Eilenberg–Ganea theorem

Short description: On constructing an aspherical CW complex whose fundamental group is a given group

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 [math]\displaystyle{ 3\le \operatorname{cd}(G)\le n }[/math]), 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 [math]\displaystyle{ G }[/math] be a group and let [math]\displaystyle{ X=K(G,1) }[/math] be the corresponding Eilenberg−MacLane space. Then we have the following singular chain complex which is a free resolution of [math]\displaystyle{ \mathbb{Z} }[/math] over the group ring [math]\displaystyle{ \mathbb{Z}[G] }[/math] (where [math]\displaystyle{ \mathbb{Z} }[/math] is a trivial [math]\displaystyle{ \mathbb{Z}[G] }[/math]-module):

[math]\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, }[/math]

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

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

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

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

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

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

Eilenberg−Ganea theorem

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

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 [math]\displaystyle{ n=2 }[/math] 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 [math]\displaystyle{ \pi_1(X)=G }[/math].

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

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. doi:10.1007/s002220050168. Bibcode: 1997InMat.129..445B. .
• 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