Classifying spaces and a subgroup of the exceptional Lie group 𝐺₂

Abstract

We consider a problem on the conditions of a compact Lie group that its loop space of the p-completed classifying space be a p-compact group, as well as some related problems. A previously obtained necessary condition is shown to be not sufficient. Our counterexample is given by a quotient group of a subgroup of the exceptional Lie group G 2 at p = 3. The K-theory of the space is isomorphic to K(BG 2 ; Z ∧ 3), though its loop space is not a 3-compact group.

Classifying spaces of compact Lie groups that are p–compact for all prime numbers

“…In a previous article [19], the classifying space BG is said to be p-compact if Ω(BG) ∧ p is a p-compact group. There are some results for a special case.…”

“…We summarize work of earlier articles [19,20] together with some basic results, in order to introduce the problem of p-compactness. For a compact Lie group G, the classifying space BG is p-compact if and only if Ω(BG) ∧ p is F p -finite.…”

“…For K = NT , the normalizer of a maximal torus T of a connected compact simple Lie group, the converse of Lemma 3.2 is true, though it doesn't hold in general. Note that π 0 Γ 2 = Z/2 and that BΓ 2 is not 3-compact [19]. Any proper closed subgroup of G which includes the normalizer NT satisfies the assumption of Theorem 2.…”

“…We recall that the p-completion of a compact Lie group G is a p-compact group if π 0 (G) is a p-group. Next, if C(ρ) denotes the centralizer of a group homomorphism ρ from a p-toral group to a compact Lie group, according to [8, Theorem 6.1], the loop space of the p-completion Ω(BC(ρ)) ∧ p is a p-compact group.In a previous article [19], the classifying space BG is said to be p-compact if Ω(BG) ∧ p…”