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

Homology groups

Over the integers

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

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

These homology groups can be obtained from the knowledge of the homology groups of cyclic group:Z2 using the Kunneth formula.

Over an abelian group $M$

The homology groups with coefficients in an abelian group $M$ (which may be equipped with additional structure as a module over a ring $R$) are given as follows:

Cohomology groups and cohomology ring

Groups over the integers

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

$H^p(\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z};\mathbb{Z}) =$

Second cohomology groups and extensions

Schur multiplier

The Schur multiplier, defined as the second cohomoogy 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.