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Pré-Publication, Document De Travail Année : 2023

On Algebraization in Low-Dimensional Topology

Résumé

In this paper, we give a new direct proof of a result by Bobtcheva and Piergallini that provides finite algebraic presentations of two categories, denoted $3\mathrm{Cob}$ and $4\mathrm{HB}$, whose morphisms are manifolds of dimension $3$ and $4$, respectively. More precisely, $3\mathrm{Cob}$ is the category of connected oriented $3$-dimensional cobordisms between connected surfaces with connected boundary, while $4\mathrm{HB}$ is the category of connected oriented $4$-dimensional $2$-handlebodies up to $2$-deformations. For this purpose, we explicitly construct the inverse of the functor $Φ: 4\mathrm{Alg} \to 4\mathrm{HB}$, where $4\mathrm{Alg}$ denotes the free monoidal category generated by a Bobtcheva--Piergallini Hopf algebra. As an application, we deduce an algebraic presentation of $3\mathrm{Cob}$ and show that it is equivalent to the one conjectured by Habiro.

Dates et versions

hal-04504357 , version 1 (14-03-2024)

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Anna Beliakova, Ivelina Bobtcheva, Marco de Renzi, Riccardo Piergallini. On Algebraization in Low-Dimensional Topology. 2024. ⟨hal-04504357⟩
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