Unitary embeddings of finite loop spaces

Abstract

Abstract. In this paper we construct faithful representations of saturated fusion systems over discrete p-toral groups and use them to find conditions that guarantee the existence of unitary embeddings of p-local compact groups. These conditions hold for the Clark-Ewing and Aguadé-Zabrodsky p-compact groups as well as some exotic 3-local compact groups. We also show the existence of unitary embeddings of finite loop spaces. IntroductionIn the theory of compact Lie groups, the existence of a faithful unitary representation for every compact Lie group is a consequence of the Peter-Weyl theorem. This paper is concerned with the existence of analogous representations for several objects in the literature which are considered to be homotopical counterparts of compact Lie groups.In 1994, W.G. Dwyer and C.W. Wilkerson [17] introduced p-compact groups. They are loop spaces which satisfy some finiteness properties at a particular prime p. For example, if G is a compact Lie group such that its group of connected components is a finite p-group, then its p-completion G , where a bijective correspondence between connected p-compact groups and reflection data over the p-adic integers was established.Many ideas from the theory of compact Lie groups have a homotopical analogue for p-compact groups. Faithful unitary representations correspond to homotopy monomorphisms at the prime p from the classifying space BX of a pcompact group into BU (n) ∧ p for some n. A homotopy monomorphism at p is map g such that the homotopy fiber F of g ∧ p is BZ/p-null, that is, the evaluation map Map(BZ/p, F ) → F is a homotopy equivalence. For simplicity, we will call such maps BX → BU (n) In this article we deal with the same question for the combinatorial structures called p-local compact groups, which encode the p-local information of some spaces at a prime p. They were introduced in [9] to model p-completed classifying spaces of compact Lie groups, p-compact groups, as well as linear torsion groups, and they 2010 Mathematics Subject Classification. 55R35, (primary), 20D20, 20C20 (secondary).

Vector bundles over classifying spaces of p‐local finite groups and Benson–Carlson duality

“…When constructing fusion‐preserving representations, it may be convenient to restrict to the orbit category and to the family of centric or centric radical subgroups in (see also Cantarero–Castellana [, Remark 3.2]). Proposition Let be a saturated fusion system over .…”

“…In order to prove Theorem , we recall the notion of homotopy monomorphism at the prime from Cantarero–Castellana . Definition A connected pointed space is ‐null if the pointed mapping space is contractible for any choice of basepoint in .…”

“…Proof By Theorem , there is a multiple of the regular representation of in the image of some . A preimage of this representation must be a homotopy monomorphism by Cantarero–Castellana [, Theorem 2.5]. Consider the map induced by the standard inclusion of in .…”

“…Consider the map induced by the standard inclusion of in . The homotopy fibre of this map is , which is ‐finite, so it is a homotopy monomorphism by [, Proposition 2.2]. Moreover, since is connected, [, Lemma 2.4(c)] implies that the composition of these two maps is a homotopy monomorphism.…”

“…Homotopy monomorphisms at the prime were also studied in by the first two authors of this article, where they are called unitary embeddings. Note that has some overlap with this paper, but in this case the Sylow subgroup is finite and that allows us to show the existence of unitary embeddings of ‐local finite groups. Moreover, in Section we prove the following improvement.…”