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

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):

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

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 $\mathbb {Z} [G]$-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 $\mathbb {Z}$ (denoted by $\operatorname {cd} _{\mathbb {Z} }(G)$) if

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

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

Eilenberg−Ganea theorem

Let $G$ be a finitely presented group and $n\geq 3$ be an integer. Suppose the cohomological dimension of $G$ with coefficients in $\mathbb {Z}$ is at most $n$, i.e., $\operatorname {cd} _{\mathbb {Z} }(G)\leq n$. 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 cd_Z(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 $\operatorname {cd} _{\mathbb {Z} }(G)=2$, there exists a 3-dimensional aspherical CW complex X with $\pi _{1}(X)=G$.

See also

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

References

• Bestvina, Mladen; Brady, Noel (1997). "Morse theory and finiteness properties of groups". Inventiones Mathematicae. 129 (3): 445–470. Bibcode:1997InMat.129..445B. doi:10.1007/s002220050168. MR 1465330. S2CID 120422255..
• 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