Hatcehr Exercise 3.2.16.

Show that if $X$ and $Y$ are finite CW complexes such that $H^∗(X;\mathbb Z)$ and $H^∗(Y;\mathbb Z)$ contain no elements of order a power of a given prime $p$, then the same is true for $X×Y$ . [Apply Theorem 3.15 with coefficients in various fields.]

The hint says we can apply

Theorem 3.15. The cross product $H^∗(X;R)⊗_R H^∗(Y;R)→H^∗(X×Y;R)$ is an isomorphism of rings if $X$ and $Y$ are CW complexes and $H^k(Y;R)$ is a finitely generated free $R$-module for all $k$.

Take an element $a\in H^∗(X×Y)$ with order $p^n$. I need to produce some elements with orders powers of $p$ in $H^∗(X;\mathbb Z)$ and $H^∗(Y;\mathbb Z)$.

How can I use his hint? I am thinking about taking $R=\mathbb Z_p$ or $\operatorname{GF}(p^n)$ but I don't see the connection to the right solution.

  • $\begingroup$ By the theorem (with $R=\Bbb Z$), we can identify $a$ with $\sum_ia_i\otimes b_i$.. $\endgroup$ – Berci Mar 19 at 21:36
  • $\begingroup$ If you need more help, if you know all the orders of the $a_i \otimes b_i$ you can compute what the order of $\Sigma_i a_i \otimes b_i$ has to divide by Lagrange's theorem. $\endgroup$ – Connor Malin Mar 20 at 0:59
  • $\begingroup$ @Berci Where am I supposed to use his hint "Apply Theorem 3.15 with coefficients in various fields"? $\endgroup$ – No One Apr 3 at 18:30

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