= Integral homology of two real projective planes
{title2=$H_*(\mathbb{RP}^2\times\mathbb{RP}^2;\mathbb Z)$}
Each <real projective plane> has cells $e_0,e_1,e_2$ with $de_2=2e_1$ and $de_1=0$. Their product <cellular chain complex> has ranks $1,2,3,2,1$. Its degree-two <cycle> $e_1\times e_1$ has order two in <homology>. The degree-three cycle $e_2\times e_1+e_1\times e_2$ also has order two, because its double is $d(e_2\times e_2)$. Consequently $H_0=\mathbb Z$, $H_1=(\mathbb Z/2)^2$, $H_2=H_3=\mathbb Z/2$ and all other groups vanish. The degree-three torsion is also the <Tor functor> term in the <Künneth theorem>.
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