Expression de la différentielle d₃ de la suite spectrale de Hochschild-Serre cohomologie Bornée Réelle

Abstract

For discrete groups, we construct two bounded cohomology classes with coefficients in the second space of the reduced real ℓ₁-homology. Precisely, we associate to any discrete group G a bounded cohomology class of degree two noted g₂ ∊ H_b^² (G,(H_₂^(l₁) ) ̅ (G,ℝ)). For G and Π groups and θ : Π → Out(C) any homomorphism we associate a bounded cohomology class of degree three noted [θ] ∊ H_b^³ (Π, ,(H_₂^(l₁) ) ̅ (G,ℝ)). When the outer homomorphism θ : Π →Out(C) induces an extension of G by Π we show that the class g₂ is Π-invariant and that the differential d3 of Hochschild-Serre spectral sequence sends the class g₂ on the class [θ] :d₃(g₂) = [θ]. Moreover, we show that for any integer n ≥ 0 the differential d₃ : E_₃^(n,2)→ E_₃^(n+3,0) of Hochschild-Serre spectral sequence in real bounded cohomology is given as a cup-product by the class [θ].