= Cohomology ring of the antipodal two-sphere mapping torus
{title2=$H^*(T_{-1}(S^2);\mathbb Z)$}
The <antipodal map> on the two-sphere reverses <orientation>, so its <mapping torus> is a nonorientable closed three-manifold. The <Wang sequence> gives $H_0=H_1=\mathbb Z$, $H_2=\mathbb Z/2$ and $H_3=0$. The <universal coefficient theorem for cohomology> therefore gives the <cohomology ring> $\mathbb Z[t,u]/(t^2,tu,u^2,2u)$ with $|t|=1$, $|u|=3$. The base-circle class satisfies $t^2=0$ by pullback; the other positive-degree products vanish for dimension reasons.
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