fundamental groups of complement of lines in $\mathbb{C}^2$

Question

Consider the 2-dimensional complex vector space $\mathbb{C}^2$. $H_1$ and $H_2$ are the 1-dimensional subspace determined by $z_1+z_2=0$ and $z_1-z_2=0$ respectively. How to compute the fundamental group and homology groups of $\mathbb{C}^2\setminus(H_1\cup H_2)$ ? I need help. Thanks a lot!

Answer 1

As Daniel points out in his comment, you may replace $H_1$ and $H_2$ by the coordinate axes $z_1=0, z_2=0$.
Deleting these leaves the space $\mathbb C^*\times \mathbb C^*$ whose fundamental group $\pi_1(\mathbb C^*\times \mathbb C^*)$ and first homology group $H_1(\mathbb C^*\times \mathbb C^*,\mathbb Z)$ are $\mathbb Z\times \mathbb Z$.
The second homology group is $H_2(\mathbb C^*\times \mathbb C^*,\mathbb Z)=\mathbb Z$ and the higher homology groups are zero.

Answer 2

Another way is to find a deformation retraction of $\mathbb{C}^2\backslash (H_{1}\cup H_{2})$ onto $\mathbb{S}^{1}\times\mathbb{S}^{1}$ (not difficult), so their groups coincide.

Here again $H_{1}:z_{1}=0$, $H_{2}:z_{2}=0$.