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

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):

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

where ${\displaystyle E}$ 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 ${\displaystyle H^{*}(G,M)}$.

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

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

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

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.

Eilenberg−Ganea theorem

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

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.[2]

Theorem: Every finitely generated group of cohomological dimension one is free.

For ${\displaystyle n=2}$ 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 ${\displaystyle \pi _{1}(X)=G}$.

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.

See also

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

References

1. • • Eilenberg, Samuel; Ganea, Tudor (1957). "On the Lusternik–Schnirelmann category of abstract groups". Annals of Mathematics. 2nd Ser. 65 (3): 517–518. doi:10.2307/1970062. MR 0085510.
2. • John R. Stallings, "On torsion-free groups with infinitely many ends", Annals of Mathematics 88 (1968), 312–334. MR0228573

• 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