Kerler-Lyubashenko functors on 4-dimensional 2-handlebodies
This paper constructs linear representations of a certain topological category whose morphisms are 2-deformation classes of 4-dimensional 2-handlebodies. More precisely, the authors define a braided monoidal functor \(J_{4}\) with source the category 4HB of connected 4-dimensional 2-handlebodies of \textit{I. Bobtcheva} and \textit{R. Piergallini} [J. Knot Theory Ramifications 21, No. 12, 1250110, 230 p. (2012; Zbl 1276.57010)] and with target a unimodular ribbon category \(\mathcal{C}\).\N\NThe synopsis of the paper goes as follows.\N\N\begin{itemize}\N\item[\S 2] recalls some basic facts about handle decompositions and signature defects.\N\N\item[\S 3] recalls the definition of the three topological categories focused on in this paper, namely, \textit{I. Bobtcheva} and \textit{R. Piergallini} [J. Knot Theory Ramifications 21, No. 12, 1250110, 230 p. (2012; Zbl 1276.57010)] category 4HB of connected 4-dimensional 2-handlebodies, and \textit{T. Kerler} and \textit{V. V. Lyubashenko}'s [Non-semisimple topological quantum field theories for 3-manifolds with corners. Berlin: Springer (2001; Zbl 0982.57013)] categories 3Cob\(^{\sigma}\) and 3Cob of connected (framed) 3-dimensional cobordisms.\N\N\item[\S 4] gives a diagrammatic description of the three topological categories of the previous section by first defining the categories of Kirby tangles KTan and of (signed) top tangles in handlebodies TTan\(^{\sigma}\) and TTan, then defining functors between them, and finally showing that the following diagram commutes:\N\[\N\begin{array} [c]{ccccc} \text{KTan} & \overset{E}\longrightarrow & \text{TTan}^{\sigma} & \overset{F_{T}}\longrightarrow & \text{TTan}\\\N^{D}\downarrow\cong & & ^{\chi^{\sigma}}\downarrow\cong & & ^{\chi}\downarrow\cong\\\N\text{4HB} & \underset{\partial}\longrightarrow & \text{3Cob}^{\sigma} & \underset{F_{C}}\longrightarrow & \text{3Cob} \end{array}\N\]\N\N\item[\S 5] summarizes the terminology needed in order to understand the target of the braided monoidal functor of the theorem stated in \S 7.\N\N\item[\S 6] recalls the algebraic presentation of the topological and diagrammatic categories of \S 3 and \S 4 by defining three categories 4Alg, 3Alg\(^{\sigma}\), and 3Alg freely generated by BPH algebras, and by extending the above commutative diagram.\N\[\N\begin{array} [c]{ccccc} \text{4Alg} & \overset{Q^{\sigma}}\longrightarrow & \text{3Alg}^{\sigma} & \overset{Q}\longrightarrow & \text{3Alg}\\\N^{K}\downarrow\cong & & ^{T^{\sigma}}\downarrow\cong & & ^T\downarrow\cong\\\N\text{KTan} & \overset{E}\longrightarrow & \text{TTan}^{\sigma} & \overset{F_{T}}\longrightarrow & \text{TTan}\\\N^{D}\downarrow\cong & & ^{\chi^{\sigma}}\downarrow\cong & & ^{\chi}\downarrow\cong\\\N\text{4HB} & \underset{\partial}\longrightarrow & \text{3Cob}^{\sigma} & \underset{F_{C}}\longrightarrow & \text{3Cob} \end{array}\N\]\N\N\item[\S 7] establishes the following main theorem:\N\NTheorem. If \(\mathcal{C}\) is a unimodular ribbon category, then there exists a braided monoidal functor\N\[\NJ_{4}:\text{4HB}\rightarrow\mathcal{C}\N\]\N\ sending the generating Hopf algebra object 4HB\ (the solid torus) to the end\N\[\N\mathcal{E}=\int_{X\in\mathcal{C}}X\otimes X^{\ast}\in\mathcal{C}\N\]\N\N\item[\S 8] presents an algorithm for the computation of the functor\N\[\NJ_{4}:\text{KTan}\rightarrow H\text{-mod}\N\]\Nassociated with a unimodular ribbon Hopf algebra \(H\), and of the functor\N\[\NJ_{3}^{\sigma}:\text{TTan}^{\sigma}\rightarrow H\text{-mod}\N\]\Nassociated with a factorizable ribbon Hopf algebra \(H\).\N\N\item[\S 9] provides concrete examples of unimodular ribbon categories, exhibiting explicit formulas required for computations.\N\N\item[Appendix A] proves that a unimodular ribbon category \ is modular iff the copairing \(w_{+}:\boldsymbol{1}\rightarrow\mathcal{E}\otimes\mathcal{E} \) of the end is nondegenerate.\N\N\item[Appendix B] gives explicit formulas for the ribbon element \(v_{+}\in u_{q}\mathfrak{sl}_{2}\) and for the monodromy matrix \(M\in u_{q} \mathfrak{sl}_{2}\otimes u_{q}\mathfrak{sl}_{2}\), as well as their inverses.\N\N\item[Appendix C] gives a short explicit proof of the instability of (potentially) 2-exotic phenomenon under boundary connected sum with \(S^{2}\times D^{2}\).\N\end{itemize}
scientific article; zbMATH DE number 7935852