 Integral homology of two real projective planes

Each real projective plane has cells $e_0,e_1,e_2$ with $d{e}_2=2e_1$ and $d{e}_1=0$. 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.