Finding the fundamental group of a subspace of $\mathbb{R}^4$ Let $C^{\times} := \mathbb{C} \backslash \{0\}$ and $Z := \{(w, z) \in \left(\mathbb{C}^{\times}\right)^2 \ | \ w^n = z\}$ for some fixed $n \geq 1$. I'm trying to find the fundamental group of $Z$. My toolbox consists pretty much out of Seifert-van Kampen and deformation retracting to a CW-complex, but I don't know how to do that here because $Z$ is 4 dimensional. Here is my work so far:
For a given $z$, the solutions are $\{w_1, ..., w_n\}$ where $w_i$ has length $\sqrt[n]{w}$ and argument $\frac{2i}{n} \cdot\pi $, and the fundamental group of $\mathbb{C}^\times$ is $\mathbb{Z}$ because it deformation retracts unto $\{z \in \mathbb{C} \ | \ |z| = 1\}$.  

We can write $Z = \{(w, w^n) \mid w \in \mathbb{C}^{\times} \}$. Let $\alpha : \mathbb{C}^{\times} \to \mathbb{C}^{\times}, \alpha(w) = w^n$. We see that $Z$ is nothing else than the graph of $\alpha$.
But for any contiunuous map $\phi : X \to Y$ between topological spaces $X,Y$ the graph $G(\phi) = \{ (x,\phi(x)) \mid x \in X \} \subset X \times Y$ is homeomorphic to $X$. In fact, define $f : X \to G(\phi), f(x) = (x,\phi(x))$ and $g : G(\phi) \to X, g(x,\phi(x)) = x$. These are continuous maps (note that $g$ is the restriction of the projection $X \times Y \to X$) such that $g \circ f = id_X$ and $f \circ g = id_{G(\phi)}$.
Hence $Z \approx \mathbb{C}^{\times}$ and $\pi_1(Z) \approx \mathbb{Z}$.
