Nowadays, there are some very useful notions of a cohomological dimension of a group.

Classically, for a torsion-free group G, the ordinary cohomological dimension cdG is equal to its

geometric dimension gdG (the minimal dimension of a free contractible G-CW-complex) provided

the cohomological dimension is not two. When cdG=2, the Eilenberg-Ganea Conjecture asserts

that gdG=2.

For groups that contain torsion, the analogues of algebraic and geometric dimensions are less

clear. This prompted K. S. Brown's question in 1977 for groups that are virtually torsion-free.

It subsequently led to new notions of cohomological dimensions and to various other

formulations of the Brown's question. I will discuss the history of this topic and the recent

joint work with Ian Leary where we give the first counterexamples to the original Brown's

question.