= Degree-one Bockstein square identity
{title2=$\rho\widehat\beta(x)=x^2,\quad x\in H^1(X;\mathbb F_2)$}
Reduction $\rho$ of the <integral Bockstein homomorphism> associated to $0\to\mathbb Z\xrightarrow{2}\mathbb Z\to\mathbb F_2\to0$ equals the <cup product> square on degree-one classes. One can see this on an ordered two-simplex: lift the values of a mod-two one-cocycle to zero or one. The half-coboundary is one precisely when both consecutive edge values are one, giving the cup-square cocycle.
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