Maps between classifying spaces

Fibrations of classifying spaces

“…The composite map /' • r\BT~ :BT~^ BG is of form Bp for a homomorphism G t/ p : F~ -v G [16]. We can show that the homotopy fibre of the map r\Bp(T~) -* X is a retract of the finite CW-complex G/T~ and hence homotopy equivalent to a finite CW-complex.…”

Classifying spaces and homotopy sets of axes of pairings

“…As shown in §1, there is a strong relationship between the set of the homotopy classes of axes f ⊥ (BK, BG) and map (BL, BG) (G). A result of Jackowski-McClure-Oliver [9] and Notbohm [15] shows that if f : BG → BG is a self-equivalence, the map BZ(G) → map(BG, BG) f is a mod p equivalence for any prime p. A related result can be found in [4].…”

“…The following is a "BG"-analog at a prime p. If a map α : BK → BG is induced by a homomorphism, let C G (α) denote the centralizer of the homomorphism. For a p-toral group K (a group extension of a torus by a finite p-group), it is known, [5] and [14], that any map α : BK → BG (at p) has the form α = Bη (α = (Bη) ∧ p ) for some homomorphism η. Let BG ∧ p denote the p-completion of BG.…”

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

“…Suppose P is a maximal /7-toral subgroup of G, [10]. The //-structure on (BG) p induces a group homomorphism P x P -» P which makes BP an H-space, [5] and [16].…”

“…Let NP be the normalizer of P in G and let W=NP/P. Since the maximal /7-toral subgroup P is abelian, the mod p cohomology H* ((BG) Notice [5] and [16] that map(BP,(BNP)*) m~B P, since the classifying space of the centralizer of P in NP = P X! W is /^-equivalent to BP.…”