I am missing some knowledge about torsion and torsion-free groups that I need to understand an example (let's say I have not seen these expression before). We have the exact sequence of abelian groups:
$0 \to H_2(\mathbb{R} P^2) \to \mathbb{Z} \stackrel{2}{\to} H_1(\mathbb{R} P^1) \stackrel{i_*}{\to} H_1(\mathbb{R} P^2) \to \mathbb{Z}$
We can see that $H_2(\mathbb{R} P^2) = 0$. We also know that the rank of $H_2(\mathbb{R}P^2)$ and $H_1(\mathbb{R}P^2)$ are equal (and hence 0).
So I know from here that abelian groups of rank 0 are torsion. (i.e. each element has finite order).
The next statement is that exactness shows that $i_*$ is surjective (because $H_1(\mathbb{R}P^2)$ is torsion and $\mathbb{Z}$ is torsion-free. Can anyone shed some light on this statement? (Or just a wiki link!)