Group cohomology of Klein four-group
This article gives specific information, namely, group cohomology, about a particular group, namely: Klein four-group.
View group cohomology of particular groups | View other specific information about Klein four-group
Classifying space and corresponding chain complex
The classifying space of the Klein four-group is the product space , where is infinite-dimensional real projective space.
A chain complex that can be used to compute the homology of this space is given as follows:
- The chain group is a sum of copies of , indexed by ordered pairs where . In other words, the chain group is:
- The boundary map is (up to some sign issues that need to be fixed!) given by adding up the following maps:
- The map is multiplication by zero if is odd and is multiplication by two if is even.
- The map is multiplication by zero if is odd and multiplication by two if is even.
Over the integers
The homology groups with coefficients in the ring of integers are given as follows:
These homology groups can be obtained from the knowledge of the homology groups of cyclic group:Z2 (see group cohomology of cyclic group:Z2) using the Kunneth formula for group homology. They can also be computed explicitly using the chain complex description above.
Here is the computation using the Kunneth formula for group homology: <toggledisplay>
We set and in the formula.
|Case on||Value of where||Value of where||Value of||Value of||Value of (sum of preceding two columns|
|in case||No such cases||0|
|odd positive||for the case and the case . 0 in all other cases, because for to be odd, at least one of and must be even, forcing the corresponding homology group to be 0.||when are both odd positive, 0 otherwise||because that's the number of ordered pairs of positive odd numbers that add up to .|
|even positive||for the cases both odd positive, 0 otherwise.||zero in all cases||0|
Over an abelian group
The homology groups with coefficients in an abelian group (which may be equipped with additional structure as a module over a ring ) are given as follows:PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]
Cohomology groups and cohomology ring
Groups over the integers
The cohomology groups with coefficients in the integers are given as below:
Second cohomology groups and extensions
See also the projective representation theory of Klein four-group.