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Group cohomology of Klein four-group

756 bytes added, 15:24, 21 July 2011
Classifying space and corresponding chain complex
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 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==
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