In mathematics, the term homology, originally introduced in algebraic topology, has three primary, closely related usages. First, there is the homology of a chain complex, a sequence of abelian groups, called homology groups, which are regarded as fundamental invariants of the chain complex. Secondly, when one can associate a chain complex to a different mathematical object, one can also associate its homology to that object. Distinct procedures of associating chain complexes to a given object are grouped into homology theories. Finally, homology is important in the study of topological spaces. Under nice conditions in which distinct homology theories for a single topological space produce the same homology groups, one can define a single homology of a topological space. This last notion of homology is closely related to topological ideas frequently discussed in popular mathematics such as the holes of a surface or the cycles of a graph. There is also a related notion of the cohomology of a cochain complex, giving rise to various cohomology theories, in addition to the notion of the cohomology of a topological space.
Homology of chain complexes We start 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):
⋯ ⟶ C n + 1 ⟶ d n + 1 C n ⟶ d n C n − 1 ⟶ d n − 1 ⋯ {\displaystyle \cdots \longrightarrow C_{n+1}{\stackrel {d_{n+1}}{\longrightarrow }}C_{n}{\stackrel {d_{n}}{\longrightarrow }}C_{n-1}{\stackrel {d_{n-1}}{\longrightarrow }}\cdots } , such that the composition of any two consecutive maps is zero:
d n ∘ d n + 1 = 0. {\displaystyle d_{n}\circ d_{n+1}=0.}
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
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