed by Dwyer and Wilkerson having the property that(the rank four Dickson algebra at the prime 2). Benson then considered, for each odd prime power q , the homotopy fixed point set of the Z-action on BDI(4) generated by an "Adams operation" ψ q constructed by Dwyer and Wilkerson. This homotopy fixed point set is denoted here BDI 4 (q).In this paper, we construct a family of 2-local finite groups, in the sense of [6], which have the 2-local structure considered by Solomon, and whose classifying spaces are homotopy equivalent to Benson's spaces BDI 4 (q). The results of [6] combined with those here allow us to make much more precise the statement that these spaces have many of the properties which the 2-completed classifying spaces of the groups studied by Solomon would have had if they existed. To explain what this means, we first recall some definitions.A fusion system over a finite p-group S is a category whose objects are the subgroups of S , and whose morphisms are monomorphisms of groups which include all those induced by conjugation by elements of S . A fusion system is saturated if it satisfies certain axioms formulated by Puig [19], and als...

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Section: Bdi(4)mentioning

confidence: 99%

Construction of 2–local finite groups of a type studied by Solomon and Benson

2002

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“…We state these properties without proof because either they can be found in the papers [7], [12], [4], [11] or they follow from straightforward computations that are left to the reader.…”

Section: The 2-compact Group G 3 and Its Maximal Torusmentioning

confidence: 99%

The torsion index of a $p$-compact group

Aguadé

2010

Proc. Amer. Math. Soc.

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“…By [11,Proposition 3.3], the Adams operation Ψ q is an equivalence and therefore it induces an isomorphism of the algebra H * (BDI(4);…”

Section: The Bockstein Spectral Sequence For Bsol(q)mentioning

confidence: 99%

“…Soon after the experts became aware of the existence of self homotopy equivalences Ψ q : BDI(4) −→ BDI(4) for each odd prime power q, which are analogues of the unstable Adams operations on classifying spaces of compact Lie groups completed away from q. The first publish proof of this fact appeared in Notbohm's work [11]. Benson observed that Dwyer-Wilkerson's finite loop space DI(4) carries a 2-local structure closely related to Solomon's fusion systems.…”

Section: Introductionmentioning

confidence: 99%