Integral cup products in the four-dimensional inversion mapping torus ($H^{\ast }(M_4;\mathbb{Z})$)

Choose the base class $a$, degree-two torsion classes ${\tau }_i=\hat{\beta }(x_i)$, and degree-two free lifts $b_{ij}$ reducing to $x_i{x}_j$. A generator $\kappa$ of the top $\mathbb{Z}/2$ reduces to $t{x}_1{x}_2{x}_3$. 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.