Mod-two intersection pairing of the Klein bottle
= Mod-two intersection pairing of the Klein bottle
{title2=$Q=\left(\begin{smallmatrix}1&1\\1&0\end{smallmatrix}\right)$}
A one-sided section loop and a two-sided fiber loop in the reflection <mapping torus> have mod-two self-intersections one and zero and meet once. Their <intersection form> is $Q$. In the evaluation-dual <cohomology> basis, the <Poincare duality pairing> has matrix $Q^{-1}$, giving $t^2=0$ and $x^2=tx\ne0$.