On the Logical Independence of (DR) and (CLA)

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 showing that the presentation 2-complex associated to the presentation is (DR) but not (CLA). In fact, we prove that if a nontrivial group G occurs as the fundamental group of a (DR) 2-complex, then there is a (DR) 2-complex with fundamental group $G$ that is not (CLA)