Logical Distinction Between Diagrammatic and Cohen–Lyndon Asphericity

Abstract

We resolve the status of the implication CLA ⇒ DA in the negative by showing that the two-dimensional model of the group presentation (a, b : a, b-1 aba-1 b-1) is DA but not CLA. This settles a question that has been addressed in [2, 7, 10, 11]. In 1941, Whitehead posed the question whether asphericity is a hereditary property for two-dimensional CW complexes. This question remains unanswered. Out of its study developed the formulation of several combinatorial properties for group presentations that are sufficient (but not necessary) for asphericity of the associated two-dimensional model. The logical relationships between these flavors of asphericity are just partially understood. In this article we show that two of these flavors of asphericity are in fact distinct. As a consequence, all of the known flavors are distinct. An argument of Lyndon and Schupp [5, III Property 10.6] shows that if a two-dimensional CW complex K is Cohen–Lyndon aspherical (CLA), then K is also diagrammatically aspherical (DA). The status of the reverse implication had been open prior to this writing. We will present an alternative proof for the implication CLA ⇒ DA and demonstrate that every spherical picture over the presentation (a, b : a, b-2 aba-1) can be reduced without insertions of dipoles, thus concluding that the presentation is DA. A straightforward argument shows that the presentation cannot be CLA. Our main tool is the theory of spherical pictures and picture moves of which we will give a short survey.