Eilenberg-Ganea conjecture

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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 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.

[edit] References

• Samuel Eilenberg, Tudor Ganea, On the Lusternik-Schnirelmann category of abstract groups, Annals of Mathematics, 2nd Ser., 65 (1957), no. 3, 517 – 518. MR0085510
• Mladen Bestvina, Noel Brady, Morse theory and finiteness properties of groups, Inventiones Mathematicae 129 (1997), no. 3, 445 – 470. MR1465330