Event Detail

Event Type: 
Friday, August 28, 2020 - 15:00 to 17:00
Zoom - If you are interested in attending this presentation, please send an email to Nikki Sullivan - nikki.sullivan@oregonstate.edu - to request Zoom log in details.

In 1941, J.H.C. Whitehead posed the question of whether asphericity is a hereditary property for 2-dimensional CW complexes. This question remains unanswered, but has led to the development of several algebraic and topological properties that are sufficient (but not necessary) for the asphericity of presentation 2-complexes. While many of the logical relationships between these flavors of asphericity are known, there remain a few to be answered. In particular, it has long been known that Cohen-Lyndon aspherical (CLA) complexes are not necessarily diagrammatically reducible (DR), but the existence of a (DR) complex which is not (CLA) remains open. We resolve this by giving a presentation that is (DR) but not (CLA). Moreover, we produce an example for any non-trivial group which admits a (DR) presentation. To produce this counterexample, we first show that the inclusion of a subcomplex in a (CLA) presentation must have free kernel. Then, we produce a (DR) presentation, for which the inclusion-kernel is not residually nilpotent, hence not free, hence the presentation is not (CLA). This recovers a result of Biskup and resolves that status of all possible implications between the major flavors of asphericity.