The Riemann surface of an nth root is the n-sheeted covering w maps to w to the nth power

For each $w\in {\mathbb{C}}^{\times }$, let $V_w:=w^{n}B(1,1\mathrm{/}2)$ and define ${\rho }_w(u):=w\exp \text{\hspace{0.17em}⁣}(\mathrm{Log}(u\mathrm{/}{w}^n)\mathrm{/}n)$ for $u\in V_w$. Because $B(1,1\mathrm{/}2)\subseteq \mathbb{C}\setminus (-\mathrm{\infty },0]$, the principal logarithm is defined there, and ${\rho }_w(u{)}^n=w^n\exp (\mathrm{Log}(u\mathrm{/}{w}^n))=u$ while ${\rho }_w(w^n)=w$. So $\mathrm{\Psi }(w):=[{\rho }_w{]}_{w^n}$ is an $n$th-root germ over $w^n$.

The germ $\mathrm{\Psi }(w)$ lies on ${\mathcal{R}}_n$. By [L2], choose $\lambda \in \mathbb{C}$ with $e^{\lambda }=w$. Along the path $\gamma (t)=e^{nt\lambda }$ from $1$ to $w^n$, refine $[0,1]$ so that successive values of $e^{t\lambda }$ are close enough for the neighboring branches ${\rho }_{e^{t_j\lambda }}$ to agree on overlaps, exactly as in the logarithm-surface construction. This continues the principal root germ at $1$ to $\mathrm{\Psi }(w)$.

If $\mathrm{\Psi }(w)=\mathrm{\Psi }(v)$, then the two germs have the same base point and the same value there, so $w=v$. Hence $\mathrm{\Psi }$ is injective.

Let $\xi =[\rho {]}_z\in {\mathcal{R}}_n$ and put $w:=\rho (z)\in {\mathbb{C}}^{\times }$. Shrink the representative domain to a connected disc $U\subseteq V_w$. On $U$ the quotient $h:=\rho \mathrm{/}{\rho }_w$ is holomorphic and satisfies $h(u{)}^n=1$ for every $u\in U$. So $h(U)$ lies in the finite set of $n$th roots of unity. Since $U$ is connected and $h(z)=1$, the function $h$ is constantly $1$. Thus $\xi =\mathrm{\Psi }(w)$, and $\mathrm{\Psi }$ is surjective.

The inverse of $\mathrm{\Psi }$ is ${\mathrm{\Lambda }}_n$. In a chart $N(\rho ,U)$ one has ${\mathrm{\Lambda }}_n\circ {\varphi }_{\rho ,U}^{-1}(u)=\rho (u)$, so [L3] makes ${\mathrm{\Lambda }}_n$ holomorphic. Near any fixed $w_0\in {\mathbb{C}}^{\times }$, the chart $N({\rho }_{w_0},V_{w_0})$ satisfies ${\varphi }_{{\rho }_{w_0},V_{w_0}}(\mathrm{\Psi }(w))=w^n$, which is holomorphic in $w$, so $\mathrm{\Psi }$ is holomorphic. Therefore ${\mathrm{\Lambda }}_n$ is a biholomorphism.

For every $[\rho {]}_z\in {\mathcal{R}}_n$ one has $p_n([\rho {]}_z)=z=\rho (z{)}^n={\mathrm{\Lambda }}_n([\rho {]}_z{)}^n$, so under ${\mathrm{\Lambda }}_n$ the projection is the power map $w\mapsto w^n$. Its fibre over a nonzero point consists of the $n$ distinct roots $w{\zeta }_n^k$ for $0\le k<n$, so this is exactly the standard $n$-sheeted covering of ${\mathbb{C}}^{\times }$.