The integral <Künneth theorem> gives a natural short exact sequence containing tensor and Tor terms. Here all cohomology groups are free abelian, so the Tor terms vanish and the external cup product is a ring isomorphism. Thus
$$
H^*(\mathbb{CP}^2\times\mathbb{CP}^2;\mathbb Z)
\cong\mathbb Z[x,y]/(x^3,y^3),
\qquad |x|=|y|=2.
$$

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Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-114/2/b/solution