I am trying to understand the solution given for the following problem here on this site:
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 $X \times Y.$
Here is the link to the solution given here:
Questions about Hatcher 3.2.16
Here are the two parts of the solution I do not understand:
1- If $p$ is a prime, the only way for $H^n(X)$ to have elements of order a power of $p$ is for $H_n(X)$ to also have elements of order a power of $p$, this time by the Universal Coefficient Theorem with $G=\mathbb{Z}$
My questions about this part is as follows:
Why is this statement true and how are we using this idea in the solution?
My questions about the comments is as follows:
Also, in the third comment, the advice was to calculate the integral cohomology groups? Why we are calculating those groups?
How can we implement the following advice given in the comments by John Palmieri "use the universal coefficient theorem to relate integral homology to rational and mod p cohomology"?
And finally, what are the correct logical sequence of steps for solving this problem?
Any help will be greatly appreciated!