(also nonabelian homological algebra)
Let $X$ be a topological space and $A \hookrightarrow X$ a subspace. Write $C_\bullet(X)$ for the chain complex of singular homology on $X$ and $C_\bullet(A) \hookrightarrow C_\bullet(X)$ for the chain map induced by the subspace inclusion.
The cokernel of this inclusion, hence the quotient $C_\bullet(X)/C_\bullet(A)$ of $C_\bullet(X)$ by the image of $C_\bullet(A)$ under the inclusion, is the chain complex of $A$-relative singular chains.
A boundary in this quotient is called an $A$-relative singular boundary,
a cycle is called an $A$-relative singular cycle.
The chain homology of the quotient is the $A$-relative singular homology of $X$
This means that a singular $(n+1)$-chain $c \in C_{n+1}(X)$ is an $A$-relative cycle if its boundary $\partial c \in C_{n}(X)$ is, while not necessarily 0, contained in the $n$-chains of $A$: $\partial c \in C_n(A) \hookrightarrow C_n(X)$. So it vanishes only “up to contributions coming from $A$”.
Let $A \stackrel{i}{\hookrightarrow} X$. The corresponding relative homology sits in a long exact sequence of the form
The connecting homomorphism $\delta_{n} \colon H_{n+1}(X, A) \to H_n(A)$ sends an element $[c] \in H_{n+1}(X, A)$ represented by an $A$-relative cycle $c \in C_{n+1}(X)$, to the class represented by the boundary $\partial^X c \in C_n(A) \hookrightarrow C_n(X)$.
This is the homology long exact sequence induced by the given short exact sequence $0 \to C_\bullet(A) \stackrel{i}{\hookrightarrow} C_\bullet(X) \to coker(i) \simeq C_\bullet(X)/C_\bullet(A) \to 0$ of chain complexes.
Let $B \hookrightarrow A \hookrightarrow X$ be a sequence of two inclusions. Then there is a long exact sequence of relative homology groups of the form
Observe that we have a (degreewise) short exact sequence of chain complexes
The corresponding homology long exact sequence is the long exact sequence in question.
Let $Z \hookrightarrow A \hookrightarrow X$ be a sequence of topological subspace inclusions such that the closure $\bar Z$ of $Z$ is still contained in the interior $A^\circ$ of $A$: $\bar Z \hookrightarrow A^\circ$.
In the above situation, the inclusion $(X-Z, A-Z) \hookrightarrow (X,A)$ induces isomorphism in relative singular homology groups
for all $n \in \mathbb{N}$.
Let $A,B \hookrightarrow X$ be two topological subspaces such that their interior is a cover $A^\circ \coprod B^\circ \to X$ of $X$.
In the above situation, the inclusion $(B, A \cap B) \hookrightarrow (X,A)$ induces isomorphisms in relative singular homology groups
for all $n \in \mathbb{N}$.
A proof is spelled out in (Hatcher, from p. 128 on).
These two propositions are equivalent to each other. To see this identify $B = X - Z$.
Relative homology is homotopy invariant in both arguments.
(…)
A topological subspace inclusion $A \hookrightarrow X$ is called a good pair if
$A$ is closed inside $X$;
$A$ has an neighbourhood in $X$ which is a deformation retract of $A$.
For $X$ a CW complex, the inclusion of any subcomplex $A \hookrightarrow X$ is a good pair (called a CW-pair $(X,A)$).
This is discussed at CW complex – Subcomplexes.
If $A \hookrightarrow X$ is a topological subspace inclusion which is good in the sense of def. 3, then the $A$-relative singular homology of $X$ coincides with the reduced singular homology of the quotient space $X/A$:
For instance (Hatcher, prop. 2.22).
By assumption we can find a neighbourhood $A \stackrel{j}{\to} U \hookrightarrow X$ such that $A \hookrightarrow U$ has a deformation retract and hence in particular is a homotopy equivalence and so induces also isomorphisms on all singular homology groups.
It follows in particular that for all $n \in \mathbb{N}$ the canonical morphism $H_n(X,A) \stackrel{H_n(id,j)}{\to} H_n(X,U)$ is an isomorphism, by prop. 2.
Given such $U$ we have an evident commuting diagram of pairs of topological spaces
Here the right vertical morphism is in fact a homeomorphism.
Applying relative singular homology to this diagram yields for each $n \in \mathbb{N}$ the commuting diagram of abelian groups
Here the left horizontal morphisms are the above isomorphims induced from the deformation retract. The right horizontal morphisms are isomorphisms by prop. 3 and the right vertical morphism is an isomorphism since it is induced by a homeomorphism. Hence the left vertical morphism is an isomorphism (2-out-of-3 for isomorphisms).
Let $X$ be a inhabited topological space and let $x \colon * \hookrightarrow X$ any point. Then the relative singular homology $H_n(X , *)$ is isomorphic to the absolute reduced singular homology $\tilde H_n(X)$ of $X$
This is the special case of prop. 5 for $A$ a point.
The reduced singular homology of the $n$-sphere $S^{n}$ equals the $S^{n-1}$-relative homology of the $n$-disk with respect to the canonical boundary inclusion $S^{n-1} \hookrightarrow D^n$: for all $n \in \mathbb{N}$
The $n$-sphere is homeomorphic to the $n$-disk with its entire boundary identified with a point:
Moreover the boundary inclusion is evidently a good pair in the sense of def. 3. Therefore the example follows with prop. 5.
If an inclusion $A \hookrightarrow X$ is such that all relative homology vanishes, $H_\bullet(X , A) \simeq 0$, then the inclusion induces isomorphisms on all singular homology groups.
Under the given assumotion the long exact sequence in prop. 1 secomposes into short exact pieces of the form
Exactness says that the middle morphism here is an isomorphism.
Let $X$ be a CW-complex and write
for its filtered topological space-structure with $X_{n+1}$ the topological space obtained from $X_n$ by gluing on $(n+1)$-cells.
The relative singular homology of the filtering degrees is
where $Cells(X)_n \in Set$ denotes the set of $n$-cells of $X$ and $\mathbb{Z}[Cells(X)_n]$ denotes the free abelian group on this set.
For instance (Hatcher, lemma 2.34).
The inclusion $X_{k-1} \hookrightarrow X_k$ is clearly a good pair in the sense of def. 3. The quotient $X_k/X_{k-1}$ is by definition of CW-complexes a wedge sum of $k$-spheres, one for each element in $kCell$. Therefore by prop. 5 we have an isomorphism $H_n(X_k , X_{k-1}) \simeq \tilde H_n( X_k / X_{k-1})$ with the reduced homology of this wedge sum. The statement then follows by the respect of reduced homology for wedge sums as discussed at Reduced homology - Respect for wedge sums.
A standard textbook account for relative singular homology is section 2.1 of