Integral Bockstein homomorphism ($\hat{\beta }:H^q(X;\mathbb{Z}/n)\to H^{q+1}(X;\mathbb{Z})$)

The integral Bockstein is the connecting homomorphism of $0\to \mathbb{Z}\stackrel{n}{\to }\mathbb{Z}\to \mathbb{Z}/n\to 0$. Lift a modulo-$n$ cocycle to an integral cochain $a$; write its coboundary as $\delta a=nb$. Then $\hat{\beta }[a]=[b]$. Its image is the subgroup of integral cohomology killed by $n$.