# Exactness Axiom of Homology Theory

Axioms we are using for Homology Theory:

1) Homotopy: if $f$ and $g$ are homotopic, then $h_{n}(f) = h_{n}(g)$

2) exactness: each map $f:(X,A)\to (Y,B)$ gives us a commuting ladder of long exact sequences (the top bar of which I have included below in my question)

3) Excision: if $(X,A)$ is a pair and $C\subset A$ with the closure of $C$ contained in the interior of $A$, then the inclusion $e:(X-C,A-C)\to (X,A)$ induces an isomorphism $h_{n}(e):h_{n}(X,A)\to (Y,B)$.

Exactness axiom:

For each $f:(X,A)\to (Y,B)$ there is a commuting ladder of long exact sequences:

$\dots \to h_{n}(A,\phi)\to h_{n}(X,\phi)\to h_{n}(X,A)\to h_{n-1}A\to ...$

My question:

Based on my notes, I can't find a definition for $h_{n-1}A$, nor the map $h_{n}(X,A)\to h_{n-1}A$ (which is, however, labelled as $\partial_{(X,A)}$)

I browsed some online sources and found that it is refered to as a boundary map.

But what is its definition? (same question for the space $h_{n-1}A$).

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$h_{n-1}(A)$ means $h_{n-1}(A,\emptyset)$. This boundary map is what ties together $h_n$ with $h_{n-1}$ (otherwise they would be independent homotopy functors... we don't want that). What are the axioms you are using for a homology theory? –  Thomas Belulovich Apr 7 '12 at 21:39
We listed 6 axioms, but I think we are only using 3 (I will add them in the question right now.) –  Kyle Schlitt Apr 7 '12 at 21:43
As far as the question goes, it doesn't quite make sense - these are axioms, describing what a homology theory is, so defining these maps is something you do for a particular homology theory. Are you asking about what the boundary map is for, say, singular homology theory? –  Martin Wanvik Apr 7 '12 at 21:49
Also, i suspect that the terms $(X,\phi)$ and $(A,\phi)$ in the long exact sequence you've written down, are supposed to be $(X,\emptyset)$ and $(A,\emptyset)$. –  Martin Wanvik Apr 7 '12 at 21:53
@borninthe80s: Right, I see that I've misunderstood somewhat above - yes, I guess the fact that you have a "commuting ladder" for each map of pairs $(X,A) \to (Y,B)$ would indeed characterize the boundary maps for a homology theory. You will find a slightly different statement of the axioms here: en.wikipedia.org/wiki/Eilenberg-Steenrod_axioms (combining the exactness axiom from that page with the requirement that $\partial$ is a natural transformation implies the existence of a commutative ladder diagram for each map of pairs). –  Martin Wanvik Apr 7 '12 at 22:18

Let me try to answer the question "what is the boundary map $\partial: H_n(X, A) \to H_{n-1}(A)$?"

As noted in the comments, this is part of the axioms of a homology theory, so actually the official answer is that it's given to you as part of the homology theory. Obviously, this is not very satisfying, so instead let me answer the question "what is the boundary map in singular homology?"

The answer is that if your given homology theory is constructed as the homology of some chain complexes (which singular homology certainly is!), the boundary map is something you get for free for abstract nonsense reasons (i.e. diagram chasing).

Given a pair $(X, A)$, for each $k$ we have the following commuting diagram

$$\begin{array}{ccccccccc} 0 &\to& C_k(A) &\xrightarrow{i_k}& C_k(X) &\xrightarrow{q_k}& C_k(X)/C_k(A) &\to& 0 \\ & & \downarrow & & \downarrow & & \downarrow & & \\ 0 &\to& C_{k-1}(A) &\xrightarrow{i_{k-1}}& C_{k-1}(X) &\xrightarrow{q_{k-1}}& C_{k-1}(X)/C_{k-1}(A) & \to & 0 \end{array}$$ where $i_k$ and $q_k$ denote inclusion and quotient, respectively, and all the downward arrows are $\partial_k$. Suppose I have some element of $H_k(X, A)$. Then I can pick some relative cycle $\tilde{c} \in C_k(X)/C_k(A)$ representing it, and since $q_k$ is surjective I can pick some $c \in C_k(X)$ such that $q_k(c) = \tilde{c}$. Consider $\partial_k c \in C_{k-1}(X)$. By commutativity, of the diagram, $q_{k-1} \partial_k(c) = \partial_k q_k(c) = \partial_k \tilde{c} = 0$, so $\partial_k c \in \ker q_{k-1}$. By exactness of the bottom row, there is some $b_{k-1} \in C_{k-1}(A)$ such that $i_{k-1}b_{k-1} = \partial_k c_k$.

So let us define a map $H_k(X,A)$ by sending $[\tilde{c}]$ to $[b_{k-1}] \in H_{k-1}(A)$. A similar diagram chase shows that this map is well-defined (i.e. independent of all the choices we made). This is the well-known connecting homomorphism, which in homology we typically denote by $\partial_k: H_k(X,A) \to H_{k-1}(A)$. This construction is usually called the snake lemma.

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First of all, thanks very much for this. But I'm really quite lost. What is $C_{k}(\cdot)$ representing? –  Kyle Schlitt Apr 10 '12 at 21:56
$C_k(\cdot)$ are the chains, with boundary map $\partial: C_k(\cdot) \to C_{k-1}(\cdot)$, so that $H_k = \ker \partial_k / \mathrm{im} \partial_{k+1}$. They depend on what homology theory you are considering, but they are always there in the background somewhere. For the singular homology of a space $X$ they are formal linear combinations of maps $\Delta^k \to X$ where $\Delta^k$ is the standard $k$-simplex. For simplicial and cellular homology there are analogous definitions. Whenever you encounter any kind of homology you should think of it as the homology of some underlying chain complex. –  Jonathan Apr 12 '12 at 1:51