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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.

Thanks!

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    @mland: this wasn't a question about the homotopy theory of branched covers, that kind of takes things in a different direction.2013-03-13

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