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I really don't think there is an actual definition of a hole in mathematics. It's just that if you have an image of n-dimensional sphere that cannot be shrunk to a point, then you have an n-dimensional hole, but where that hole is exactly and what it is, or how many holes there are, there is no good answer to. You might first look at the rank of homology or homotopy groups, but the torsion parts fuck everything up.
