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

Abstract

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 $H^{2}(W,∂W;G)$, 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 $U=U_{q}sl_{2}$ at a root of unity $q$ of even order, and for its braided extension $\widetilde{U}=\widetilde{U}_{q}sl_{2}$, 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 $\bar{U}=\bar{U}_{q}sl_{2}$ 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 $ω=0$.