Fundamental group of the following disc

Question

What is the fundamental group of the following space in $\mathbf C^n$?

This is the topological space given by $$\{(x_1,\ldots,x_n)\in \mathbf C^n-\{0\} : \vert x_1\vert < 1, \ldots, \vert x_n\vert <1\} - \{x_1\cdot \ldots \cdot x_n =0\}.$$

For $n=1$, I know it's $\mathbf Z$. I'm guessing it should be $\mathbf Z^n$ in general. Is this true? And why?

Answer 1

Your space deformation retracts to the $n$-torus $\prod_i\{\vert x_i\vert=\frac {1}{2} \}$, so has the same fundamental group : $$\mathbb Z^n $$