# Hatcher Exercise 3.2.16

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.

• By the theorem (with $R=\Bbb Z$), we can identify $a$ with $\sum_ia_i\otimes b_i$.. – Berci Mar 19 at 21:36
• 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. – Connor Malin Mar 20 at 0:59
• @Berci Where am I supposed to use his hint "Apply Theorem 3.15 with coeﬃcients in various ﬁelds"? – No One Apr 3 at 18:30