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, 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 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 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 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 a 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 = G, then cd_Z ≤ 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.