Group cohomology of 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.

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 homology groups are given below:

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:

We set $$G_1 = G_2 = \mathbb{Z}/2\mathbb{Z}$$ and $$M = \mathbb{Z}$$ in the formula.

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

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

Here, $$M/2M$$ is the quotient of $$M$$ by $$2M = \{ 2x \mid x \in M \}$$ and $$\operatorname{Ann}_M(2) = \{ x \in M \mid 2x = 0 \}$$.

These homology groups can be computed in terms of the homology groups over integers using the universal coefficients theorem for group homology.

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

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

The first few cohomology groups are given below:

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

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

Here, $$M/2M$$ is the quotient of $$M$$ by $$2M = \{ 2x \mid x \in M \}$$ and $$\operatorname{Ann}_M(2) = \{ x \in M \mid 2x = 0 \}$$.

These can be deduced from the homology groups with coefficients in the integers using the dual universal coefficients theorem for group cohomology.

The first few groups are given below:

Over a 2-divisible ring
If $$R$$ is a 2-divisible unital ring, then it is also a uniquely 2-divisible ring. In this case, all cohomology groups in positive degrees vanish, and $$H^*(\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z};R)$$ is isomorphic to $$R$$, occurring in the $$H^0$$.

In particular, this includes the case $$R$$ a field of characteristic not 2, as well as $$R$$ a ring (not necessarily a field) of finite positive characteristic.

Schur multiplier and Schur covering groups
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.

There are two possibilities for the Schur covering group: dihedral group:D8 and quaternion group. These belong to the Hall-Senior family $$\Gamma_2$$ (up to isoclinism). They are precisely the stem extensions where the acting group is the Klein four-group and the base group is its Schur multiplier, namely cyclic group:Z2. For more, see second cohomology group for trivial group action of V4 on Z2.

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

Second cohomology groups for trivial group action
 