# Group cohomology of 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 $\mathbb{R}\mathbb{P}^\infty \times \mathbb{R}\mathbb{P}^\infty$, where $\mathbb{R}\mathbb{P}^\infty$ 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 $n^{th}$ chain group is a sum of $n + 1$ copies of $\mathbb{Z}$, indexed by ordered pairs $(i,j)$ where $i + j = n$. In other words, the $n^{th}$ chain group is:

$\mathbb{Z}_{(0,n)} \oplus \mathbb{Z}_{(1,n-1)} \oplus \dots \mathbb{Z}_{(n,0)}$

• The boundary map is (up to some sign issues that need to be fixed!) given by adding up the following maps:
• The map $\mathbb{Z}_{(i,j)} \to \mathbb{Z}_{(i,j-1)}$ is multiplication by zero if $j$ is odd and is multiplication by two if $j$ is even.
• The map $\mathbb{Z}_{(i,j)} \to \mathbb{Z}_{(i-1,j)}$ is multiplication by zero if $i$ is odd and multiplication by two if $i$ is even.

## Homology groups

### Over the integers

The homology groups with coefficients in the ring of integers $\mathbb{Z}$ are given as follows:

$H_p(\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z};\mathbb{Z}) = \left\lbrace\begin{array}{rl} (\mathbb{Z}/2\mathbb{Z})^{(p + 3)/2}, & p = 1,3,5,\dots \\ (\mathbb{Z}/2\mathbb{Z})^{p/2}, & p = 2,4,6,\dots \\ \mathbb{Z}, & p = 0 \\\end{array}\right.$

The first few homomology groups are given below:

$p$ $\! 0$ $\! 1$ $\! 2$ $\! 3$ $\! 4$ $\! 5$
$H_p$ $\mathbb{Z}$ $\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$ $\mathbb{Z}/2\mathbb{Z}$ $\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$ $\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$ $\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$

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: [SHOW MORE]

### Over an abelian group

The homology groups with coefficients in an abelian group $M$ are given as follows:

$H_p(\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z};M) = \left\lbrace\begin{array}{rl} (M/2M)^{(p+3)/2} \oplus (\operatorname{Ann}_M(2))^{(p-1)/2}, & \qquad p = 1,3,5,\dots\\ (M/2M)^{p/2} \oplus (\operatorname{Ann}_M(2))^{(p+2)/2}, & \qquad p = 2,4,6,\dots \\ M, & \qquad p = 0 \\\end{array}\right.$

## Cohomology groups and cohomology ring

### Groups over the integers

The cohomology groups with coefficients in the integers are given as below:

$H^p(\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z};\mathbb{Z}) = \left\lbrace \begin{array}{rl} (\mathbb{Z}/2\mathbb{Z})^{(p+1)/2}, & p = 1,3,5,\dots \\ (\mathbb{Z}/2\mathbb{Z})^{(p+2)/2}, & p = 2,4,6,\dots \\ \mathbb{Z}, & p = 0 \\\end{array}\right.$

## Second cohomology groups and extensions

### Schur multiplier

The Schur multiplier, defined as the second cohomology group for trivial group action $H^2(G,\mathbb{C}^\ast)$ and also as the second homology group $H_2(G,\mathbb{Z})$, is isomorphic to cyclic group:Z2.