= Mod-two cohomology ring of the antipodal two-sphere mapping torus
{title2=$\mathbb F_2[t,y]/(t^2,y^2),\quad|t|=1,\ |y|=2$}
Modulo two the <antipodal map> acts trivially on fiber <cohomology>. The <Wang sequence> supplies a class $y$ restricting to the generator of $H^2(S^2;\mathbb F_2)$. The <Leray-Hirsch theorem> with classes $1,y$ gives a basis $1,t,y,ty$, so $ty\ne0$. Pullback from the circle gives $t^2=0$, while dimension gives $y^2=0$. This <cohomology ring> agrees with that of $S^1\times S^2$, although the original manifold is nonorientable.
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