Group cohomology of free abelian groups

This article gives specific information, namely, group cohomology, about a family of groups, namely: free abelian group.
View group cohomology of group families | View other specific information about free abelian group

This article describes the homology and cohomology groups of the free abelian group $\mathbb {Z} ^{d}$ with $d$ generators.

Classifying space and corresponding chain complex

The free abelian group $\mathbb {Z} ^{d}$ has classifying space equal to the $d$-fold torus, i.e., the space $S^{1}\times S^{1}\times \dots \times S^{1}$. A chain complex that can be used to compute the homology for the classifying space and hence also the group is below:

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Homology groups for trivial group action

FACTS TO CHECK AGAINST (homology group for trivial group action):
First homology group: first homology group for trivial group action equals tensor product with abelianization
Second homology group: formula for second homology group for trivial group action in terms of Schur multiplier and abelianization|Hopf's formula for Schur multiplier
General: universal coefficients theorem for group homology|homology group for trivial group action commutes with direct product in second coordinate|Kunneth formula for group homology

Over the integers

$H_{q}(\mathbb {Z} ^{d};\mathbb {Z} )=(\mathbb {Z} )^{\binom {d}{q}}$

Here, ${\binom {d}{q}}$ denotes the binomial coefficient: the number of subsets of size $q$ in a set of size $d$. In particular, the homology group is $0$ for $q>d$.

Note that all homology groups themselves are free abelian groups. The homology groups for small values of $d$ and $q$ are given below. Note that missing cells correspond to zero homology groups:

	$q=0$	$q=1$	$q=2$	$q=3$	$q=4$	$q=5$
$d=0$	$\mathbb {Z}$
$d=1$	$\mathbb {Z}$	$\mathbb {Z}$
$d=2$	$\mathbb {Z}$	$\mathbb {Z} ^{2}$	$\mathbb {Z}$
$d=3$	$\mathbb {Z}$	$\mathbb {Z} ^{3}$	$\mathbb {Z} ^{3}$	$\mathbb {Z}$
$d=4$	$\mathbb {Z}$	$\mathbb {Z} ^{4}$	$\mathbb {Z} ^{6}$	$\mathbb {Z} ^{4}$	$\mathbb {Z}$

Over an abelian group

The homology groups with coefficients in an arbitrary abelian group are given below:

$H_{q}(\mathbb {Z} ^{d};M)=M^{\binom {d}{q}}$