Fundamental group of $\mathbb{C}^2\setminus \Delta?$

Question

Let $\Delta=\{(z,z): z\in \mathbb{C}\}$. What will be the fundamental group of $\mathbb{C}^2\setminus \Delta?$

I am unable to proceed at all. Please give me a hint.

Answer 1

By using a linear transformation, we see that $\Bbb C^2\setminus\Delta$ is homeomorphic to $\Bbb C^2\setminus(\Bbb C \times\{0\})$. This is homotopy equivalent to $\Bbb C\setminus\{0\}$ which in turn is homotopy equivalent to the circle $S^1$.