Meaning of relative homology

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?

Solution 1:

See page 115 of Hatcher: Elements of $H_n(X,A)$ are represented by $n$-chains $\alpha \in C_n(X)$ such that $\partial \alpha \in C_{n-1}(A) \subset C_{n-1}(X)$. So you can think of an element of $H_n(X,A)$ as being an $n$-thing in $X$ whose boundary (an $(n-1)$-thing) lies in $A$.

Since the relative homology groups fit into the long exact sequence $$\cdots \to H_n(A) \to H_n(X) \to H_n(X,A) \to H_{n-1}(A) \to \cdots,$$ you can think of them as giving a crude measurement of, e.g., the non-surjectiveness or non-injectiveness of the maps $H_n(A) \to H_n(X)$.

Solution 2:

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:

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