Question on relative homology
Solution 1:
Hints:
- Show that the elements of $Z_0(X,Y)$ are just linear combinations of points of $X$.
- Show that $x-y\in B_0(X)$ if and only if there is path from $x$ to $y$.
Solution 2:
Use that relative homology splits over its path components and assume that $X$ has a single path-component. Then for any pair of chains $p\in C_0(X) = Z_0(X,Y)$ and $p^\prime \in C_0(Y)$ we can find an element $s\in Z_1(X)$ such that $\partial s = p-p^\prime$. Hence $[p] = [p^\prime]$ but $[p^\prime] = 0$.