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\begin{document}

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\section*{cellular 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{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{cwcomplex}{CW-Complex}\dotfill \pageref*{cwcomplex} \linebreak
\noindent\hyperlink{CellularHomology}{Cellular homology}\dotfill \pageref*{CellularHomology} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{cellular_chains}{Cellular chains}\dotfill \pageref*{cellular_chains} \linebreak
\noindent\hyperlink{relation_to_singular_homology}{Relation to singular homology}\dotfill \pageref*{relation_to_singular_homology} \linebreak
\noindent\hyperlink{RelationToSpectralSequenceOfFilteredSingularComplex}{Relation to the spectral sequence of the filtered singular complex}\dotfill \pageref*{RelationToSpectralSequenceOfFilteredSingularComplex} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

Cellular homology is a very efficient tool for computing the [[ordinary homology|ordinary]] [[homology groups]] of [[topological spaces]] which are [[CW complexes]], based on the [[relative singular homology]] of their [[cell complex]]-decomposition and using [[degree of a continuous function|degree]]-computations.

Hence cellular homology uses the combinatorial structure of a CW-complex to define, first a [[chain complex]] of \emph{celluar chains} and then the corresponding [[chain homology]]. The resulting \emph{cellular homology} of a CW-complex is [[isomorphism|isomorphic]] to its [[singular homology]], hence to its [[ordinary homology]] as a topological space, and hence provides an efficient method for computing the latter.

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

\hypertarget{cwcomplex}{}\subsubsection*{{CW-Complex}}\label{cwcomplex}

For definiteness and to fix notation which we need in the following, we recall the definition of \emph{[[CW-complex]]}. The actual definition of cellular homology is \hyperlink{CellularHomology}{below}.

For $n \in \mathbb{N}$ write

\begin{itemize}%
\item $S^n \in$ [[Top]] for the standad $n$-[[sphere]];


\item $D^n \in$ [[Top]] for the standard $n$-[[disk]];


\item $S^n \hookrightarrow D^{n+1}$ for the [[continuous function]] that includes the $n$-sphere as the [[boundary]] of the $(n+1)$-disk.



\end{itemize}
Write furthermore $S^{-1} \coloneqq \emptyset$ for the [[empty set|empty]] topological space and think of $S^{-1} \to D^0 \simeq *$ as the boundary inclusion of the (-1)-sphere into the 0-disk, which is the [[point]].

\begin{defn}
\label{CWComplex}\hypertarget{CWComplex}{}
A \textbf{[[CW complex]] of [[dimension]] $(-1)$} is the [[empty set|empty]] [[topological space]].

By [[induction]], for $n \in \mathbb{N}$ a \textbf{[[CW complex]] of [[dimension]] $n$} is a [[topological space]] $X_{n}$ obtained from

\begin{enumerate}%
\item a $CW$-complex $X_{n-1}$ of dimension $n-1$;


\item an index set $Cell(X)_n \in Set$;


\item a set of [[continuous maps]] (the \textbf{attaching maps}) $\{ f_i \colon S^{n-1} \to X_{n-1}\}_{i \in Cell(X)_n}$



\end{enumerate}
as the [[pushout]] $X_n \simeq \coprod_{j \in Cell(X)_n} D^n \coprod_{j \in Cell(X)_n S^{n-1}} X_n$

\begin{displaymath}
\itexarray{
    \coprod_{j \in Cell(X)_{n}} S^{n-1} &\stackrel{(f_j)}{\to}& X_{n-1}
    \\
    \downarrow && \downarrow
    \\
    \coprod_{j \in Cell(X)_{n}} D^{n} &\to& X_{n} 
  }
  \,.
\end{displaymath}
By this construction an $n$-dimensional CW-complex is canonical a [[filtered topological space]] with filter inclusion maps

\begin{displaymath}
\emptyset \hookrightarrow X_0 \hookrightarrow X_1 \hookrightarrow \cdots \hookrightarrow X_{n-1} \hookrightarrow X_n
\end{displaymath}
the right vertical morphisms in these pushout diagrams.

A general \textbf{[[CW complex]]} $X$ is a [[topological space]] given as the [[sequential colimit]] over a [[tower]] [[diagram]] each of whose morphisms is such a filter inclusion

\begin{displaymath}
\emptyset \hookrightarrow X_0 \hookrightarrow X_1 \hookrightarrow \cdots \hookrightarrow X
  \,.
\end{displaymath}
\end{defn}
For the following a CW-complex is all this data: the chosen [[filtered topological space|filtering]] with the chosen attaching maps.

\hypertarget{CellularHomology}{}\subsubsection*{{Cellular homology}}\label{CellularHomology}

We define ``[[ordinary homology|ordinary]]'' cellular homology with [[coefficients]] in the [[group]] $\mathbb{Z}$ of [[integers]]. The analogous definition for other [[coefficients]] is immediate.

\begin{defn}
\label{CellularChainComplex}\hypertarget{CellularChainComplex}{}
For $X$ a [[CW-complex]], def. \ref{CWComplex}, its \textbf{cellular chain complex} $H_\bullet^{CW}(X) \in Ch_\bullet$ is the [[chain complex]] such that for $n \in \mathbb{N}$

\begin{itemize}%
\item the [[abelian group]] of [[chains]] is the [[relative singular homology]] group of $X_n \hookrightarrow X$ relative to $X_{n-1} \hookrightarrow X$:

\begin{displaymath}
H_n^{CW}(X) \coloneqq H_n(X_n, X_{n-1})
  \,,
\end{displaymath}

\item the [[differential]] $\partial^{CW}_{n+1} \colon H_{n+1}^{CW}(X) \to H_n^{CW}(X)$ is the [[composition]]

\begin{displaymath}
\partial^{CW}_n 
  \colon
  H_{n+1}(X_{n+1}, X_n) 
   \stackrel{\partial_n}{\to} 
  H_n(X_n) 
   \stackrel{i_n}{\to} 
  H_n(X_n, X_{n-1})
  \,,
\end{displaymath}
where $\partial_n$ is the [[boundary]] map of the [[singular chain complex]] and where $i_n$ is the morphism on [[relative homology]] induced from the canonical inclusion of pairs $(X_n, \emptyset) \to (X_n, X_{n-1})$.



\end{itemize}
\end{defn}
\begin{prop}
\label{}\hypertarget{}{}
The composition $\partial^{CW}_{n} \circ \partial^{CW}_{n+1}$ of two differentials in def. \ref{CellularChainComplex} is indeed zero, hence $H^{CW}_\bullet(X)$ is indeed a [[chain complex]].

\end{prop}
\begin{proof}
On representative singular [[chains]] the morphism $i_n$ acts as the identity and hence $\partial^{CW}_{n} \circ \partial^{CW}_{n+1}$ acts as the double singular boundary, $\partial_{n} \circ \partial_{n+1} = 0$.

\end{proof}
\begin{remark}
\label{}\hypertarget{}{}
By the discussion at \emph{\href{relative%20homology#RelationToQuotientTopologicalSpaces}{Relative homology - Relation to reduced homology of quotient spaces}} the relative homology group $H_n(X_n, X_{n-1})$ is isomorphic to the the [[reduced homology]] $\tilde H_n(X_n/X_{n-1})$ of $X_n/X_{n-1}$.

This implies in particular that

\begin{itemize}%
\item a \textbf{cellular $n$-chain} is a singular $n$-chain required to sit in filtering degree $n$, hence in $X_n \hookrightarrow X$;


\item a \textbf{cellular $n$-cycle} is a singular $n$-chain whose singular boundary is not necessarily 0, but is contained in filtering degree $(n-2)$, hence in $X_{n-2} \hookrightarrow X$.



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

\hypertarget{cellular_chains}{}\subsubsection*{{Cellular chains}}\label{cellular_chains}

\begin{prop}
\label{}\hypertarget{}{}
For every $n \in \mathbb{N}$ we have an [[isomorphism]]

\begin{displaymath}
H^{CW}_n(X)
  \coloneqq
  H_n(X_n, X_{n-1}) 
  \simeq
  \mathbb{Z}(Cell(X)_n)
\end{displaymath}
that the group of cellular $n$-chains with the [[free abelian group]] whose set of [[basis]] elements is the set of $n$-[[disks]] attached to $X_{n-1}$ to yield $X_n$.

\end{prop}
This is discussed at \emph{\href{relative%20homology#RelativeHomologyOfCWComplexes}{Relative homology - Homology of CW-complexes}}.

\begin{remark}
\label{}\hypertarget{}{}
Thus, each cellular differential $\partial^{CW}_n$ can be described as a [[matrix]] $M$ with [[integer]] entries $M_{i j}$. Here an index $j$ refers to the attaching map $f_j \colon S^n \to X_n$ for the $j^{th}$ disk $D^{n+1}$. The integer entry $M_{i j}$ corresponds to a map

\begin{displaymath}
\mathbb{Z} \cong H_{n+1}(D^{n+1}, S^n) \to H_n(S^n) \to H_n(D^n, S^{n-1}) \cong H_n(S^n) \cong \mathbb{Z}
\end{displaymath}
and is computed as the [[degree of a continuous function]]

\begin{displaymath}
S^n \stackrel{f_j}{\to} X_n \to X_n/(X_n - D^n) \cong D^n/S^{n-1} \cong S^n
\end{displaymath}
where the inclusion $X_n - D^n \hookrightarrow X_n$ corresponds to the attaching map for the $i^{th}$ disk $D^n$.

\end{remark}
\hypertarget{relation_to_singular_homology}{}\subsubsection*{{Relation to singular homology}}\label{relation_to_singular_homology}

\begin{theorem}
\label{}\hypertarget{}{}
For $X$ a [[CW-complex]], its cellular homology $H^{CW}_\bullet(X)$ agrees with its [[singular homology]] $H_\bullet(X)$:

\begin{displaymath}
H^{CW}_\bullet(X)
  \simeq
  H_\bullet(X)
  \,.
\end{displaymath}
\end{theorem}
This appears for instance as (\hyperlink{Hatcher}{Hatcher, theorem 2.35}). A proof is below as the proof of cor. \ref{CelluarEquivalentToSingularFromSpectralSequence}.

\hypertarget{RelationToSpectralSequenceOfFilteredSingularComplex}{}\subsubsection*{{Relation to the spectral sequence of the filtered singular complex}}\label{RelationToSpectralSequenceOfFilteredSingularComplex}

The structure of a [[CW-complex]] on a [[topological space]] $X$, def. \ref{CWComplex} naturally induces on its [[singular simplicial complex]] $C_\bullet(X)$ the structure of a [[filtered chain complex]]:

\begin{defn}
\label{}\hypertarget{}{}
For $X_0 \hookrightarrow X_1 \hookrightarrow \cdots \hookrightarrow X$ a [[CW complex]], and $p \in \mathbb{N}$, write

\begin{displaymath}
F_p C_\bullet(X) \coloneqq C_\bullet(X_p)
\end{displaymath}
for the [[singular chain complex]] of $X_p \hookrightarrow X$. The given [[topological subspace]] inclusions $X_p \hookrightarrow X_{p+1}$ induce [[chain map]] inclusions $F_p C_\bullet(X) \hookrightarrow F_{p+1} C_\bullet(X)$ and these equip the singular chain complex $C_\bullet(X)$ of $X$ with the structure of a bounded [[filtered chain complex]]

\begin{displaymath}
0 
  \hookrightarrow
  F_0 C_\bullet(X)
  \hookrightarrow
  F_1 C_\bullet(X)
  \hookrightarrow
  F_2 C_\bullet(X)
  \hookrightarrow 
  \cdots
  \hookrightarrow 
  F_\infty C_\bullet(X)
  \coloneqq
  C_\bullet(X)
  \,.
\end{displaymath}
(If $X$ is of finite [[dimension]] $dim X$ then this is a bounded filtration.)

Write $\{E^r_{p,q}(X)\}$ for the [[spectral sequence of a filtered complex]] corresponding to this filtering.

\end{defn}
We identify various of the pages of this spectral sequences with structures in singular homology theory.

\begin{prop}
\label{PagesInTheSpectralSequenceOfTheFilteredSingularComplex}\hypertarget{PagesInTheSpectralSequenceOfTheFilteredSingularComplex}{}
\begin{itemize}%
\item $r = 0$ -- $E^0_{p,q}(X) \simeq C_{p+q}(X_p)/C_{p+q}(X_{p-1})$ is the group of $X_{p-1}$-[[relative homology|relative (p+q)-chains]] in $X_p$;


\item $r = 1$ -- $E^1_{p,q}(X) \simeq H_{p+q}(X_p, X_{p-1})$ is the $X_{p-1}$-[[relative singular homology]] of $X_p$;


\item $r = 2$ -- $E^2_{p,q}(X) \simeq \left\{ \itexarray{ H_p^{CW}(X) & for\; q = 0 \\ 0 & otherwise }  \right.$


\item $r = \infty$ -- $E^\infty_{p,q}(X) \simeq F_p H_{p+q}(X) / F_{p-1} H_{p+q}(X)$.



\end{itemize}
\end{prop}
\begin{proof}
(\ldots{})

\end{proof}
This now directly implies the isomorphism between the cellular homology and the singular homology of a CW-complex $X$:

\begin{cor}
\label{CelluarEquivalentToSingularFromSpectralSequence}\hypertarget{CelluarEquivalentToSingularFromSpectralSequence}{}
\begin{displaymath}
H^{CW}_\bullet(X) \simeq H_\bullet(X)
\end{displaymath}
\end{cor}
\begin{proof}
By the third item of prop. \ref{PagesInTheSpectralSequenceOfTheFilteredSingularComplex} the $(r = 2)$-page of the spectral sequence $\{E^r_{p,q}(X)\}$ is concentrated in the $(q = 0)$-row. This implies that all [[differentials]] for $r \gt 2$ vanish, since their domain and codomain groups necessarily have different values of $q$. Accordingly we have

\begin{displaymath}
E^\infty_{p,q}(X) \simeq E^2_{p,q}(X)
\end{displaymath}
for all $p,q$. By the third and fourth item of prop. \ref{PagesInTheSpectralSequenceOfTheFilteredSingularComplex} this is equivalently

\begin{displaymath}
G_p H_{p}(X) \simeq H^{CW}_p(X)
  \,.
\end{displaymath}
Finally observe that $G_p H_p(X) \simeq H_p(X)$ by the definition of the filtering on the homology as $F_p H_p(X) \coloneq image(H_p(X_p) \to H_p(X))$ and by standard properties of singular homology of [[CW complexes]] discusses at \emph{\href{CW+complex#SingularHomology}{CW complex -- singular homology}}.

\hypertarget{software}{}\subsection*{{Software}}\label{software}

There are convenient software implementations for large-scale computations of cellular homology: one may use \href{http://www.linalg.org}{LinBox}, \href{http://chomp.rutgers.edu}{CHomP} or \href{http://www.sas.upenn.edu/~vnanda/perseus}{Perseus}.

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

A standard textbook account is from p. 139 on in

\begin{itemize}%
\item [[Allen Hatcher]], \emph{\href{http://www.math.cornell.edu/~hatcher/AT/ATpage.html}{Algebraic topology}}, Cambridge Univ. Press 2002,

\end{itemize}
Lecture notes include

\begin{itemize}%
\item Lisa Jeffrey, \emph{Homology of CW-complexes and Cellular homology} (\href{http://www.math.toronto.edu/~mat1300/oldnotes/cellular-homology.pdf}{pdf})

\end{itemize}
[[!redirects cellular chain complex]]

[[!redirects cellular cohomology]]

[[!redirects cellular cochain complex]]



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