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Group cohomology of Klein four-group

Revision as of 22:58, 12 October 2011 by Vipul (talk | contribs) (Over an abelian 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

Contents

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.

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

Second cohomology groups and extensions