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Eilenberg–Ganea theorem

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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 3 ≤ cd ⁡ ( G ) ≤ n {\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.

Definitions

Group cohomology: Let G {\displaystyle G} be a group and let X = K ( G , 1 ) {\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 Z {\displaystyle \mathbb {Z} } over the group ring Z [ G ] {\displaystyle \mathbb {Z} } (where Z {\displaystyle \mathbb {Z} } is a trivial Z [ G ] {\displaystyle \mathbb {Z} } -module):

⋯ → δ n + 1 C n ( E ) → δ n C n − 1 ( E ) → ⋯ → C 1 ( E ) → δ 1 C 0 ( E ) → ε Z → 0 , {\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 } \mathbb {Z} \rightarrow 0,}

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

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

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

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

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

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

Eilenberg−Ganea theorem

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

It is known that given a group G with cd Z ⁡ ( G ) = 2 {\displaystyle \operatorname {cd} _{\mathbb {Z} }(G)=2} , there exists a 3-dimensional aspherical CW complex X with π 1 ( X ) = G {\displaystyle \pi _{1}(X)=G} .

See also

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. JSTOR 1970062. MR 0085510.
  2. * John R. Stallings, "On torsion-free groups with infinitely many ends", Annals of Mathematics 88 (1968), 312–334. MR0228573
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