Group cohomology of free abelian groups

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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:


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}}

Cohomology groups for trivial group action