Bockstein linking invariant of a three-dimensional lens space
= Bockstein linking invariant of a three-dimensional lens space
{c}
{title2=$t(a)=\langle a\smile\beta(a),[L(p)]\rangle$}
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 $n$.