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Is there a quick and simple proof of the fact that the homology group of a nice (say with piecewise smooth boundary) planar domain is free abelian with a basis corresponding to the holes in the domain? Can you direct me to literature that might be relevant, i.e., with explicit examples similar to this? In particular, I would like to know an exact condition under which the above fact is true. For instance, it seems that one cannot take simply an open subset of the plane, as it appears that at least the homotopy of general planar sets is an active research field today.

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If you already are at the point where you can talk about the "holes" in the domain, then you can apply Mayer-Vietoris to the union of the planar domain and discs capping off the holes. –  Andy Putman Oct 12 '10 at 5:22
    
Thanks a lot! The holes are defined using the ambient plane simply as the bounded connected components of the complement (I am not sure this is the definition you had in mind). I was thinking of an argument that can be given in a complex analysis class. There is a complex analysis argument to this but another topological argument would be nice if it can be presented in say 20 minutes. –  timur Oct 12 '10 at 5:31
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I suppose I was assuming the Schönflies theorem (which implies that each of those bounded components is homeomorphic to a disc). The homological corollary of the Schönflies theorem is an immediate corollary of Alexander duality, but given that I don't have any idea of the background of your students I can't recommend an easy way of teaching the special case you need. –  Andy Putman Oct 12 '10 at 5:49
    
Thanks! Can you direct me to literature that might be relevant, i.e., with explicit examples similar to this? I particular, I would like to know an exact condition under which the above fact is true. For instance, it seems that one cannot take simply an open subset of the plane, as it appears that at least the homotopy of general planar sets is an active research field today. –  timur Oct 12 '10 at 19:06

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