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 with generators.
Classifying space and corresponding chain complex
The free abelian group has classifying space equal to the -fold torus, i.e., the space . A chain complex that can be used to compute the homology for the classifying space and hence also the group is below:PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]
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
Here, denotes the binomial coefficient: the number of subsets of size in a set of size . In particular, the homology group is for .
Note that all homology groups themselves are free abelian groups. The homology groups for small values of and are given below. Note that missing cells correspond to zero homology groups:
Over an abelian group
The homology groups with coefficients in an arbitrary abelian group are given below: