Let $\mathcal{O}(-1)$ be the Hopf bundle over $\mathbb{C}\mathbb{P}^\infty$. 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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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?

Sorry if this question is more appropriate for Mathematics Stack Exchange.