= Cohomology ring of a Hopf attachment with a sphere summand
{title2=$X_{d,f}=(S^2\vee S^3)\cup_{d\eta+f\iota_3}D^4$}
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.
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