Homology (mathematics)

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In mathematics, the term homology, originally introduced in algebraic topology, has three primary, closely related usages. First, there is the homology of a chain complex, a sequence of abelian groups, called homology groups, which are regarded as fundamental invariants of the chain complex. Secondly, when one can associate a chain complex to a different mathematical object, one can also associate its homology to that object. Distinct procedures of associating chain complexes to a given object are grouped into homology theories. Finally, homology is important in the study of topological spaces. Under nice conditions in which distinct homology theories for a single topological space produce the same homology groups, one can define a single homology of a topological space. This last notion of homology is closely related to topological ideas frequently discussed in popular mathematics such as the holes of a surface or the cycles of a graph. There is also a related notion of the cohomology...

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