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 (namely $$3\le \operatorname{cd}(G)\le n$$), 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.

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


 * $$\cdots \xrightarrow{\delta_n+1} C_n(E)\xrightarrow{\delta_n} C_{n-1}(E)\rightarrow \cdots \rightarrow C_1(E)\xrightarrow{\delta_1} C_0(E)\xrightarrow{\varepsilon} \Z\rightarrow 0,$$

where $$E$$ is the universal cover of $$X$$ and $$C_k(E)$$ is the free abelian group generated by the singular $$k$$-chains on $$E$$. The group cohomology of the group $$G$$ with coefficient in a $$\Z[G]$$-module $$M$$ is the cohomology of this chain complex with coefficients in $$M$$, and is denoted by $$H^*(G,M)$$.

Cohomological dimension: A group $$G$$ has cohomological dimension $$n$$ with coefficients in $$\Z$$ (denoted by $$\operatorname{cd}_{\Z}(G)$$) if
 * $$n=\sup \{k : \text{There exists a }\Z[G]\text{ module }M\text{ with }H^{k}(G,M)\neq 0\}. $$

Fact: If $$G$$ has a projective resolution of length at most $$n$$, i.e., $$\Z$$ as trivial $$\Z[G]$$ module has a projective resolution of length at most $$n$$ if and only if $$H^i_{\Z}(G,M)=0$$ for all $$\Z$$-modules $$M$$ and for all $$i>n$$.

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

The 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.

Eilenberg−Ganea theorem
Let $$G$$ be a finitely presented group and $$n\ge 3$$ be an integer. Suppose the cohomological dimension of $$G$$ with coefficients in $$\Z$$ is at most $$n$$, i.e., $$\operatorname{cd}_{\Z}(G)\le n$$. Then there exists an $$n$$-dimensional aspherical CW complex $$X$$ such that the fundamental group of $$X$$ is $$G$$, i.e., $$\pi_1(X)=G$$.

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(X) = G, then cdZ(G) ≤ 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 $$n=2$$ 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 $$\pi_1(X)=G$$.

It is known that given a group G with $$\operatorname{cd}_{\Z}(G)=2$$, there exists a 3-dimensional aspherical CW complex X with $$\pi_1(X)=G$$.