A New Finite Loop Space at the Prime Two

Dwyer, W. G.; Wilkerson, Clarence

doi:10.2307/2152794

Cited by 33 publications

(59 citation statements)

References 11 publications

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“…The result as stated above is much more elementary (and also implicit in [9]), so we sketch the proof here.…”

Section: − − − − − − → X Bdi(4)mentioning

confidence: 82%

“…We now want to examine the relation between the spaces BSol(q) which we have just constructed, and the space BDI(4) constructed by Dwyer and Wilkerson in [9]. Recall that this is a 2-complete space characterized by the property that its cohomology is the Dickson algebra in four variables over F 2 ; ie, the ring of invariants (2) .…”

Section: Relation With the Dwyer-wilkerson Spacementioning

confidence: 99%

“…Here, η 0 is the homotopy equivalence of [13], induced by some fixed choice of embedding of the Witt vectors for F q into C, while δ(q ∞ ) is the union of the inclusions |L cc [9].…”

Section: − − − − − − − → Bdi(4)mentioning

confidence: 99%

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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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“…The result as stated above is much more elementary (and also implicit in [9]), so we sketch the proof here.…”

Section: − − − − − − → X Bdi(4)mentioning

confidence: 82%

Section: Relation With the Dwyer-wilkerson Spacementioning

confidence: 99%

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

2002

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“…It is the "classifying space" BX that carries all of the information about the loop space. Amazingly enough, apart from compact Lie groups, there are only a few families of exotic p-compact groups and they have been recently completely classified by Andersen, Grodal, Møller, and Viruel: see [3] for the odd prime case and [2], [29], [30] for the prime 2 (the only exotic 2-compact group is basically the space DI (4) constructed by Dwyer and Wilkerson [16]). …”

Section: Introductionmentioning

confidence: 99%

Noetherian loop spaces

Castellana¹,

Crespo²,

Scherer³

2011

J. Eur. Math. Soc.

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Abstract. The class of loop spaces of which the mod p cohomology is Noetherian is much larger than the class of p-compact groups (for which the mod p cohomology is required to be finite). It contains Eilenberg-Mac Lane spaces such as CP ∞ and 3-connected covers of compact Lie groups. We study the cohomology of the classifying space BX of such an object and prove it is as small as expected, that is, comparable to that of BCP ∞ . We also show that BX differs basically from the classifying space of a p-compact group in a single homotopy group. This applies in particular to 4-connected covers of classifying spaces of compact Lie groups and sheds new light on how the cohomology of such an object looks like.

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“…The remaining important small prime cases were constructed by Zabrodsky (G 12 , p = 3) [46], Dwyer-Wilkerson (G 24 , p = 2) [12], and Notbohm-Oliver (G(·, ·, ·), p small) [31]. Since G 24 and G 12 are the only finite simple Q p -reflection groups which do not come from compact Lie groups, for p = 2 and 3 respectively, any counterexample will have to involve these groups.…”

Section: Introductionmentioning

confidence: 99%

A finite loop space not rationally equivalent to a compact Lie group

et al. 2004

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Abstract. We construct a connected finite loop space of rank 66 and dimension 1254 whose rational cohomology is not isomorphic as a graded vector space to the rational cohomology of any compact Lie group, hence providing a counterexample to a classical conjecture. Aided by machine calculation we verify that our counterexample is minimal, i.e., that any finite loop space of rank less than 66 is in fact rationally equivalent to a compact Lie group, extending the classical known bound of 5.

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A New Finite Loop Space at the Prime Two

Cited by 33 publications

References 11 publications

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

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

Noetherian loop spaces

A finite loop space not rationally equivalent to a compact Lie group

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