I want to compute the cohomology ring of the space $S^1 \times \mathbb{C} P^{\infty}/(S^1 \times\{x_0\})$.
By Künneth formula, the cohomology ring of $S^1 \times \mathbb{C}P^{\infty}$ is $\mathbb{Z}[\alpha, \beta]/(\beta^2,\alpha \beta)$,where $\alpha$ is the generator of $H^2(\mathbb{C}P^{\infty})$ and $\beta$ is a generator of circle. But how to compute the quotient space really puzzles me. I also know that there is a formula $H_i(S^1 \times X)=H_i(X)\oplus H_{i-1}(X)$. However I don’t know whether this formula helps? Maybe we can use the long exact sequence for pairs, but it’s still complicated to compute. Hope someone could help. Thanks!
Best Answer
Your space is the reduced suspension of $\mathbb{C}P^{\infty} \sqcup * $. This implies the multiplication is trivial, and Mayer-Vietoris says the additive structure of a suspension is just the cohomology shifted up.
The key geometric picture to have is that your space is the union of two cones on projective space, joined at the base and tip.