Editing Chain complex (section)

===Chain homotopy===
{{See also|Homotopy category of chain complexes}}
A chain homotopy offers a way to relate two chain maps that induce the same map on homology groups, even though the maps may be different. Given two chain complexes ''A'' and ''B'', and two chain maps {{nowrap|''f'', ''g'' : ''A'' → ''B''}}, a '''chain homotopy''' is a sequence of homomorphisms {{nowrap|''h''<sub>''n''</sub> : ''A''<sub>''n''</sub> → ''B''<sub>''n''+1</sub>}} such that {{nowrap|1=''hd''<sub>''A''</sub> + ''d''<sub>''B''</sub>''h'' = ''f'' − ''g''}}. The maps may be written out in a diagram as follows, but this diagram is not commutative.

:[[Image:Chain homotopy between chain complexes.svg|650 px|class=skin-invert]]

The map ''hd''<sub>''A''</sub> + ''d''<sub>''B''</sub>''h'' is easily verified to induce the zero map on homology, for any ''h''. It immediately follows that ''f'' and ''g'' induce the same map on homology. One says ''f'' and ''g'' are '''chain homotopic''' (or simply '''homotopic'''), and this property defines an [[equivalence relation]] between chain maps.

Let ''X'' and ''Y'' be topological spaces. In the case of singular homology, a [[homotopy]] between continuous maps {{nowrap|''f'', ''g'' : ''X'' → ''Y''}} induces a chain homotopy between the chain maps corresponding to ''f'' and ''g''. This shows that two homotopic maps induce the same map  on singular homology. The name "chain homotopy" is motivated by this example.