Eilenberg–Ganea theorem

Definitions

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

${\displaystyle \cdots \stackrel{{\delta }_n+1}{\to }{C}_n(E)\stackrel{{\delta }_n}{\to }{C}_{n-1}(E)\to \cdots \to C_1(E)\stackrel{{\delta }_1}{\to }{C}_0(E)\stackrel{\epsilon }{\to }\mathbb{Z}\to 0,}$

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

Cohomological dimension: A group ${\displaystyle G}$ has cohomological dimension ${\displaystyle n}$ with coefficients in ${\displaystyle \mathbb{Z}}$ (denoted by ${\displaystyle {\mathrm{cd}}_{\mathbb{Z}}(G)}$) if

${\displaystyle n=sup\{k:\text{There exists a }\mathbb{Z}[G]\text{ module }M\text{ with }{H}^k(G,M)\ne 0\}.}$

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

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

The cohomological dimension of G with coefficient in ${\displaystyle \mathbb{Z}}$ is the smallest n (possibly infinity) such that G has a projective resolution of length n, i.e., ${\displaystyle \mathbb{Z}}$ has a projective resolution of length n as a trivial ${\displaystyle \mathbb{Z}[G]}$ module.