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

TL;DR

This paper refines invariants of 4-dimensional 2-handlebodies using algebraic structures like ribbon Hopf G-coalgebras, relating them to quantum groups and decomposing original invariants into refined components.

Contribution

It introduces refined invariants of 4D 2-handlebodies incorporating cohomology classes, extending previous invariants with algebraic and topological insights.

Findings

Decomposition formulas relate original and refined invariants.

Refined invariants incorporate cohomology classes and quantum group structures.

Identification of non-refined invariants with refined ones for specific quantum groups.

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 $\omega=0$.