Eilenbergā€“Ganea theorem

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In mathematics, particularly in homological algebra and algebraic topology, the Eilenbergā€“Ganea theorem states for every finitely generated group G with certain conditions on its cohomological dimension (namely 3 ā‰¤ cd(G) ā‰¤ 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 Eilenbergāˆ’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 ā‰¤ n, i.e. Z as trivial Z[G] module has a projective resolution of length ā‰¤ n if and only if HiZ(G,M) = 0 for all Z module M and for all i > n.[citation needed]

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.

Eilenbergāˆ’Ganea theorem

Let G be a finitely presented group and n ā‰„ 3 be an integer. Suppose cohomological dimension of G with coefficients in Z, i.e. cdZ(G) ā‰¤ n. Then there exists an n-dimensional aspherical CW complex X such that the fundamental group of X is G i.e. Ļ€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 Ļ€1(X) = G, then cdZ(G) ā‰¤ 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 Eilenbergā€“Ganea conjecture.

Eilenbergāˆ’Ganea Conjecture: If a group G has cohomological dimension 2 then there is a 2-dimensional aspherical CW complex X with Ļ€1(X) = G.

It is known that given a group G with cdZ(G) = 2 there exists a 3-dimensional aspherical CW complex X with Ļ€1(X) = G.

See also

References

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