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,
Cohomological dimension of G with coefficient in Z is the smallest n such that G has a projective resolution of length n, i.e., Z has a projective resolution of length n as a trivial Z 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 = G, then cdZ ≤ 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 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 cdZ = 2 there exists a 3-dimensional aspherical CW complex X with π1 = G.