Bockstein isomorphism for a three-dimensional lens space
= Bockstein isomorphism for a three-dimensional lens space
{c}
{title2=$\beta:H^1(L(p);\mathbb F_p)\xrightarrow{\sim}H^2(L(p);\mathbb F_p)$}
For the three-dimensional <lens space> $L(p)$, the integral coefficient sequence shows that the integral connecting map $H^1(L(p);\mathbb F_p)\to H^2(L(p);\mathbb Z)$ is an isomorphism between groups of order $p$. Reduction modulo $p$ is also an isomorphism in degree two, so their composite Bockstein is an isomorphism.