I want to show that $$f: \pi_1(\mathbb{C}-\{0\},1) \to \mathbb Z$$ with $$[a] \to n(a,0)$$ ($n(a,0)$ being the winding number of $a$ in $0$) is an isomorphism.
So basically, all I have to do is to show the following things:
1) $\pi_1(\mathbb{C}-\{0\},1)$ with $[a][b]=[ab]$ and $\mathbb Z$ with $+$ are groups.
2) $f$ is well-defined.
I've already shown 1) and 2), but now I'm having a problem with the other properties:
3) $f$ is injective.
Let $f([a])=f([b])$ for $[a],[b] \in \pi_1(\mathbb{C}-\{0\},1)$. This implies that the winding numbers of a and b are the same, so basically $n(a,0)=n(b,0)$. I know that for all closed curves the winding number is $0$. Does this already show 3)?
4) $f$ is surjective.
5) $f$ is a homomorphism.
Could someone help me with the last three properties?