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\begin{document}

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\section*{relative homology}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{homological_algebra}{}\paragraph*{{Homological algebra}}\label{homological_algebra}

[[!include homological algebra - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{in_singular_homology}{In singular homology}\dotfill \pageref*{in_singular_homology} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{LongExactSequences}{Long exact sequences}\dotfill \pageref*{LongExactSequences} \linebreak
\noindent\hyperlink{Excision}{Excision}\dotfill \pageref*{Excision} \linebreak
\noindent\hyperlink{HomotopyInvariance}{Homotopy invariance}\dotfill \pageref*{HomotopyInvariance} \linebreak
\noindent\hyperlink{RelationToQuotientTopologicalSpaces}{Relation to reduced homology of quotient topological spaces}\dotfill \pageref*{RelationToQuotientTopologicalSpaces} \linebreak
\noindent\hyperlink{RelationToReducedHomology}{Relation to reduced homology}\dotfill \pageref*{RelationToReducedHomology} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{basic_examples}{Basic examples}\dotfill \pageref*{basic_examples} \linebreak
\noindent\hyperlink{detecting_homology_isomorphisms}{Detecting homology isomorphisms}\dotfill \pageref*{detecting_homology_isomorphisms} \linebreak
\noindent\hyperlink{RelativeHomologyOfCWComplexes}{Relative homology of CW-complexes}\dotfill \pageref*{RelativeHomologyOfCWComplexes} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\hypertarget{in_singular_homology}{}\subsubsection*{{In singular homology}}\label{in_singular_homology}

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.

\begin{defn}
\label{}\hypertarget{}{}
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 \textbf{chain complex of $A$-relative singular chains}.

\begin{itemize}%
\item A [[boundary]] in this quotient is called an \textbf{$A$-relative singular boundary},


\item a [[cycle]] is called an \textbf{$A$-relative singular cycle}.


\item The [[chain homology]] of the quotient is the \textbf{$A$-relative singular homology of $X$}

\begin{displaymath}
H_n(X , A)\coloneqq H_n(C_\bullet(X)/C_\bullet(A))
  \,.
\end{displaymath}


\end{itemize}
\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
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 $A$''.

\end{remark}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{LongExactSequences}{}\subsubsection*{{Long exact sequences}}\label{LongExactSequences}

\begin{prop}
\label{RelativeHomologyLongExactSequence}\hypertarget{RelativeHomologyLongExactSequence}{}
Let $A \stackrel{i}{\hookrightarrow} X$. The corresponding relative homology sits in a [[long exact sequence]] of the form

\begin{displaymath}
\cdots
  \to 
  H_n(A)
   \stackrel{H_n(i)}{\to}
  H_n(X)
  \to 
  H_n(X, A)
   \stackrel{\delta_{n-1}}{\to}
  H_{n-1}(A)
   \stackrel{H_{n-1}(i)}{\to}
  H_{n-1}(X)
  \to 
  H_{n-1}(X, A)
  \to 
  \cdots
  \,.
\end{displaymath}
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]] $\partial^X c \in C_n(A) \hookrightarrow C_n(X)$.

\end{prop}
\begin{proof}
This is the \emph{[[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.

\end{proof}
\begin{prop}
\label{RelativeTripleHomologyLongExactSequence}\hypertarget{RelativeTripleHomologyLongExactSequence}{}
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

\begin{displaymath}
\cdots \to H_n(A , B) \to H_n(X , B) \to H_n(X , A ) \to H_{n-1}(A , B) \to \cdots
  \,.
\end{displaymath}
\end{prop}
\begin{proof}
Observe that we have a (degreewise) [[short exact sequence]] of chain complexes

\begin{displaymath}
0 \to C_\bullet(A)/C_\bullet(B) \to C_\bullet(X)/C_\bullet/B) \to C_\bullet(X)/C_\bullet(A) \to 0
  \,.
\end{displaymath}
The corresponding [[homology long exact sequence]] is the long exact sequence in question.

\end{proof}
\hypertarget{Excision}{}\subsubsection*{{Excision}}\label{Excision}

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 $A$: $\bar Z \hookrightarrow A^\circ$.

\begin{prop}
\label{ExcisionA}\hypertarget{ExcisionA}{}
In the above situation, the inclusion $(X-Z, A-Z) \hookrightarrow (X,A)$ induces [[isomorphism]] in relative singular homology groups

\begin{displaymath}
H_n(X-Z, A-Z) \stackrel{\simeq}{\to} H_n(X,A)
\end{displaymath}
for all $n \in \mathbb{N}$.

\end{prop}
Let $A,B \hookrightarrow X$ be two [[topological subspaces]] such that their [[interior]] is a [[cover]] $A^\circ \coprod B^\circ \to X$ of $X$.

\begin{prop}
\label{ExcisionB}\hypertarget{ExcisionB}{}
In the above situation, the inclusion $(B, A \cap B) \hookrightarrow (X,A)$ induces [[isomorphisms]] in relative singular homology groups

\begin{displaymath}
H_n(B, A \cap B) \stackrel{\simeq}{\to} H_n(X,A)
\end{displaymath}
for all $n \in \mathbb{N}$.

\end{prop}
A proof is spelled out in (\hyperlink{Hatcher}{Hatcher, from p. 128 on}).

\begin{remark}
\label{}\hypertarget{}{}
These two propositions are equivalent to each other. To see this identify $B = X - Z$.

\end{remark}
\hypertarget{HomotopyInvariance}{}\subsubsection*{{Homotopy invariance}}\label{HomotopyInvariance}

\begin{defn}
\label{HomotopyInvariance}\hypertarget{HomotopyInvariance}{}
Relative homology is homotopy invariant in both arguments.

\end{defn}
(\ldots{})

\hypertarget{RelationToQuotientTopologicalSpaces}{}\subsubsection*{{Relation to reduced homology of quotient topological spaces}}\label{RelationToQuotientTopologicalSpaces}

\begin{defn}
\label{GoodPair}\hypertarget{GoodPair}{}
A [[topological subspace]] inclusion $A \hookrightarrow X$ is called a \textbf{good pair} if

\begin{enumerate}%
\item $A$ is [[closed subset|closed]] inside $X$;


\item $A$ has an [[neighbourhood]] in $X$ which is a [[deformation retract]] of $A$.



\end{enumerate}
\end{defn}
\begin{example}
\label{}\hypertarget{}{}
For $X$ a [[CW complex]], the inclusion of any subcomplex $A \hookrightarrow X$ is a good pair (called a \emph{[[CW-pair]]} $(X,A)$).

\end{example}
This is discussed at \emph{\href{CW+complex#Subcomplexes}{CW complex -- Subcomplexes}}.

\begin{prop}
\label{HomologyOfQuotientSpace}\hypertarget{HomologyOfQuotientSpace}{}
If $A \hookrightarrow X$ is a [[topological subspace]] inclusion which is \emph{good} in the sense of def. \ref{GoodPair}, then the $A$-relative singular homology of $X$ coincides with the [[reduced singular homology]] of the [[quotient space]] $X/A$:

\begin{displaymath}
H_n(X , A)
  \simeq
  \tilde H_n(X/A)
 \,.
\end{displaymath}
\end{prop}
For instance (\hyperlink{Hatcher}{Hatcher, prop. 2.22}).

\begin{proof}
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. \ref{HomotopyInvariance}.

Given such $U$ we have an evident [[commuting diagram]] of pairs of [[topological spaces]]

\begin{displaymath}
\itexarray{
    (X,A) &\stackrel{(id,j)}{\to}& (X,U) &\leftarrow& (X-A, U - A)
    \\
    \downarrow && \downarrow && \downarrow^{\mathrlap{\simeq}}
    \\
    (X/A, A/A) &\stackrel{(id,j/A)}{\to}& (X/A, U/A) &\leftarrow& (X/A - A/A, U/A - A/A)
  }
  \,.
\end{displaymath}
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

\begin{displaymath}
\itexarray{
    H_n(X,A) &\underoverset{\simeq}{H_n(id,j)}{\to}& H_n(X,U) &\stackrel{\simeq}{\leftarrow}& H_n(X-A, U - A)
    \\
    \downarrow && \downarrow && \downarrow^{\mathrlap{\simeq}}
    \\
    H_n(X/A, A/A) &\underoverset{\simeq}{H_n(id,j/A)}{\to}& H_n(X/A, U/A) &\stackrel{\simeq}{\leftarrow}& H_n(X/A - A/A, U/A - A/A)
  }
  \,.
\end{displaymath}
Here the left horizontal morphisms are the above isomorphims induced from the deformation retract. The right horizontal morphisms are isomorphisms by prop. \ref{ExcisionA} 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).

\end{proof}
\hypertarget{RelationToReducedHomology}{}\subsubsection*{{Relation to reduced homology}}\label{RelationToReducedHomology}

\begin{prop}
\label{}\hypertarget{}{}
Let $X$ be a [[inhabited set|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 $X$

\begin{displaymath}
H_n(X , *) \simeq \tilde H_n(X)
  \,.
\end{displaymath}
\end{prop}
\begin{proof}
This is the special case of prop. \ref{HomologyOfQuotientSpace} for $A$ a point.

\end{proof}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{basic_examples}{}\subsubsection*{{Basic examples}}\label{basic_examples}

\begin{example}
\label{}\hypertarget{}{}
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 $n \in \mathbb{N}$

\begin{displaymath}
\tilde H_\bullet(S^n) \simeq H_\bullet(D^n, S^{n-1})
 \,.
\end{displaymath}
\end{example}
\begin{proof}
The $n$-[[sphere]] is [[homeomorphism|homeomorphic]] to the $n$-[[disk]] with its entire [[boundary]] identified with a point:

\begin{displaymath}
S^n \simeq D^n/S^{n-1}
  \,.
\end{displaymath}
Moreover the boundary inclusion is evidently a \emph{good pair} in the sense of def. \ref{GoodPair}. Therefore the example follows with prop. \ref{HomologyOfQuotientSpace}.

\end{proof}
\hypertarget{detecting_homology_isomorphisms}{}\subsubsection*{{Detecting homology isomorphisms}}\label{detecting_homology_isomorphisms}

\begin{example}
\label{}\hypertarget{}{}
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.

\end{example}
\begin{proof}
Under the given assumotion the long exact sequence in prop. \ref{RelativeHomologyLongExactSequence} secomposes into short exact pieces of the form

\begin{displaymath}
0 \to H_n(A) \to H_n(X) \to 0
  \,.
\end{displaymath}
Exactness says that the middle morphism here is an [[isomorphism]].

\end{proof}
\hypertarget{RelativeHomologyOfCWComplexes}{}\subsubsection*{{Relative homology of CW-complexes}}\label{RelativeHomologyOfCWComplexes}

Let $X$ be a [[CW-complex]] and write

\begin{displaymath}
X_0 \hookrightarrow X_1 \hookrightarrow X_2 \hookrightarrow \cdots \hookrightarrow X
\end{displaymath}
for its [[filtered topological space]]-structure with $X_{n+1}$ the topological space obtained from $X_n$ by gluing on $(n+1)$-cells.

\begin{prop}
\label{}\hypertarget{}{}
The relative singular homology of the filtering degrees is

\begin{displaymath}
H_n(X_k , X_{k-1})
  \simeq
  \left\{
    \itexarray{
       \mathbb{Z}[Cells(X)_n] & if\; k = n
       \\
       0 & otherwise
    }
  \right.
  \,,
\end{displaymath}
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.

\end{prop}
For instance (\hyperlink{Hatcher}{Hatcher, lemma 2.34}).

\begin{proof}
The inclusion $X_{k-1} \hookrightarrow X_k$ is clearly a \emph{good pair} in the sense of def. \ref{GoodPair}. 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. \ref{HomologyOfQuotientSpace} 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 \emph{\href{reduced+homology#RelationToWedgeSums}{Reduced homology - Respect for wedge sums}}.

\end{proof}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[relative cohomology]]


\item [[cellular homology]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

A standard textbook account for relative singular homology is section 2.1 of

\begin{itemize}%
\item [[Allen Hatcher]], \emph{\href{http://www.math.cornell.edu/~hatcher/AT/ATpage.html}{Algebraic Topology}}

\end{itemize}
[[!redirects relative singular homology]]



\end{document}