Eilenberg–Ganea theorem

In mathematics, particularly in homological algebra and algebraic topology, the Eilenberg–Ganea theorem states for every finitely generated group G with certain conditions on its cohomological dimension (namely 3 ≤ cd(G) ≤ n), one can construct an aspherical CW complex X of dimension n whose fundamental group is G. The theorem is named after Polish mathematician Samuel Eilenberg and Romanian mathematician Tudor Ganea. The theorem was first published in a short paper in 1957 in the Annals of Mathematics.

Group cohomology: Let G be a group and X = K(G, 1) is the corresponding Eilenberg−MacLane space. Then we have the following singular chain complex which is a free resolution of Z over the group ring Z[G] (where Z is a trivial Z[G] module).

where E is the universal cover of X and C_k(E) is the free abelian group generated by singular k chains. Group cohomology of the group G with coefficient in G module M is the cohomology of this chain complex with coefficient in M and is denoted by H^*(G, M).

Cohomological dimension: G has cohomological dimension n with coefficients in Z (denoted by cd_Z(G)) if

Fact: If G has a projective resolution of length ≤ n, i.e. Z as trivial Z[G] module has a projective resolution of length ≤ n if and only if Hⁱ_Z(G,M) = 0 for all Z module M and for all i > n.

...
Wikipedia