Complex structure of branched cover over Riemann surface [closed]

Suppose $$X$$ is a Riemann surface, $$Y$$ is a Hausdorff topological space and $$p: Y\to X$$ is a local homeomorphism. Then there is a unique complex structure on $$Y$$ such that $$p$$ is holomorphic. Now if $$\pi:Y\to X$$ is a branched cover, can we also lift a complex structure to $$Y$$, s.t. $$\pi$$ is holomorphic?

closed as unclear what you're asking by Moishe Kohan, Yanior Weg, Xander Henderson, Adrian Keister, José Carlos SantosMay 3 at 18:17

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• In the local homeomorphism case $\pi$ is a local chart making $Y$ a Riemann surface. Then it suffices to check what happens at isolated branch points where the degree is finite. – reuns Apr 21 at 18:19
• @reuns sorry can you explain how you check? Can you give an example? – Danny Apr 22 at 6:55
• You should first define what a "branched cover" means when $Y$ is a Hausdorff topological space. I know how to define it when $Y$ is a surface.... – Moishe Kohan Apr 23 at 1:32