In the definition of relative homology group, $C_n(X,A)$ is defined to be the quotient group $C_n(X)/C_n(A)$, while $X$ is a space and $A\subset X$ is a subspace of $X$. It means chains in $A$ should be taken trivial in $C_n(X,A)$. I was wondering how should we treat a singular chain like the one in the picture. It sits both in $A$ and $X-A$. Any ideas? Thanks.
Best Answer
Well if it doesn't sit entirely within $A$, it's not killed in the quotient.
In your situation (if I understand the drawing well), though, if you take its boundary for instance, you see that the bottom face will completely die, and the two other faces still live inbetween $A$ and $X\setminus A$.
$C_*(X,A)$ is exactly $C_*(X)$ except that the chains which lie entirely in $A$ die.