[Math] Relative version of Kunneth Theorem for $CW$ pairs.


I am working on problem 3.2.12 in Hatcher's algebraic topology which asks to show that the spaces $(S^1 \times \mathbb{C}P^{\infty})/(S^1 \times \{x_0\})$ and $S^3 \times \mathbb{C}P^{\infty}$ have isomorphic cohomology rings.

I have found the generators of $H^*(S^3 \times \mathbb{C} P^{\infty})$. I want to show these spaces are isomorphic finding a homomorphism mapping generators to generators.

I would like to use a variant of Kunneth Formula on the cohomology ring $H^*(S^1 \times \mathbb{C}P^{\infty})/(S^1 \times \{x_0\})$ but I can't find anything straightforward in Hatcher describing this.

Is there a relative version of the Kunneth formulas for the $CW$ pair $(S^1 \times \mathbb{C} P^{\infty}, S^1 \times \{x_0\} )$?

Best Answer

The relative Kunneth formula gives (under appropriate hypotheses) an isomorphism $H^*(X,A)\otimes H^*(Y,B)\to H^*(X\times Y, A\times Y\cup X\times B)$ (or more generally, a short exact sequence that also involves a Tor term); see Theorem 3.18 in Hatcher. In your case, you can apply this with $(X,A)=(S^1,\emptyset)$ and $(Y,B)=(\mathbb{C}P^\infty,\{x_0\})$.

(Alternatively, you can just use the long exact sequence in cohomology for the pair $(S^1\times\mathbb{C}P^\infty,S^1\times\{x_0\})$. To use this, you need to understand the restriction map $H^*(S^1\times\mathbb{C}P^\infty)\to H^*(S^1\times\{x_0\})$, which you can compute using the naturality of the Kunneth isomorphism.)

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