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 ), 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 be a group and let be the corresponding Eilenberg−MacLane space. Then we have the following singular chain complex which is a free resolution of over the group ring (where is a trivial -module):

where is the universal cover of and is the free abelian group generated by the singular -chains on . The group cohomology of the group with coefficient in a -module is the cohomology of this chain complex with coefficients in , and is denoted by .

Cohomological dimension: A group has cohomological dimension with coefficients in (denoted by ) if

Fact: If has a projective resolution of length at most , i.e., as trivial module has a projective resolution of length at most if and only if for all -modules and for all .

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

The cohomological dimension of G with coefficient in is the smallest n (possibly infinity) such that G has a projective resolution of length n, i.e., has a projective resolution of length n as a trivial module.

Eilenberg−Ganea theorem

Let be a finitely presented group and be an integer. Suppose the cohomological dimension of with coefficients in is at most , i.e., . Then there exists an -dimensional aspherical CW complex such that the fundamental group of is , i.e., .

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.

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 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 .

It is known that given a group G with , there exists a 3-dimensional aspherical CW complex X with .

See also

References

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