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 ≤ cd(G) ≤ 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 G be a group and X = K(G, 1) is the corresponding Eilenberg−MacLane space. Then we have the following singular chain complex which is a free resolution of Z over the group ring Z[G] (where Z is a trivial 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 }}Z\rightarrow 0,

where E is the universal cover of X and C_k(E) is the free abelian group generated by singular k chains. Group cohomology of the group G with coefficient in G module M is the cohomology of this chain complex with coefficient in M and is denoted by H^*(G, M).

Cohomological dimension: G has cohomological dimension n with coefficients in Z (denoted by cd_Z(G)) if

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

Fact: If G has a projective resolution of length ≤ n, i.e. Z as trivial Z[G] module has a projective resolution of length ≤ n if and only if Hⁱ_Z(G,M) = 0 for all Z module M and for all 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 G be a finitely presented group and n ≥ 3 be an integer. Suppose cohomological dimension of G with coefficients in Z, i.e. cd_Z(G) ≤ n. Then there exists an n-dimensional aspherical CW complex X such that the fundamental group of X is G i.e. π₁(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 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 π₁(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.

References

↑
- 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.
↑
- John R. Stallings, "On torsion-free groups with infinitely many ends", Annals of Mathematics 88 (1968), 312–334. MR 0228573

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. MR 1324339. ISBN 0-387-90688-6

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