Algebraic characterisation of the category of cobordisms of 2-dimensional CW-complexes and the Andrews-Curtis conjecture

TL;DR

This paper establishes an algebraic framework for 2-dimensional cobordisms using Hopf algebras, revealing deep structural equivalences and connections to the Andrews-Curtis conjecture in topology.

Contribution

It characterizes the category of 2-dimensional cobordisms algebraically via Hopf algebras and shows its equivalence to a free symmetric monoidal category, linking topology and algebra.

Findings

$S^1$ is a unimodular, cocommutative Hopf algebra in $CW^{1+1}$

The category $CW^{1+1}$ is equivalent to a free symmetric monoidal category generated by such a Hopf algebra

The algebraic structure of 4-dimensional cobordisms refines the structure of their spines.

Abstract

We prove that $S^1$ is a unimodular, cocommutative Hopf algebra in the category $CW^{1+1}$ of 2-equivalence classes of cobordisms of 2-dimensional CW-complexes and that $CW^{1+1}$ is actually equivalent to the symmetric monoidal category freely generated by such Hopf algebra. Moreover, we show that the algebraic structure the category ${\cal C}hb^{3+1}$ of cobordisms of orientable relative 4-dimensional 2-handlebody cobordisms up to 2-deformations described in \cite{BP12}, is a refinement of the algebraic structure of their spines.