I have the following problem:
Let $n>m>0$, show that every map $f:\mathbb{RP}^n\to\mathbb{RP}^m$ induces the trivial map on the fundamental groups.
I paste the given solution below:
Now, this solution looks wrong to me (or at least missing a lot of details). The reason is that the Smith sequence exists only if we take $\mathbb{Z}_2$ as coefficient ring, but we are trying to show this fact with coefficients in $\mathbb{Z}$ (where we have the Hurwics isomorphism, as stated).
Is there a way to save this solution as it is? Or an alternative proof? I tried working one out by myself, but I didn't find anything for the moment. I will update if I find something.