# Changes

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.