Kerler-Lyubashenko functors on 4-dimensional 2-handlebodies
type: Item
instance of: scholarly article
title: Kerler-Lyubashenko functors on 4-dimensional 2-handlebodies
zbMATH Open document ID: 1548.57041
author: Anna Beliakova
author: Marco De Renzi
published in: IMRN. International Mathematics Research Notices
publication date: 2024-10-30T00:00:00Z
review text: 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 …
reviewed by: Hirokazu Nishimura
Mathematics Subject Classification ID: 57K40
Mathematics Subject Classification ID: 57R65
zbMATH DE Number: 7935852
MaRDI profile type: Publication
DOI: 10.1093/IMRN/RNAC039
description: scientific article; zbMATH DE number 7935852