A semigroup theoretic approach to Whitehead's asphericity question

TL;DR

This paper introduces a novel semigroup-theoretic approach to the Whitehead asphericity problem, providing a new combinatorial characterization and conditions for subpresentations of aspherical group presentations.

Contribution

It applies semigroup theory to formulate a new characterization of asphericity and establishes necessary and sufficient conditions for subpresentations to be aspherical.

Findings

New combinatorial characterization of asphericity using weak dominion of submonoids

Provides necessary and sufficient conditions for subpresentations to be aspherical

Bridges combinatorial and topological approaches to the Whitehead problem

Abstract

The Whitehead asphericity problem, regarded as a problem of combinatorial group theory, asks whether any subpresentation of an aspherical group presentation is also aspherical. This is a long standing open problem which has attracted a lot of attention. Related to it, throughout the years there have been given several useful characterizations of asphericity which are either combinatorial or topological in nature. The aim of this paper is two fold. First, it brings in methods from semigroup theory to give a new combinatorial characterization of asphericity in terms of what we define here to be the weak dominion of a submonoid of a monoid, and uses this to give a sufficient and necessary condition under which a subpresentation of an aspherical group presentation is aspherical.