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 $(W,\omega )$, 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 $\omega =0$.