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

(→Cohomology ring with coefficients in integers) |
(→Classifying space and corresponding chain complex) |
||

Line 8: | Line 8: | ||

The classifying space of the [[Klein four-group]] is the product space <math>\mathbb{R}\mathbb{P}^\infty \times \mathbb{R}\mathbb{P}^\infty</math>, where <math>\mathbb{R}\mathbb{P}^\infty</math> is infinite-dimensional real projective space. | The classifying space of the [[Klein four-group]] is the product space <math>\mathbb{R}\mathbb{P}^\infty \times \mathbb{R}\mathbb{P}^\infty</math>, where <math>\mathbb{R}\mathbb{P}^\infty</math> is infinite-dimensional real projective space. | ||

− | A chain complex that can be used to compute the homology of this space is | + | A chain complex that can be used to compute the homology of this space is given as follows: |

+ | |||

+ | * The <math>n^{th}</math> chain group is a sum of <math>n + 1</math> copies of <math>\mathbb{Z}</math>, indexed by ordered pairs <math>(i,j)</math> where <math>i + j = n</math>. In other words, the <math>n^{th}</math> chain group is: | ||

+ | |||

+ | <math>\mathbb{Z}_{(0,n)} \oplus \mathbb{Z}_{(1,n-1)} \oplus \dots \mathbb{Z}_{(n,0)}</math> | ||

+ | |||

+ | * The boundary map is given by adding up the following maps: | ||

+ | |||

+ | ** The map <math>\mathbb{Z}_{(i,j)} \to \mathbb{Z}_{(i,j-1)}</math> is multiplication by zero if <math>j</math> is odd and is multiplication by two if <math>j</math> is even. | ||

+ | ** The map <math>\mathbb{Z}_{(i,j)} \to \mathbb{Z}_{(i-1,j)}</math> is multiplication by zero if <math>i</math> is odd and multiplication by two if <math>i</math> is even. | ||

+ | |||

==Homology groups== | ==Homology groups== | ||

## Revision as of 15:24, 21 July 2011

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 , where 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 chain group is a sum of copies of , indexed by ordered pairs where . In other words, the chain group is:

- The boundary map is given by adding up the following maps:

- The map is multiplication by zero if is odd and is multiplication by two if is even.
- The map is multiplication by zero if is odd and multiplication by two if is even.

## Homology groups

### Over the integers

The homology groups with coefficients in the ring of integers are given as follows:

These homology groups can be obtained from the knowledge of the homology groups of cyclic group:Z2 using the Kunneth formula.

### Over an abelian group

The homology groups with coefficients in an abelian group (which may be equipped with additional structure as a module over a ring ) are given as follows:

**PLACEHOLDER FOR INFORMATION TO BE FILLED IN**: [SHOW MORE]

## Cohomology groups and cohomology ring

### Groups over the integers

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

## Second cohomology groups and extensions

### Schur multiplier

The Schur multiplier, defined as the second cohomoogy group for trivial group action and also as the second homology group , is isomorphic to cyclic group:Z2.

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