Editing Chain complex (section)

===Chain maps===
A '''chain map''' ''f'' between two chain complexes <math>(A_\bullet, d_{A,\bullet})</math> and <math>(B_\bullet, d_{B,\bullet})</math> is a sequence <math>f_\bullet</math> of homomorphisms <math>f_n : A_n \rightarrow B_n</math> for each ''n'' that commutes with the boundary operators on the two chain complexes, so <math> d_{B,n} \circ f_n = f_{n-1} \circ d_{A,n}</math>. This is written out in the following [[commutative diagram]].
:[[Image:Chain map.svg|650 px|class=skin-invert]]

A chain map sends cycles to cycles and boundaries to boundaries, and thus induces a map on homology <math>(f_\bullet)_*:H_\bullet(A_\bullet, d_{A,\bullet}) \rightarrow H_\bullet(B_\bullet, d_{B,\bullet})</math>.

A continuous map ''f'' between topological spaces ''X'' and ''Y'' induces a chain map between the singular chain complexes of ''X'' and ''Y'', and hence induces a map ''f''<sub>*</sub> between the singular homology of ''X'' and ''Y'' as well. When ''X'' and ''Y'' are both equal to the [[n-sphere|''n''-sphere]], the map induced on homology defines the [[Degree of a continuous mapping#From Sn to Sn|degree]] of the map ''f''.

The concept of chain map reduces to the one of boundary through the construction of the [[Mapping cone (homological algebra)|cone]] of a chain map.