Editing Homology (mathematics) (section)

== Types of homology ==
The different types of homology theory arise from functors mapping from various categories of mathematical objects to the category of chain complexes. In each case the composition of the functor from objects to chain complexes and the functor from chain complexes to homology groups defines the overall homology functor for the theory.<ref>{{Harvnb|Spanier|1966|p=156}}</ref>

=== Simplicial homology ===
{{main|Simplicial homology}}

The motivating example comes from [[algebraic topology]]: the '''[[simplicial homology]]''' of a [[simplicial complex]] ''X''. Here the chain group ''C<sub>n</sub>'' is the [[free abelian group]] or [[free module]] whose generators are the ''n''-dimensional oriented simplexes of ''X''. The orientation is captured by ordering the complex's [[vertex (geometry)|vertices]] and expressing an oriented simplex <math>\sigma</math> as an ''n''-tuple <math>(\sigma[0], \sigma[1], \dots, \sigma[n])</math> of its vertices listed in increasing order (i.e. <math>\sigma[0] < \sigma[1] < \cdots < \sigma[n]</math> in the complex's vertex ordering, where <math>\sigma[i]</math> is the <math>i</math>th vertex appearing in the tuple). The mapping <math>\partial_n</math> from ''C<sub>n</sub>'' to ''C<sub>n−1</sub>'' is called the {{em|boundary mapping}} and sends the simplex
: <math>\sigma = (\sigma[0], \sigma[1], \dots, \sigma[n])</math>
to the [[formal sum]]
: <math>\partial_n(\sigma) = \sum_{i=0}^n (-1)^i \left (\sigma[0], \dots, \sigma[i-1], \sigma[i+1], \dots, \sigma[n] \right ),</math>\
which is evaluated as 0 if <math>n = 0.</math> This behavior on the generators induces a homomorphism on all of ''C<sub>n</sub>'' as follows. Given an element <math>c \in C_n</math>, write it as the sum of generators <math display="inline">c = \sum_{\sigma_i \in X_n} m_i \sigma_i,</math> where <math>X_n</math> is the set of ''n''-simplexes in ''X'' and the ''m<sub>i</sub>'' are coefficients from the ring ''C<sub>n</sub>'' is defined over (usually integers, unless otherwise specified). Then define
: <math>\partial_n(c) = \sum_{\sigma_i \in X_n} m_i \partial_n(\sigma_i).</math>

The dimension of the ''n''-th homology of ''X'' turns out to be the number of "holes" in ''X'' at dimension ''n''. It may be computed by putting [[Matrix (mathematics)|matrix]] representations of these boundary mappings in [[Smith normal form]].

=== Singular homology ===
{{main|Singular homology}}

Using simplicial homology example as a model, one can define a ''singular homology'' for any [[topological space]] ''X''. A chain complex for ''X'' is defined by taking ''C<sub>n</sub>'' to be the free abelian group (or free module) whose generators are all [[Continuous function (topology)|continuous]] maps from ''n''-dimensional [[Simplex|simplices]] into ''X''. The homomorphisms ∂<sub>''n''</sub> arise from the boundary maps of simplices.

=== Group homology ===
{{Main|Group cohomology}}

In [[abstract algebra]], one uses homology to define [[derived functor]]s, for example the [[Tor functor]]s. 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 <math>F_1</math> and a [[surjective]] homomorphism <math>p_1 : F_1 \to X.</math> Then one finds a free module <math>F_2</math> and a surjective homomorphism <math>p_2 : F_2 \to \ker\left(p_1\right).</math> Continuing in this fashion, a sequence of free modules <math>F_n</math> and homomorphisms <math>p_n</math> can be defined. By applying the functor ''F'' to this sequence, one obtains a chain complex; the homology <math>H_n</math> of this complex depends only on ''F'' and ''X'' and is, by definition, the ''n''-th derived functor of ''F'', applied to ''X''.

A common use of group (co)homology <math>H^2(G, M)</math> is to classify the possible [[Group extension|extension groups]] ''E'' which contain a given ''G''-module ''M'' as a [[normal subgroup]] and have a given [[quotient group]] ''G'', so that <math>G = E / M.</math>

=== Other homology theories ===

{{div col}}
* [[Borel–Moore homology]]
* [[Cellular homology]]
* [[Cyclic homology]]
* [[Hochschild homology]]
* [[Floer homology]]
* [[Intersection homology]]
* [[K-homology]]
* [[Khovanov homology]]
* [[Morse homology]]
* [[Persistent homology]]
* [[Steenrod homology]]
{{Div col end}}