Let S and S' be closed [Edit: orientable] surfaces, then it is well known that for S' to cover S it is necessary and sufficient that chi(S)|chi(S'). (Here 'chi' denotes the Euler characteristic).


However, if S and S' are punctured surfaces then the above condition is necessary but no longer sufficient. 



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> Is the question of determining necessary and sufficient criteria for S' to cover S answered in the research or expository literature?
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I think that I know such criteria and want to use them a paper that I'm working on but (a) may be deceiving myself and (b) want to know whether I should write up a proof anew or whether there's a suitable reference. Surely the relevant criteria have been rediscovered many times, but I've never seen them discussed in writing.


Edit: Thanks Pete, I should have demanded that my surfaces be orientable



Necessary and sufficient criteria for a surface to cover a surface
In a [recent paper](http://arxiv.org/abs/1003.0411) by Calegari, Sun, and Wang, the authors cite 


W.S. Massey, Finite covering spaces of 2-manifolds with boundary. Duke Math. J. 41 (1974),
no. 4, 875-887.


which proves that Dmitri's conditions (1) and (2) are sufficient if the S and S' are compact, orientable with non-empty boundary. 


(Dmitri's condition (3) is redundant as Massey points out in the second page of his article: writing chi(S') = 2 - 2g(S') - p(S') and chi(S) = 2 - 2g(S) - p(S) Dimitri's condition (1) gives


2 - 2g(S') - p(S') = d(2 - 2g(S) - p(S)) 


and reducing (mod 2) gives that p(S') and dp(S) have the same parity. But the proof therein is longer than the one that Dimitri for the case that he treats (if one takes the Ore result as given).


