Passing from Homology relative integers to $\mathbb{Z}_p$ loses topological information

Question

I just read that passing from Homology using the integers as cofficients to $\mathbb{Z}_p$ can mean embedding less topological information into the homology groups. Can somebody give me an example of a simplicial structure or CW-complex that whose homology groups lose information when these alternate coefficients are considered?

Answer 1

The Klein bottle and the torus.

The first isn't orientable, so its $2$nd integral homology group isn't isomorphic to $\Bbb Z$. The torus is orientable however...

Meanwhile if you use $\Bbb Z_2$ coefficients, their $2$nd homology groups are the same ($\Bbb Z_2$).