= Integral cup products in the four-dimensional inversion mapping torus
{title2=$H^*(M_4;\mathbb Z)$}
Choose the base class $a$, degree-two torsion classes $\tau_i=\widehat\beta(x_i)$, and degree-two free lifts $b_{ij}$ reducing to $x_i x_j$. A generator $\kappa$ of the top $\mathbb Z/2$ reduces to $t x_1x_2x_3$. The <mod-two cohomology ring of an inversion mapping torus> determines products in top integral degree, since reduction there is injective. The products of distinct $b_{12},b_{13},b_{23}$ equal $\kappa$; so does $b_{ij}\tau_k$ when $i,j,k$ are all distinct. Their other degree-four products vanish. Also $a\tau_i=0$ because degree-three <integral cohomology> is torsion-free.
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