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 $3\le \operatorname(G)\le n$), 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 let $X=K(G,1)$ be the corresponding Eilenberg−MacLane space. Then we have the following singular chain complex which is a free resolution of $\mathbb$ over the group ring \mathbb /math> (where $\mathbb$ is a trivial \mathbb /math>-module):

:$\cdots \xrightarrow C_n(E)\xrightarrow C_(E)\rightarrow \cdots \rightarrow C_1(E)\xrightarrow C_0(E)\xrightarrow \Z\rightarrow 0,$

where $E$ is the universal cover of $X$ and $C_k(E)$ is the free abelian group generated by the singular $k$-chains on $E$. The group cohomology of the group $G$ with coefficient in a \Z /math>-module $M$ is the cohomology of this chain complex with coefficients in $M$, and is denoted by $H^*(G,M)$.

Cohomological dimension: A group $G$ has cohomological dimension $n$ with coefficients in $\Z$ (denoted by $\operatorname_(G)$) if
:$n=\sup \.$

Fact: If $G$ has a projective resolution of length at most $n$, i.e., $\Z$ as trivial \Z /math> module has a projective resolution of length at most $n$ if and only if $H^i_(G,M)=0$ for all $\Z$-modules $M$ and for all $i>n$.

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

''The 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 /math> ''module.''

Eilenberg−Ganea theorem

Let $G$ be a finitely presented group and $n\ge 3$ be an integer. Suppose the cohomological dimension of $G$ with coefficients in $\Z$ is at most $n$, i.e., $\operatorname_(G)\le n$. Then there exists an $n$-dimensional aspherical CW complex $X$ such that the fundamental group of $X$ is $G$, i.e., $\pi_1(X)=G$.

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 ''π''₁(''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.* John R. Stallings, "On torsion-free groups with infinitely many ends", ''Annals of Mathematics'' 88 (1968), 312–334.

Theorem: Every finitely generated group of cohomological dimension one is free.

For $n=2$ 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 $\pi_1(X)=G$.

It is known that given a group ''G'' with $\operatorname_(G)=2$, there exists a 3-dimensional aspherical CW complex ''X'' with $\pi_1(X)=G$.

See also

* Eilenberg–Ganea conjecture
* Group cohomology
* Cohomological dimension
* Stallings theorem about ends of groups

References

*.
* Kenneth S. Brown, ''Cohomology of groups'', Corrected reprint of the 1982 original, Graduate Texts in Mathematics, 87, Springer-Verlag, New York, 1994. .

{{DEFAULTSORT:Eilenberg-Ganea theorem
Homological algebra
Theorems in algebraic topology