Bockstein isomorphism for a three-dimensional lens space ($\beta :H^1(L(p);{\mathbb{F}}_p)\stackrel{\sim }{\to }{H}^2(L(p);{\mathbb{F}}_p)$)

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.

For a generator $a\in H^1(L(p);{\mathbb{F}}_p)$, Poincare duality and the Bockstein isomorphism for a three-dimensional lens space make $t(a)$ nonzero. Replacing $a$ by $na$ multiplies $t(a)$ by $n^2$. An orientation-reversing homotopy equivalence multiplies the evaluation by $-1$, so its existence forces $n^2=-1$ in ${\mathbb{F}}_p$ for some $n$.