Thinning (morphology)

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.

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).

$\cdots \stackrel{{\delta }_n+1}{\to }{C}_n(E)\stackrel{{\delta }_n}{\to }{C}_{n-1}(E)\to \cdots \to C_1(E)\stackrel{{\delta }_1}{\to }{C}_0(E)\stackrel{\epsilon }{\to }Z\to 0,$

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

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

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.Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park.

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.

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.

See also

• Eilenberg−Ganea conjecture
• Group cohomology
• Cohomological dimension
• Stallings theorem about ends of groups

References

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• Kenneth S. Brown, Cohomology of groups, Corrected reprint of the 1982 original, Graduate Texts in Mathematics, 87, Springer-Verlag, New York, 1994. Template:MathSciNet. ISBN 0-387-90688-6