TITLE: Example where $H_1 (X, A) \not\cong H_1 (X / A)$
QUESTION [6 upvotes]: I'm trying to get a better understanding of relative homology, and I have a hard time visualizing the distinction between $H_1(X,A)$ and $H_1(X/A)$. I assume there's no way they are equal (otherwise why even define $H(X,A)$), but I can't come up with an example.

REPLY [8 votes]: Let $X=S^1$ and let $A=S^1\setminus\{p\}$ for some point $p\in S^1$.  Then $A$ is contractible, so $H_1(X,A)\cong H_1(X)\cong\mathbb{Z}$.  However, the quotient space $X/A$ has only two points, and in fact is contractible (that's a nice exercise).  So $H_1(X/A)$ is trivial.