I'm reading something about Hopf manifolds. We define the Hopf manifold in the following way. Let $z\in \mathbb{C}$ a non-zero complex number which lies in the open unit disk centered at the origin. So we consider the action of $\mathbb{Z}$ in $\mathbb{C^m}\setminus\{0\}$ given by $$(v,k)\in \mathbb{C^m}\setminus\{0\}\times \mathbb{Z} \mapsto z^k v.$$ We say that $\mathbb{C}^m \setminus \{0\} / \mathbb{Z}$ is a Hopf manifold. At some point the author says that it's known it exists a diffeomorphism between $\mathbb{C}^m \setminus \{0\} / \mathbb{Z} $ and $S^{2m-1} \times S^1$.
How can it be build?
Consider the map $$\varphi : \mathbb{S}^1\times \mathbb{S}^{2n-1} \to X,\quad (t , x_1,y_1,...,x_n,y_n) \mapsto [z^{t}((x_1 + iy_1),...,(x_n + iy_n))] $$ with inverse $$\varphi^{-1} : X \to \mathbb{S}^1\times \mathbb{S}^{2n-1}, [((s_1 + it_1),...,(s_n + it_n))] \mapsto (q-k, |z|^{k-q}(s'_1,t'_1,..., s'_n,t'_n)), $$ where $q\in \mathbb{R}$ is such that $$ \sum_j |z|^q(s_j^2 + t_j^2) = 1 $$ and $k\in \mathbb{Z}$ such that $s_j = z^{k}s_j'$, $t_j = z^{k}t_j'$ and $q-k \in \mathbb{S}^1$.