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 ), 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. Definitions
Group cohomology: Let  be a group and let  be the corresponding Eilenberg−MacLane space. Then we have the following singular chain complex which is a free resolution of  over the group ring  (where  is a trivial -module):

where  is the universal cover of  and  is the free abelian group generated by the singular -chains on . The group cohomology of the group  with coefficient in a -module  is the cohomology of this chain complex with coefficients in , and is denoted by . Cohomological dimension: A group  has cohomological dimension  with coefficients in  (denoted by ) if 

Fact: If  has a projective resolution of length at most , i.e.,  as trivial  module has a projective resolution of length at most  if and only if  for all -modules  and for all . Therefore, we have an alternative definition of cohomological dimension as follows,

The cohomological dimension of G with coefficient in  is the smallest n (possibly infinity) such that G has a projective resolution of length n, i.e.,  has a projective resolution of length n as a trivial  module. Eilenberg−Ganea theorem
Let  be a finitely presented group and  be an integer. Suppose the cohomological dimension of  with coefficients in  is at most , i.e., . Then there exists an -dimensional aspherical CW complex  such that the fundamental group of  is , i.e., . Converse
Converse of this theorem is an consequence of cellular homology, and the fact that every free module is projective. Theorem: Let X be an aspherical n-dimensional CW complex with π1(X) = G, then cdZ(G) ≤ n.

Related results and conjectures
For n = 1 the result is one of the consequences of Stallings theorem about ends of groups. Theorem: Every finitely generated group of cohomological dimension one is free. For  the statement is  known as the Eilenberg–Ganea conjecture. Eilenberg−Ganea Conjecture: If a group G has cohomological dimension 2 then there is a 2-dimensional aspherical CW complex X with . It is known that given a group G with , there exists a 3-dimensional aspherical CW complex X with . See also 
 Eilenberg–Ganea conjecture
 Group cohomology
 Cohomological dimension
 Stallings theorem about ends of groups

References 

. Kenneth S. Brown, Cohomology of groups, Corrected reprint of the 1982 original, Graduate Texts in Mathematics, 87, Springer-Verlag, New York, 1994. . Homological algebra
Theorems in algebraic topology