TITLE: Meaning of relative homology
QUESTION [42 upvotes]: It is a bit easier to understand the homology $H_1(X, \mathbb Z)$ for various compact surfaces in analogy with handles and so on. There seems to be a nice intuitive picture with handles, holes, etc to think of the first homology group, and similar heuristics for higher homology groups.
But almost all axiomatic treatment of homology groups uses instead the relative homology. But it is not so intuitively clear how to visualize the relative homology groups.
What are some intuitive crutches for dealing with these relative homology groups, particularly for surfaces?

REPLY [21 votes]: This question was asked a long time ago. But, it may be still relevant.
Intuitively, relative homology of $H(K, K_0)$ is the homology of the space when we identify all the points that separate $K_0$ from $K$ to be a single point. Here $K_0$ is a subcomplex of $K$.
Figure from Relative Homology chapter Edelsbrunner book- p.107
For example, a relative $1-$cycle can result from:

Its all edges reside in the space $K-K_0$. This cycle is not affected by the relative homology computation.
It was not a cycle in $K-K_0$, but its two endpoints are in $K_0$.