Résumé
In this paper we refine our recently constructed invariants of 4-dimensional 2-handlebodies up to 2-deformations. More precisely, we define invariants of pairs of the form (W,ω), where W is a 4-dimensional 2-handlebody, ω is a relative cohomology class in H2(W,∂W;G), and G is an abelian group. The algebraic input required for this construction is a unimodular ribbon Hopf G-coalgebra. We study these refined invariants for the restricted quantum group U=Uqsl2 at a root of unity q of even order, and for its braided extension U~=U~qsl2, which fits in this framework for G=Z/2Z, and we relate them to our original invariant. We deduce decomposition formulas for the original invariants in terms of the refined ones, generalizing splittings of the Witten-Reshetikhin-Turaev invariants with respect to spin structures and cohomology classes. Moreover, we identify our non-refined invariant associated with the small quantum group U¯=U¯qsl2 at a root of unity q whose order is divisible by 4 with the refined one associated with the restricted quantum group U for the trivial cohomology class ω=0.