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What is definition of branched covering in the page 10 of following paper ?

In Hatcher, Allen; Lochak, Pierre; Schneps, Leila, On the Teichmüller tower of mapping class groups, J. Reine Angew. Math. 521, 1-24 (2000). ZBL0953.20030..

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    $\begingroup$ I cannot access the paper, but it seems to me that the definition of branched covering is quite standardized. It is a map which is a homeomorphism on a (usually closed ) subset and a covering map (usually finite degree) on its complement. Is there any reason why it would differ from that? $\endgroup$
    – Nick L
    Commented May 11, 2022 at 19:04
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    $\begingroup$ I'm used to a slightly different definition than what @NickL suggests. I do not think the map has to be one-to-one on the branch set. en.wikipedia.org/wiki/Branched_covering $\endgroup$
    – Ryan Budney
    Commented May 11, 2022 at 19:52
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    $\begingroup$ For motivation, my "mental shortcut" definition is that branched covers look locally like non-constant complex analytic maps (of complex curves). The branch points are where the derivative is zero. $\endgroup$
    – Ryan Budney
    Commented May 12, 2022 at 5:26

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At the top of page ten of this paper the authors write "the standard two-sheeted branched covering of the sphere by the torus, branched over four points which become the four boundary circles of the (0,4) surface." This branched cover is elegantly illustrated on the cover of the book A Topological Picturebook by George Francis.

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    $\begingroup$ This branched covering is indeed an example of "a covering map everywhere except for a nowhere dense set known as the branch set," the definition (from Wikipedia en.wikipedia.org/wiki/Branched_covering) Ryan Budney gave (mathoverflow.net/questions/422318/…), so I think it's reasonable to assume that's the correct definition (since no other branched covering appears in the paper). $\endgroup$
    – Will Sawin
    Commented May 11, 2022 at 20:20
  • $\begingroup$ I can't understand why this picture is branded covering? Can you give me more details? $\endgroup$
    – 1200785626
    Commented May 15, 2022 at 13:55
  • $\begingroup$ I can’t give details at the level I think you require. Perhaps you will find useful the material in Section 1.1 of Stillwell’s book “Classical topology and combinatorial group theory”. $\endgroup$
    – Sam Nead
    Commented May 15, 2022 at 20:39