Homology (mathematics)

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In mathematics (especially algebraic topology and abstract algebra), homology (in Greek ὁμός homos "identical") is a certain general procedure to associate a sequence of abelian groups or modules with a given mathematical object such as a topological space or a group. See homology theory for more background, or singular homology for a concrete version for topological spaces, or group cohomology for a concrete version for groups.

For a topological space, the homology groups are generally much easier to compute than the homotopy groups, and consequently one usually will have an easier time working with homology to aid in the classification of spaces.

The original motivation for defining homology groups is the commonplace observation that one aspect of the shape of an object is its holes. But because a hole is "not there", it is not immediately obvious how to define a hole, or how to distinguish between different kinds of holes. Homology is a rigorous mathematical method for detecting and categorizing holes in a shape. (As it turns out, there exist subtle kinds of holes that homology cannot "see" -- in which case homotopy groups may be what is needed.)

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Construction of homology groups

The procedure works as follows. Given an object such as a topological space X\,, one first defines a chain complex C(X)\, encoding information about X\,. A chain complex is a sequence of abelian groups or modules C_0, C_1, C_2, \dots connected by homomorphisms  \partial_n \colon C_n \to C_{n-1}, which we call boundary operators. That is,

where 0\, denotes the trivial group and C_i\equiv0 for i\ < 0. We also require the composition of any two consecutive boundary operators to be zero. That is, for all n\,,

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