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Group cohomology of Klein four-group

Revision as of 22:12, 12 October 2011 by Vipul (talk | contribs) (Over the integers)
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


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.

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: <toggledisplay>

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

Case on p Value of H_i(G_1;M) \otimes H_j(G_2;M) where i + j = p Value of \operatorname{Tor}_{\mathbb{Z}}^1(H_u(G_1;M),H_v(G_2;M)) where u + v = p -1 Value of \bigoplus_{i+j = p} H_i(G_1;M) \otimes H_j(G_2;M) Value of \bigoplus_{u + v = p - 1} \operatorname{Tor}_{\mathbb{Z}}^1(H_u(G_1;M),H_v(G_2;M)) Value of H_p(G_1 \times G_2;M) (sum of preceding two columns
p = 0 \mathbb{Z} in case i = j = 0 No such cases \mathbb{Z} 0 \mathbb{Z}
p odd positive \mathbb{Z}/2\mathbb{Z} for the case i = 0, j = p and the case i = p, j = 0. 0 in all other cases, because for i + j to be odd, at least one of i and j must be even, forcing the corresponding homology group to be 0. \mathbb{Z}/2\mathbb{Z} when u,v are both odd positive, 0 otherwise \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z} (\mathbb{Z}/2\mathbb{Z})^{(p-1)/2} because that's the number of ordered pairs of positive odd numbers that add up to p - 1. (\mathbb{Z}/2\mathbb{Z})^{(p+3)/2}
p even positive \mathbb{Z}/2\mathbb{Z} for the cases i,j both odd positive, 0 otherwise. zero in all cases (\mathbb{Z}/2\mathbb{Z})^{p/2} 0 (\mathbb{Z}/2\mathbb{Z})^{p/2}

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}) = \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