= Low-degree integral homology of a mod-two Eilenberg–MacLane space
{title2=$H_{n,n+2,n+3}(K(\mathbb Z/2,n);\mathbb Z)=\mathbb Z/2\quad(n\geq4)$}
For $n\geq4$, the first four possible degrees have integral groups $\mathbb Z/2,0,\mathbb Z/2,\mathbb Z/2$ in degrees $n,n+1,n+2,n+3$. Mod-two Betti numbers count cyclic summands, while the nonzero first Bockstein actions on $u,\operatorname{Sq}^2u,\operatorname{Sq}^2\operatorname{Sq}^1u$ show that their orders are two. This prevents mistaking a nonzero mod-two group for a higher-order integral cyclic group.
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