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Let $\mathcal{O}(-1)$ be the Hopf bundle over $\mathbb{C}\mathbb{P}^\infty$. Let $\mathcal{O}$ be the trivial rank one bundle. Consider the projectivization of the rank two bundle $\mathcal{O}(-1)\oplus \mathcal{O}$. This is a bundle over $\mathbb{C}\mathbb{P}^\infty$ with fiber $\mathbb{C}\mathbb{P}^1$.

How to describe explicitly the cohomology ring of this space?

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    $\begingroup$ Here $\mathcal O$ is the trivial rank 1 bundle? So your sum has Chern class 1-x. The projective bundle formula (en.wikipedia.org/wiki/…) gives you that your total space has cohomology ring $\Bbb Z[x, \zeta]/(\zeta^2-x\zeta)$. If you instead meant the projectivization of $\mathcal O(-1) \oplus \mathcal O(1)$, the formula instead gives $\Bbb Z[x, \zeta]/(\zeta^2-x^2)$. In any case, the Gysin sequence implies that $H^*(E)$ is a free module of rank 2 over $\Bbb Z[x]$. $\endgroup$
    – mme
    Commented Dec 26, 2023 at 10:57
  • $\begingroup$ Yes, this is the trivial rank one bundle. Thank you. $\endgroup$
    – asv
    Commented Dec 26, 2023 at 11:05

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