# relative homology groups

If $X$ is a topological space, and $A$ a subspace, then the inclusion map $A\hookrightarrow X$ makes $C_{n}(A)$ into a subgroup of $C_{n}(X)$. Since the boundary map on $C_{*}(X)$ restricts to the boundary map on $C_{*}(A)$, we can take the quotient complex $C_{*}(X,A)$,

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The homology groups of this complex $H_{n}(X,A)$, are called the relative homology groups of the pair $(X,A)$. Under relatively mild hypotheses, $H_{n}(X,A)=H_{n}(X/A)$ where $X/A$ is the set of equivalence classes of the relation $x\sim y$ if $x=y$ or if $x,y\in A$, given the quotient topology (this is essentially $X$, with $A$ reduced to a single point). Relative homology groups are important for a number of reasons, principally for computational ones, since they fit into long exact sequences, which are powerful computational tools in homology.

Title relative homology groups RelativeHomologyGroups 2013-03-22 13:14:47 2013-03-22 13:14:47 bwebste (988) bwebste (988) 5 bwebste (988) Definition msc 55N10