Homotopy equivalence of torus and $\mathbb{C^2}$ without coordinate cross.

Question

I have difficulties with one intermediate result. I know it is right, but it is not obviously for me. I'm trying to prove homotopy equivalence of $\mathbb{C^2}\setminus\{(a,b)\in\mathbb{C^2}\mid a=0\vee b=0\}$ and torus $\mathbb{T}=S^1\times S^1$.

I would be happy if you give me some hints or if you tell me how to construct this homotopy equivalence.

Answer 1

Well, $S^1$ is clearly homotopy equivalent to $X = \mathbb{C}\setminus \{0\}$. Then $S^1 \times S^1$ is homotopy equivalent to $X \times X$, and $X \times X$ is precisely $\mathbb{C}^2$ without the coordinate cross.

Looking at this reasoning, it's not hard to build the equivalence explicitly.