With the indicated points as basepoints, $X$ is the <smash product> $\mathbb{CP}^2\wedge\mathbb{CP}^2$. The quotient map identifies its positive-degree cohomology with the relative cohomology of the product modulo the wedge, which under the Künneth isomorphism is the ideal generated by $xy$. Thus
$$
\widetilde H^*(X;\mathbb Z)
=\mathbb Z\{a,b,c,d\},
\qquad
a=xy, b=x^2y, c=xy^2, d=x^2y^2,
$$
in degrees $4,6,6,8$. The only nonzero product of positive-degree basis elements is
$$
a^2=d.
$$
Together with the unit in degree zero, this determines the cohomology ring.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 114 / 2 / c / Solution

{title2=$X\wedge Y$}
{wiki}

For based spaces, the smash product is
$$
X\wedge Y=(X\times Y)/(X\vee Y),
$$
where the wedge $X\vee Y$ consists of pairs having at least one basepoint coordinate.


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