Algebraic Presentation of 4-Dimensional 2-Handlebodies and 3-Dimensional Cobordisms

2023/12/26 by Anna Beliakova, Ivelina Bobtcheva, Beliakova, Anna +5 · 1 citation
Mathematics · #16T05 #18C40 #18M15 #57K16 #57K40 #57R56 #57R65 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Geometric Topology (math.GT) #Homotopy and Cohomology in Algebraic Topology #Quantum Algebra (math.QA)

paper · pdf · doi:10.48550/arxiv.2312.15986

openalex publication_date 2023/12/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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 3Cob and 4HB, whose morphisms are manifolds of dimension 3 and 4, respectively. More precisely, 3Cob is the category of connected oriented 3-dimensional cobordisms between connected surfaces with connected boundary, while 4HB 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: 4Alg → 4HB, where 4Alg denotes the free monoidal category generated by a Bobtcheva--Piergallini Hopf algebra. As an application, we deduce an algebraic presentation of 3Cob and show that it is equivalent to the one conjectured by Habiro.

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• Representation theory of non-factorizable ribbon Hopf algebras

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