The Eilenberg-Ganea conjecture

Theorem 1.1: there is a finitely generated residually finite group $G$ with $\mathrm{cd}\,G=2$ and $\mathrm{gd}\,G=3$; no two-dimensional $K(G,1)$ exists regardless of the number of cells. $G$ is a Bestvina-Brady group (the kernel of the height map of a right-angled Artin group); it is of type FL and FP$_\infty$ but not finitely presented. The obstruction compares an acyclic two-dimensional level complex with any hypothetical aspherical presentation, using small SU(2) labelings and a degree argument. It does not give a finitely presented counterexample, and by itself it does not decide Whitehead's asphericity conjecture.

Theorem 1.1: there is a finitely generated residually finite group $G$ with $\mathrm{cd}\,G=2$ and $\mathrm{gd}\,G=3$; no two-dimensional $K(G,1)$ exists regardless of the number of cells. $G$ is a Bestvina-Brady group (the kernel of the height map of a right-angled Artin group); it is of type FL and FP$_\infty$ but not finitely presented. The obstruction compares an acyclic two-dimensional level complex with any hypothetical aspherical presentation, using small SU(2) labelings and a degree argument. It does not give a finitely presented counterexample, and by itself it does not decide Whitehead's asphericity conjecture.