Homotopy monomorphisms and H-splitting in loop space fibrations

Let F → i E → p B be a fibration, ρ : F × Ω B → F the holonomy action of this fibration and ∂ : Ω B → F the connecting map. It is shown that if the fibre F admits an H-structure ν such that ρ ≃ ν ○ ( 1 × ∂ ) (principal fibrations of all kinds satisfy such a condition), then i is a monomorphism if and only if it is weak monomorphism, the latter is equivalent to that Ωp has a homotopy right inverse Γ. If in addition Γ is an H-map, then ΩE has the same H-type as Ω B × Ω F .