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In mathematics, particularly in homological algebra and algebraic topology, Eilenberg-Ganea theorem states that given a finitely generated group G with certain conditions on its cohomological dimension (namely 3 ≤ cd(G) ≤ n), we can construct an aspherical CW complex X of dimension n. The theorem is named after Polish mathematician Samuel Eilenberg and Romannian mathematician Tudor Ganea. The theorem was first published in a short paper in 1957 in the Annals of Mathematics^[1].

Group cohomology: Let G be a group and X = K(G,1) is the corresponding Eilenberg-Maclane space. Then we have the following singular chain complex which is a free resolution of Z over the group ring Z[G] (where Z is a trivial Z[G] module).

${\displaystyle \cdots {\xrightarrow {\delta _{n}+1}}C_{n}(E){\xrightarrow {\delta _{n}}}C_{n-1}(E)\rightarrow \cdots \rightarrow C_{1}(E){\xrightarrow {\delta _{1}}}C_{0}(E){\xrightarrow {\epsilon }}Z\rightarrow 0,}$

where E is the universal cover of X and C_k(E) is the free abelian group generated by singular k chains. Group cohomology of the group G with coefficient in G module M is the cohomology of this chain complex with coefficient in M and is denoted by H^*(G,M).

Cohomological dimension: G has cohomological dimension n with coefficients in Z (denoted by cd_Z(G)) if

${\displaystyle n=\,\,{\text{sup}}_{k}\,\,\lbrace There\,\,\,exists\,\,\,a\,\,\,Z[G]\,\,\,module\,\,\,M\,\,\,with\,\,\,H^{k}(G,M)\neq 0\,\,\,\rbrace .}$

Fact: If G has a projective resolution of length ≤n, i.e. Z as trivial Z[G] module has a projective resolution of length ≤ n if and only if Hⁱ_Z(G,M)=0 for all Z module M and for all i>n.

Therefore we have an alternative definition of cohomological dimension as follows,

Cohomological dimension of G with coefficient in Z is the smallest n (possibly infinity) such that G has a projective resolution of length n, i.e. Z has a projective resolution of length n as a trivial Z[G] module.

Let G be a finitely presented group and n≥3 be an integer. Suppose cohomological dimension of G with coefficients in Z, i.e. cd_Z(G)≤n. Then there exists an n-dimensional aspherical CW complex X such that the fundamental group of X is G i.e. π₁(X)=G.

Converse of this theorem is an consequence of cellular homology, and the fact that every free module is projective.

Theorem: Let X be an n-dimensional CW complex with π₁(X)=G, then cd_Z(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 n=2 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 π₁(X)=G.

It is known that given a group G with cd_Z(G)=2 there exists a 3-dimensional aspherical CW complex X with π₁(X)=G.

• Eilenberg−Ganea conjecture
• Group cohomology
• Cohomological dimension
• Stallings theorem about ends of groups

• Bestvina, Mladen; Brady, Noel (1997). "Morse theory and finiteness properties of groups". Inventiones Mathematicae. 129 (3): 445–470. doi:10.1007/s002220050168. MR 1465330..

• Kenneth S. Brown, Cohomology of groups, Corrected reprint of the 1982 original, Graduate Texts in Mathematics, 87, Springer-Verlag, New York, 1994. MR1324339. ISBN 0-387-90688-6