In mathematics, particularly in homological algebra and algebraic topology, the EilenbergâGanea 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. [1]
Group cohomology: Let be a group and let be the corresponding EilenbergâMacLane 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 .[ citation needed]
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.
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 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 Ď1(X) = G, then cdZ(G) ⤠n.
For n = 1 the result is one of the consequences of Stallings theorem about ends of groups. [2]
Theorem: Every finitely generated group of cohomological dimension one is free.
For the statement is known as the EilenbergâGanea conjecture.
EilenbergâGanea 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 .