I was having a hard time imagine the picture of a branched covering, the locally $z^k$ description is more "real" or "1-dimension" to me. Anybody have a good picture/example to visualize this?
2026-04-03 14:24:59.1775226299
Visualization of branched covering map between Riemann surface
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This picture (https://upload.wikimedia.org/wikipedia/commons/9/9c/Riemann_sqrt.svg) gives you a pretty good idea of what $z \mapsto z^2$ looks like near $z=0$. For the general case $z \mapsto z^k$, there will be $k$ distinct sheets away from the origin, but they will all come together at $0$.