Inversion mapping torus of a torus ($M_n=(T^{n-1}\times [0,1])/((x,0)\sim (-x,1))$)

This mapping torus fibers over the circle with fiber $T^{n-1}$ and inversion as monodromy. The Wang sequence computes its additive cohomology. Coordinate projections to the Klein bottle expose its mod-two cup products.

The base class $t$ and coordinate classes $x_i$ all have degree one. Pullback of the mod-two intersection pairing of the Klein bottle gives $x_i^2=t{x}_i$. The Leray-Hirsch theorem gives the basis consisting of square-free products $x_I$ and $t{x}_I$. Thus the additive dimensions match those of a torus, but the degree-one squares differ for $n\ge 2$.

For $r=n-1$, the Wang sequence gives an additive splittingThe kernel is free, so the splitting exists as groups. The map in even degree is zero; the map in odd degree is multiplication by minus two.

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.