Editing Relative homology (section)

== Definition ==
Given a subspace <math>A\subseteq X</math>, one may form the [[short exact sequence]]

:<math>0\to C_\bullet(A) \to C_\bullet(X)\to 
C_\bullet(X) /C_\bullet(A)  \to 0 ,</math>

where <math>C_\bullet(X)</math> denotes the [[singular chain]]s on the space ''X''. The boundary map on <math>C_\bullet(X)</math> descends{{ref|a|a}} to <math>C_\bullet(A)</math>  and therefore induces a boundary map <math>\partial'_\bullet</math> on the quotient. If we denote this quotient by <math>C_n(X,A):=C_n(X)/C_n(A)</math>, we then have a complex
:<math>\cdots\longrightarrow C_n(X,A) \xrightarrow{\partial'_n} C_{n-1}(X,A) \longrightarrow \cdots .</math>

By definition, the '''{{var|n}}<sup>th</sup> relative homology group''' of the pair of spaces <math>(X,A)</math> is

:<math>H_n(X,A) := \ker\partial'_n/\operatorname{im}\partial'_{n+1}.</math>

One says that relative homology is given by the '''relative cycles''', chains whose boundaries are chains on ''A'', modulo the '''relative boundaries''' (chains that are homologous to a chain on ''A'', i.e., chains that would be boundaries, modulo ''A'' again).<ref>{{Cite book|title=Algebraic topology|first=Allen|last=Hatcher|authorlink=Allen Hatcher|date=2002|publisher=[[Cambridge University Press]]|isbn=9780521795401|location=Cambridge, UK|oclc=45420394}}</ref>