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Group cohomology of Klein four-group

Revision as of 15:25, 21 July 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 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}) = \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 (see group cohomology of cyclic group:Z2) using the Kunneth formula. They can also be computed explicitly using the chain complex description above.

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