Bockstein factorization through integral cohomology ($\beta =\rho \hat{\beta }$)

The Bockstein homomorphism for $0\to \mathbb{Z}/n\to \mathbb{Z}/n^2\to \mathbb{Z}/n\to 0$, with injection $[a]\mapsto [na]$, is coefficient reduction $\rho$ after the integral Bockstein homomorphism. The commuting maps of coefficient sequences prove this by naturality of the connecting homomorphism.

Reduction $\rho$ of the integral Bockstein homomorphism associated to $0\to \mathbb{Z}\stackrel{2}{\to }\mathbb{Z}\to {\mathbb{F}}_2\to 0$ 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.

Exactness of the integral coefficient sequence gives $\hat{\beta }\rho =0$. The Bockstein factorization through integral cohomology therefore gives ${\beta }^2=\rho \hat{\beta }\rho \hat{\beta }=0$, for every modulus $n\ge 2$, including composite moduli.

The degree-one Bockstein homomorphism makes modulo-$n$ cohomology into a cochain complex. Its cohomology is the Bockstein cohomology. For prime $n$ this is a graded vector space; for composite $n$ it is a graded $\mathbb{Z}/n$-module. For ${\mathbb{RP}}^3$ at modulus two, its only nonzero groups are ${\mathbb{F}}_2$ in degrees zero and three. This is the cohomological counterpart of Bockstein homology.