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

≤

cd

⁡

(

G

)

≤

n

{\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.

Definitions

Group cohomology: Let

G

{\displaystyle G}

be a group and let

X

=

K

(

G

,

1

)

{\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

Z

{\displaystyle \mathbb {Z} }

over the group ring

Z

[

G

]

{\displaystyle \mathbb {Z} [G]}

(where

Z

{\displaystyle \mathbb {Z} }

is a trivial

Z

[

G

]

{\displaystyle \mathbb {Z} [G]}

-module):

⋯

→

δ

n

+

1

C

n

(

E

)

→

δ

n

C

n

−

1

(

E

)

→

⋯

→

C

1

(

E

)

→

δ

1

C

0

(

E

)

→

ε

Z

→

0

,

{\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

E

{\displaystyle E}

is the universal cover of

X

{\displaystyle X}

and

C

k

(

E

)

{\displaystyle C_{k}(E)}

is the free abelian group generated by the singular

k

{\displaystyle k}

-chains on

E

{\displaystyle E}

. The group cohomology of the group

G

{\displaystyle G}

with coefficient in a

Z

[

G

]

{\displaystyle \mathbb {Z} [G]}

-module

M

{\displaystyle M}

is the cohomology of this chain complex with coefficients in

M

{\displaystyle M}

, and is denoted by

H

∗

(

G

,

M

)

{\displaystyle H^{*}(G,M)}

.

Cohomological dimension: A group

G

{\displaystyle G}

has cohomological dimension

n

{\displaystyle n}

with coefficients in

Z

{\displaystyle \mathbb {Z} }

(denoted by

cd

Z

⁡

(

G

)

{\displaystyle \operatorname {cd} _{\mathbb {Z} }(G)}

) if

n

=

sup

{

k

:

There exists a

Z

[

G

]

module

M

with

H

k

(

G

,

M

)

≠

0

}

.

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

Fact: If

G

{\displaystyle G}

has a projective resolution of length at most

n

{\displaystyle n}

, i.e.,

Z

{\displaystyle \mathbb {Z} }

as trivial

Z

[

G

]

{\displaystyle \mathbb {Z} [G]}

module has a projective resolution of length at most

n

{\displaystyle n}

if and only if

H

Z

i

(

G

,

M

)

=

0

{\displaystyle H_{\mathbb {Z} }^{i}(G,M)=0}

for all

Z

{\displaystyle \mathbb {Z} }

-modules

M

{\displaystyle M}

and for all

i

>

n

{\displaystyle i>n}

.

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

The cohomological dimension of G with coefficient in

Z

{\displaystyle \mathbb {Z} }

is the smallest n (possibly infinity) such that G has a projective resolution of length n, i.e.,

Z

{\displaystyle \mathbb {Z} }

has a projective resolution of length n as a trivial

Z

[

G

]

{\displaystyle \mathbb {Z} [G]}

module.

Eilenberg−Ganea theorem

Let

G

{\displaystyle G}

be a finitely presented group and

n

≥

3

{\displaystyle n\geq 3}

be an integer. Suppose the cohomological dimension of

G

{\displaystyle G}

with coefficients in

Z

{\displaystyle \mathbb {Z} }

is at most

n

{\displaystyle n}

, i.e.,

cd

Z

⁡

(

G

)

≤

n

{\displaystyle \operatorname {cd} _{\mathbb {Z} }(G)\leq n}

.