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
{\displaystyle 3\leq \operatorname {cd} (G)\leq 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.
[1] Group cohomology: Let
be a group and let
Then we have the following singular chain complex which is a free resolution of
over the group ring
is the universal cover of
is the free abelian group generated by the singular
The group cohomology of the group
is the cohomology of this chain complex with coefficients in
Cohomological dimension: A group
has cohomological dimension
has a projective resolution of length at most
module has a projective resolution of length at most
[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
be a finitely presented group and
Suppose the cohomological dimension of
-dimensional aspherical CW complex
such that the fundamental group of
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. 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.
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