Let $p:\tilde{X}\rightarrow X$ be a covering. If this is an unbranched covering, deck transformations are determined by their action on one point.

If this is a branched covering, is a deck transformation determined by its action on any point that is not a preimage of a branch point? That is, if one removes the branch points and their preimages, is any deck transformation determined by its action on one point?

And does anyone know of good resources that discuss lifting lemmas and deck transformation actions on branched covers? The resources on orbifold covering theory I've looked at don't seem to address these questions.


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    $\begingroup$ Remove the branch points from a branched covering and you have an unbranched cover so your question is answered by your first assertion. This is generally how it goes: branched covers are such a mild change from regular covering spaces that you adapt the tools from plain covering space theory to fit your situation, rather than state theorems for branched covers. In part this is because there are various types of branching you may or may not want to consider, given any application. $\endgroup$ – Ryan Budney Apr 13 '11 at 23:39
  • $\begingroup$ Don't you also need to be concerned how the branch points are permuted? $\endgroup$ – user641 Apr 13 '11 at 23:46
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    $\begingroup$ Thanks Ryan, I thought so, I just wasn't sure if I was missing a subtlety. And Steve, I think you do, but if you delete them, the deck transformation will take a neighborhood of the branch point to a neighborhood of a branch point which will determine how they are permuted by continuity. $\endgroup$ – Becca Winarski Apr 13 '11 at 23:54
  • $\begingroup$ @Ryan: but the homotopy theory of such branched covers is very different from the usual theory, right? For example there is no uniqueness in path lifting and hence there should be no connection between such covers and subgroups of the fundamental group?! Eg there are branched covers over simply connected spaces, I believe complex geometers like to look at branched double covers of a K3 Kummer surface. Do you know something about the deck transformations of a branched cover (for unramified covers given in terms of quotients of the fundamental group?) $\endgroup$ – mland Mar 13 '13 at 18:11
  • $\begingroup$ @mland: this wasn't a question about the homotopy theory of branched covers, that kind of takes things in a different direction. $\endgroup$ – Ryan Budney Mar 13 '13 at 18:27

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