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Homology Group

Definition: The term "homology group" usually means a singular homology group, which is an Abelian group which partially counts the number of holes in a topological space. In particular, singular homology groups form a measure of the hole structure of a space, but they are one particular measure and they don't always pick up everything. In addition, there are "generalized homology groups" which are not singular homology groups.
Source: http://mathworld.wolfram.com/HomologyGroup.html
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