The Eilenberg–Ganea conjecture is a claim in algebraic topology. It was formulated by Samuel Eilenberg and Tudor Ganea in 1957, in a short, but influential paper. It states that if a group G has cohomological dimension 2, then it has a 2-dimensional Eilenberg–MacLane space \( {\displaystyle K(G,1)} \). For n different from 2, a group G of cohomological dimension n has an n-dimensional Eilenberg–MacLane space. It is also known that a group of cohomological dimension 2 has a 3-dimensional Eilenberg−MacLane space.
In 1997, Mladen Bestvina and Noel Brady constructed a group G so that either G is a counterexample to the Eilenberg–Ganea conjecture, or there must be a counterexample to the Whitehead conjecture; in other words, not both conjectures can be true.
References
Eilenberg, Samuel; Ganea, Tudor (1957). "On the Lusternik–Schnirelmann category of abstract groups". Annals of Mathematics. 2nd Ser. 65 (3): 517–518. doi:10.2307/1970062. MR 0085510.
Bestvina, Mladen; Brady, Noel (1997). "Morse theory and finiteness properties of groups". Inventiones Mathematicae. 129 (3): 445–470. doi:10.1007/s002220050168. MR 1465330.
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