Editing Chain complex (section)

===Singular homology===
{{main|Singular homology}}
Let ''X'' be a topological space. Define ''C''<sub>''n''</sub>(''X'') for [[Natural number|natural]] ''n'' to be the [[free abelian group]] formally generated by [[singular homology|singular n-simplices]] in ''X'', and define the boundary map <math>\partial_n: C_n(X) \to C_{n-1}(X)</math> to be

::<math>\partial_n : \, (\sigma: [v_0,\ldots,v_n] \to X) \mapsto (\sum_{i=0}^n (-1)^i \sigma: [v_0,\ldots, \hat v_i, \ldots, v_n] \to X)</math>

where the hat denotes the omission of a [[vertex (geometry)|vertex]]. That is, the boundary of a singular simplex is the alternating sum of restrictions to its faces. It can be shown that ∂<sup>2</sup> = 0, so <math>(C_\bullet, \partial_\bullet)</math> is a chain complex; the '''singular homology''' <math>H_\bullet(X)</math> is the homology of this complex.

Singular homology is a useful invariant of topological spaces up to [[homotopy#homotopy equivalence|homotopy equivalence]]. The degree zero homology group is a free abelian group on the [[connected space#Path connectedness|path-components]] of ''X''.