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.
