information:extraction, linguistic:voice_change, information:classification, linguistic:paraphrasing
Homology of Chain Complexes

To take the homology of a chain complex, one starts with a chain complex, which is a sequence (C ∙, d ∙) {\displaystyle (C_{\bullet },d_{\bullet })} of abelian groups C n {\displaystyle C_{n}} (whose elements are called chains) and group homomorphisms d n {\displaystyle d_{n}} (called boundary maps) such that the composition of any two consecutive maps is zero:

The n {\displaystyle n} th homology group H n {\displaystyle H_{n}} of this chain complex is then the quotient group H n = Z n / B n {\displaystyle H_{n}=Z_{n}/B_{n}} of cycles modulo boundaries, where the n {\displaystyle n} th group of cycles Z n {\displaystyle Z_{n}} is given by the kernel subgroup Z n:= ker d n:= { c ∈ C n | d n (c) = 0 } {\displaystyle Z_{n}:=\ker d_{n}:=\{c\in C_{n}\,|\;d_{n}(c)=0\}}, and the n {\displaystyle n} th group of boundaries B n {\displaystyle B_{n}} is given by the image subgroup B n:= i m d n + 1:= { d n + 1 (c) | c ∈ C n + 1 } {\displaystyle B_{n}:=\mathrm {im} \,d_{n+1}:=\{d_{n+1}(c)\,|\;c\in C_{n+1}\}}. One can optionally endow chain complexes with additional structure, for example by additionally taking the groups C n {\displaystyle C_{n}} to be modules over a coefficient ring R {\displaystyle R}, and taking the boundary maps d n {\displaystyle d_{n}} to be R {\displaystyle R} - module homomorphisms, resulting in homology groups H n {\displaystyle H_{n}} that are also quotient modules. Tools from homological algebra can be used to relate homology groups of different chain complexes.