# Difference between revisions of "Group cohomology of Klein four-group"

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## 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 homology groups are given below:

$p$ $\! 0$ $\! 1$ $\! 2$ $\! 3$ $\! 4$ $\! 5$
$H_p$ $\mathbb{Z}$ $\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$ $\mathbb{Z}/2\mathbb{Z}$ $\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$ $\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$ $\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$
rank of $H_p$ as an elementary abelian 2-group -- 2 1 3 2 4

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:

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

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.

### Important case types for abelian groups

Case on $M$ Conclusion about odd-indexed homology groups, i.e., $H_p, p = 1,3,5,\dots$ Conclusion about even-indexed homology groups, i.e., $H_p, p = 2,4,6,\dots$
$M$ is uniquely 2-divisible, i.e., every element of $M$ has a unique half. This includes the case that $M$ is a field of characteristic not 2. all zero groups all zero groups
$M$ is 2-torsion-free, i.e., no nonzero element of $M$ doubles to zero $(M/2M)^{(p+3)/2}$ $(M/2M)^{p/2}$
$M$ is 2-divisible, but not necessarily uniquely so, e.g., $M = \mathbb{Q}/\mathbb{Z}$ $(\operatorname{Ann}_M(2))^{(p-1)/2}$ $(\operatorname{Ann}_M(2))^{(p+2)/2}$
$M = \mathbb{Z}/2^n\mathbb{Z}$, $n$ any natural number $(\mathbb{Z}/2\mathbb{Z})^{p+1}$ $(\mathbb{Z}/2\mathbb{Z})^{p+1}$
$M$ is a finite abelian group isomorphic to $(\mathbb{Z}/2\mathbb{Z})^{r(p + 1)}$ where $r$ is the rank (i.e., minimum number of generators) for the 2-Sylow subgroup of $M$ isomorphic to $(\mathbb{Z}/2\mathbb{Z})^{r(p + 1)}$ where $r$ is the rank (i.e., minimum number of generators) for the 2-Sylow subgroup of $M$
$M$ is a finitely generated abelian group all isomorphic to $(\mathbb{Z}/2\mathbb{Z})^{r(p + 1) + s(p + 3)/2}$ where $r$ is the rank for the 2-Sylow subgroup of the torsion part of $M$ and $s$ is the free rank (i.e., the rank as a free abelian group of the torsion-free part) of $M$ all isomorphic to $(\mathbb{Z}/2\mathbb{Z})^{r(p + 1) + sp/2}$ where $r$ is the rank for the 2-Sylow subgroup of $M$ and $s$ is the free rank (i.e., the rank as a free abelian group of the torsion-free part) of $M$

## Cohomology groups for trivial group action

### 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:

$p$ $\! 0$ $\! 1$ $\! 2$ $\! 3$ $\! 4$ $\! 5$
$H^p$ $\mathbb{Z}$ 0 $\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$ $\mathbb{Z}/2\mathbb{Z}$ $\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$ $\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$
rank of $H^p$ as an elementary abelian 2-group -- 0 2 1 3 2

### 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:

$p$ $0$ $1$ $2$ $3$ $4$ $5$
$H^p$ $M$ $(\operatorname{Ann}_M(2))^2$ $(\operatorname{Ann}_M(2)) \oplus (M/2M)^2$ $(\operatorname{Ann}_M(2))^3 \oplus (M/2M)$ $(\operatorname{Ann}_M(2))^2 \oplus (M/2M)^3$ $(\operatorname{Ann}_M(2))^4 \oplus (M/2M)^2$

### Important case types for abelian groups

Case on $M$ Conclusion about odd-indexed cohomology groups, i.e., $H^p, p = 1,3,5,\dots$ Conclusion about even-indexed homology groups, i.e., $H^p, p = 2,4,6,\dots$
$M$ is uniquely 2-divisible, i.e., every element of $M$ has a unique half. This includes the case that $M$ is a field of characteristic not 2. all zero groups all zero groups
$M$ is 2-torsion-free, i.e., no nonzero element of $M$ doubles to zero $(M/2M)^{(p-3)/2}$ $(M/2M)^{(p+2)/2}$
$M$ is 2-divisible, but not necessarily uniquely so, e.g., $M = \mathbb{Q}/\mathbb{Z}$ $(\operatorname{Ann}_M(2))^{(p+3)/2}$ $(\operatorname{Ann}_M(2))^{p/2}$
$M = \mathbb{Z}/2^n\mathbb{Z}$, $n$ any natural number $(\mathbb{Z}/2\mathbb{Z})^{p+1}$ $(\mathbb{Z}/2\mathbb{Z})^{p+1}$
$M$ is a finite abelian group isomorphic to $(\mathbb{Z}/2\mathbb{Z})^{r(p + 1)}$ where $r$ is the rank (i.e., minimum number of generators) for the 2-Sylow subgroup of $M$ isomorphic to $(\mathbb{Z}/2\mathbb{Z})^{r(p + 1)}$ where $r$ is the rank (i.e., minimum number of generators) for the 2-Sylow subgroup of $M$
$M$ is a finitely generated abelian group all isomorphic to $(\mathbb{Z}/2\mathbb{Z})^{r(p + 1) + s(p - 1)/2}$ where $r$ is the rank for the 2-Sylow subgroup of the torsion part of $M$ and $s$ is the free rank (i.e., the rank as a free abelian group of the torsion-free part) of $M$ all isomorphic to $(\mathbb{Z}/2\mathbb{Z})^{r(p + 1) + s(p + 3)/2}$ where $r$ is the rank for the 2-Sylow subgroup of $M$ and $s$ is the free rank (i.e., the rank as a free abelian group of the torsion-free part) of $M$

## Cohomology ring

### 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.

## Second cohomology groups and extensions

### 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.