Eilenberg–Ganea Conjecture

A torsion-free group $G$ has cohomological dimension $\operatorname{cd}(G)$ defined by projective resolutions of the trivial $\mathbb ZG$-module, and geometric dimension $\operatorname{gd}(G)$ defined by the smallest dimension of a classifying space $K(G,1)$. Eilenberg and Ganea proved that these dimensions agree except possibly when $\operatorname{cd}(G)=2$, where their construction yields only $\operatorname{gd}(G)\le3$ [EilenbergGanea1957Category]; Brown gives the standard group-cohomological framework [Brown1982CohomologyGroups].

Bestvina and Brady exposed a sharp link with Whitehead asphericity: for a specific group of cohomological dimension two, either its geometric dimension is three, giving an Eilenberg–Ganea counterexample, or there is a nonaspherical subcomplex of an aspherical two-complex, giving a Whitehead counterexample [BestvinaBrady1997Morse]. Thus the two conjectures cannot both be true, although the construction does not determine which one fails.

What remains is to decide whether every group of cohomological dimension $2$ has a $2$-dimensional $K(G,1)$, or instead to construct a confirmed group with $\operatorname{cd}(G)=2$ and $\operatorname{gd}(G)=3$. Analogous failures for other notions of dimension do not resolve this classical integral version.