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 Ck(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 cdZ(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 HiZ(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. cdZ(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 cdZ(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 cdZ(G) = 2 there exists a 3-dimensional aspherical CW complex X with π1(X) = G.

See also

References

  1. ↑
  2. ↑