Is $1=e^{i2n\pi}$ always true?

Question

Suppose I have two Riemann sheets corresponding to the branches of $f(z)=z^{\frac{1}{2}}$, are the top and bottom sheets completely identical? i.e are $z_1$ and $z_2$ equal if $z_1$ and $z_2$ coincide as seen from the top of the Riemann surface? Or should we say that in the top sheet $z$ is of the form $re^{i\theta}$ where $\theta \in [0,2\pi)$ and in the bottom sheet $\theta \in [2\pi,4\pi)$? Because if the latter is true, $e^{2\pi i}$ would map to $-1$ and $e^{4\pi i}$ would map to $+1$, which are different points. And so, the points need to be "different".