Homology (mathematics)

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

Examples

The motivating example comes from algebraic topology: the simplicial homology of a simplicial complex X. Here An is the free abelian group or module whose generators are the n-dimensional

Related Topics:
Algebraic topology - Simplicial complex - Free abelian group

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oriented simplexes of X. The mappings are called the boundary mappings and send the simplex with vertices

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:(a, a, ..., a)

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to the sum of

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:(−1)i (a, ..., a, a, ..., a) from i = 0 to i = n.

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If we take the modules to be over a field, then the dimension

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of the n-th homology of X turns out to be the number of "holes" in X at dimension n.

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Using this example as a model, one can define a simplicial homology for any topological space X. We define a chain complex for X by taking An to be the free abelian group (or free module) whose generators are all continuous maps from n-dimensional simplices into X. The homomorphisms dn arise from the boundary maps of simplices.

Related Topics:
Topological space - Continuous - Simplices

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In abstract algebra, one uses homology to define derived functors, for example the Tor functors. Here one starts with some covariant additive functor F and some module X. The chain complex for X is defined as follows: first find a free module F1 and a surjective homomorphism p1 : F1 -> X. Then one finds a free module F2 and a surjective homomorphism p2 : F2 -> ker(p1). Continuing in this fashion, a sequence of free modules Fn and homomorphisms pn can be defined. By applying the functor F to this sequence, one obtains a chain complex; the homology Hn of this complex depends only on F and X and is, by definition, the n-th derived functor of F, applied to X.

Related Topics:
Abstract algebra - Derived functor - Tor functor - Surjective

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