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I am self-studying covering spaces of topological spaces. The following question comes to my mind.

In the case of topological covering spaces, we have a nice relation between the fundamental group of base space and that of total space, these are nicely given in Hatcher's Algebraic Topology book or any other algebraic topology books. We can do a classification of covering spaces using group theory.

Now my question is does these types of studies are done for branched covers? Especially in direction of the classification of branched covers via some algebraic techniques.

Can anyone suggest any references for these?

Thanks in advance.

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    $\begingroup$ This is a very large subject. For compact Riemann surfaces, it is part of Hurwitz theory, see e.g. Riemann Surfaces and Algebraic Curves: A First Course in Hurwitz Theory by Cavalieri and Miles for an accessible reference. $\endgroup$
    – Balazs
    Commented Apr 25, 2023 at 9:14

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