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 ${\displaystyle 3\leq \operatorname {cd} (G)\leq n}$), 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 ${\displaystyle G}$ be a group and let ${\displaystyle X=K(G,1)}$ be the corresponding Eilenberg−MacLane space. Then we have the following singular chain complex which is a free resolution of ${\displaystyle \mathbb {Z} }$ over the group ring ${\displaystyle \mathbb {Z} [G]}$ (where ${\displaystyle \mathbb {Z} }$ is a trivial ${\displaystyle \mathbb {Z} [G]}$-module):

${\displaystyle \cdots \xrightarrow {\delta _{n}+1} C_{n}(E)\xrightarrow {\delta _{n}} C_{n-1}(E)\rightarrow \cdots \rightarrow C_{1}(E)\xrightarrow {\delta _{1}} C_{0}(E)\xrightarrow {\varepsilon } \mathbb {Z} \rightarrow 0,}$

where ${\displaystyle E}$ is the universal cover of ${\displaystyle X}$ and ${\displaystyle C_{k}(E)}$ is the free abelian group generated by the singular ${\displaystyle k}$-chains on ${\displaystyle E}$. The group cohomology of the group ${\displaystyle G}$ with coefficient in a ${\displaystyle \mathbb {Z} [G]}$-module ${\displaystyle M}$ is the cohomology of this chain complex with coefficients in ${\displaystyle M}$, and is denoted by ${\displaystyle H^{*}(G,M)}$.

Cohomological dimension: A group ${\displaystyle G}$ has cohomological dimension ${\displaystyle n}$ with coefficients in ${\displaystyle \mathbb {Z} }$ (denoted by ${\displaystyle \operatorname {cd} _{\mathbb {Z} }(G)}$) if

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

Fact: If ${\displaystyle G}$ has a projective resolution of length at most ${\displaystyle n}$, i.e., ${\displaystyle \mathbb {Z} }$ as trivial ${\displaystyle \mathbb {Z} [G]}$ module has a projective resolution of length at most ${\displaystyle n}$ if and only if ${\displaystyle H_{\mathbb {Z} }^{i}(G,M)=0}$ for all ${\displaystyle \mathbb {Z} }$-modules ${\displaystyle M}$ and for all ${\displaystyle i>n}$.^{[ citation needed ]}

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

The cohomological dimension of G with coefficient in${\displaystyle \mathbb {Z} }$is the smallest n (possibly infinity) such that G has a projective resolution of lengthn, i.e., ${\displaystyle \mathbb {Z} }$has a projective resolution of lengthnas a trivial${\displaystyle \mathbb {Z} [G]}$module.

Eilenberg−Ganea theorem

Let ${\displaystyle G}$ be a finitely presented group and ${\displaystyle n\geq 3}$ be an integer. Suppose the cohomological dimension of ${\displaystyle G}$ with coefficients in ${\displaystyle \mathbb {Z} }$ is at most ${\displaystyle n}$, i.e., ${\displaystyle \operatorname {cd} _{\mathbb {Z} }(G)\leq n}$. Then there exists an ${\displaystyle n}$-dimensional aspherical CW complex ${\displaystyle X}$ such that the fundamental group of ${\displaystyle X}$ is ${\displaystyle G}$, i.e., ${\displaystyle \pi _{1}(X)=G}$.

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 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. ^[2]

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

For ${\displaystyle n=2}$ 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 ${\displaystyle \pi _{1}(X)=G}$.

It is known that given a group G with ${\displaystyle \operatorname {cd} _{\mathbb {Z} }(G)=2}$, there exists a 3-dimensional aspherical CW complex X with ${\displaystyle \pi _{1}(X)=G}$.

See also

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

References

1. • • Eilenberg, Samuel; Ganea, Tudor (1957). "On the Lusternik–Schnirelmann category of abstract groups". Annals of Mathematics . 2nd Ser. 65 (3): 517–518. doi:10.2307/1970062. JSTOR   1970062. MR   0085510.
2. • John R. Stallings, "On torsion-free groups with infinitely many ends", Annals of Mathematics 88 (1968), 312–334. MR 0228573

• Bestvina, Mladen; Brady, Noel (1997). "Morse theory and finiteness properties of groups". Inventiones Mathematicae . 129 (3): 445–470. Bibcode:1997InMat.129..445B. doi:10.1007/s002220050168. MR   1465330. S2CID   120422255..
• Kenneth S. Brown, Cohomology of groups, Corrected reprint of the 1982 original, Graduate Texts in Mathematics, 87, Springer-Verlag, New York, 1994. MR 1324339. ISBN   0-387-90688-6