Some remarks on the Whitehead asphericity conjecture
type: Item
instance of: scholarly article
title: Some remarks on the Whitehead asphericity conjecture
zbMATH Open document ID: 0941.57007
published in: Illinois Journal of Mathematics
publication date: 1999-12-13T00:00:00Z
review text: The Whitehead asphericity conjecture claims that every subcomplex of an aspherical 2-complex (trivial second homotopy group) is aspherical; in group-theoretical terms, it states that any subpresentation (with the same set of generators) of an aspherical group presentation is aspherical (i.e. the canonical 2-complex associated to the group presentation is aspherical). Using ideas of small cancellation theory, it is proved in the present paper that if the Whitehead asphericity conjecture is false then there is a counterexample which arises by deleting from an aspherical presentation of the trivial group a single relator which is just one of the generators. Here the generating set is finite or countable; it can be chosen finite if there is a finite presentation giving a counterexample to the Whitehead asphericity conjecture. Also, it is proved that any finite aspherical presentation is Andrews-Curtis equivalent to a finite aspherical presentation with the property that also every subpresentation is aspherical; as a consequence, every finite aspherical 2-complex can be 3-deformed to a finite 2-complex all of whose subcomplexes are aspherical.
Mathematics Subject Classification ID: 57M20
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 20F06
Mathematics Subject Classification ID: 20F65
zbMATH DE Number: 1361058
zbMATH Keywords: aspherical 2-complex
zbMATH Keywords: aspherical group presentation
author: S. V. Ivanov
reviewed by: Bruno Zimmermann
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1361058