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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?

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What definition of $H_k(X,\mathbb Z)$ and $H_k(X,A,\mathbb Z)$ are you using? Could you give us some details on why your intuition works in one case and not in the other? –  Ryan Budney Nov 25 '10 at 1:44
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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 $H_n(A) \to H_n(X)$.

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One way to think about this is to look at the chain level. Recall that the singular chains of a pair $C_n(X, Y)$ are sometimes defined as $C_n(X,Y) = C_n(X)/C_n(Y)$. So one way of thinking about this is that we are taking the space $X$ and attaching a cone to $Y$ so that any chain that lives in $Y$ can be shrunk to a point and forgotten about. Relative homology is then defined to be the relative cycles modulo the relative boundaries, i.e. we look at cycles AND chains living in $Y$, and we mod out not only by things that are boundaries, but things that are boundaries plus maybe a chain in $Y$.

This intuition works especially well for nice pairs $(X,Y)$ where $H_n(X,Y) = H_n(X/Y)$. In other words, we really are forgetting about all the stuff happening in $Y$ since we are shrinking it to a point.

A wonderful treatment of all of this is given in Hatcher's book- he's very good about providing intuition.

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First, let me say why all axiomatic treatments insist on talking about homology of pairs. One of the (most important/useful?) axioms is the Long exact sequence axiom, for any pair $A$ contained in $X$ we get the long exact sequence of homology groups. This is crucial for making computations. The concept of the homology of a pair is amazingly well behaved. Much better behaved than that of a quotient space, for homotopical purposes. So the best formulation of the axioms are in terms of pairs, because they require the least assumptions on the types of quotients you will allow.

As Dylan alludes to, we really want to think of $(X,A)$ as the cofiber of the inclusion $A \to X$. This cofiber construction (quotient of the mapping cylinder, or rather the mapping cone) is a much better behaved notion. Its homotopy type depends only on the homotopy type of the inclusion. This is not the case for the standard quotient, unless the subspace is "nice." For surfaces though, probably every subspace you are looking at is nice, so I think it would probably be okay if you just compute $H_*(X,A)$ as $H_*(X/A)$.

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