# 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.

## Homology groups

### 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

### Schur multiplier

The Schur multiplier, defined as the second cohomology group for trivial group action and also as the second homology group , is isomorphic to cyclic group:Z2.

See also the projective representation theory of Klein four-group.