Integral Bockstein homomorphism
= Integral Bockstein homomorphism
{title2=$\widehat\beta:H^q(X;\mathbb Z/n)\to H^{q+1}(X;\mathbb Z)$}
The integral Bockstein is the <connecting homomorphism> of $0\to\mathbb Z\xrightarrow{n}\mathbb Z\to\mathbb Z/n\to0$. Lift a modulo-$n$ cocycle to an integral cochain $a$; write its coboundary as $\delta a=nb$. Then $\widehat\beta[a]=[b]$. Its image is the subgroup of integral <cohomology> killed by $n$.