Refined Bobtcheva-Messia invariants of 4-dimensional 2-handlebodies (Q6150782)

Summary: 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 \(\tilde{U} = \tilde{U}_q \mathfrak{sl}_2\) 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 \(\omega=0\). For the entire collection see [Zbl 1519.57002].