Editing Homology (mathematics) (section)

=== 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]].