The action of the group $\Gamma=\mathbb{Z}$ on the manifold $\mathbb{C}^n-\{0\}$

Question

Let $\Gamma=\mathbb{Z}$ be the additive group of integers and give it the discrete topology. Suppose $\Gamma$ acts continuously on the topological n-manifold $\mathbb{C}^n-\{0\}$ by the map $x \mapsto 2x$. The action is free and properly discontinuous.

Claim: $M/\Gamma$ is homeomorphic to $\mathbb{S}^{2n-1}\times \mathbb{S}^1$

My Attempt: for n=1, the action is given by $z\mapsto 2z\mapsto2^2z\mapsto2^3z\mapsto\dots$

$n.z=2^nz$

which is basically the $\mathbb{Z}$ action on $\mathbb{R}^2$ and hence $M/\Gamma \simeq\mathbb{S}^1\times\mathbb{S}^1$. But I don't see how this idea really works for higher dimensions?

Answer 1

First note there is a change of coordinates $\mathbb R^{2n} \setminus \{0\} = S^{2n-1} \times \mathbb R_{>0}$ where the second coordinate is the radius. What does the group action look like in these coordinates?