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 $h_{n-1}A$).
Thanks so much in advance!