 Cohomology ring of a Hopf attachment with a sphere summand

For the Hopf map $\eta$, the indicated four-cell attachment has cellular boundary multiplication by $f$ from dimension four to three. Its integral cohomology is $\mathbb{Z}$ in degrees zero and two, $\ker (f:\mathbb{Z}\to \mathbb{Z})$ in degree three, and $\mathbb{Z}/f\mathbb{Z}$ in degree four. If $x$ generates degree two and $z$ is the four-cell cochain class, then $x^2=dz$, with every other positive-degree product zero. Collapsing the three-sphere reduces the cup-square calculation to the Hopf invariant; the induced map on degree-four cohomology reduces its integer coefficient modulo $f$. When $f=0$ there is an extra free degree-three class, whose products still vanish by dimension.