Refined Bobtcheva-Messia Invariants of 4-Dimensional 2-Handlebodies

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 $(W,\omega)$, where $W$ is a $4$-dimensional $2$-handlebody, $\omega$ is a relative cohomology class in $H^2(W,\partial W;G)$, 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 $U = U_q \mathfrak{sl}_2$ at a root of unity $q$ of even order, and for its braided extension $\tilde{U} = \tilde{U}_q \mathfrak{sl}_2$, which fits in this framework for $G=\mathbb{Z}/2\mathbb{Z}$, 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 $\bar{U} = \bar{U}_q \mathfrak{sl}_2$ 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 $\omega=0$.