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 $C_{*}(X,A)$ ,

$\begin{CD}@<{\partial}<{}<C_{n}(X)/C_{n}(A)@<{\partial}<{}<C_{n+1}(X)/C_{n+1}(% A)@<{\partial}<{}<\end{CD}$

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
Canonical name	RelativeHomologyGroups
Date of creation	2013-03-22 13:14:47
Last modified on	2013-03-22 13:14:47
Owner	bwebste (988)
Last modified by	bwebste (988)
Numerical id	5
Author	bwebste (988)
Entry type	Definition
Classification	msc 55N10