Fibrations of classifying spaces

Abstract

Abstract. We investigate fibrations of the form Z -» Y -► X, where two of the three spaces are classifying spaces of compact connected Lie groups. We obtain certain finiteness conditions on the third space which make it also a classifying space. Our results allow to express some of the basic notions in group theory in terms of homotopy theory, i.e., in terms of classifying spaces. As an application we prove that every retract of the classifying space of a compact connected Lie group is again a classifying space.

Classifying spaces and a subgroup of the exceptional Lie group 𝐺₂

“…We will prove Theorem 1 and Theorem 2 in this section. Our proof of Theorem 1 uses a result of [15]. This result implies that there is no fibration of the type BZ/3 ¹ (BG 2 ) ∧ 3 ¹ Y , despite the existence of a map f : (BG 2 ) ∧…”

“…The result [15,Theorem 4] shows that this is a contradiction, since the center of the exceptional Lie group G 2 is trivial.…”

Pairings of $p$-Compact Groups and $H$-Structures on the Classifying Spaces of Finite Loop Spaces

“…In [9] Here we make a remark analogous to the one in [8]. Taking Y=Z and f=id, our problem asks possible BX-actions on BY.…”

Classifying spaces and homotopy sets of axes of pairings

“…The main theorem deals with the case that the map f : BL → BG is a weak epimorphism studied in [8]. (We recall the definition in §2.)…”