>>13828311A cycle in a homology group that is not a boundary.
Of course, whether this formal definition actually captures the various meanings that we intuitively assign to this word is another question. I think the answer to that, is clearly no, since formally speaking, a sock does not have a whole, at least under the definition used in mathematics, but intuitively, I think in most empirical/pragmatic contexts, we would be correct to say that a sock has a hole.
