(also nonabelian homological algebra)
Cellular homology is a very efficient tool for computing the ordinary homology groups of topological spaces which are CW complexes, based on the relative singular homology of their cell complex-decomposition and using degree-computations.
Hence cellular homology uses the combinatorial structure of a CW-complex to define, first a chain complex of celluar chains and then the corresponding chain homology. The resulting cellular homology of a CW-complex is isomorphic to its singular homology, hence to its ordinary homology as a topological space, and hence provides an efficient method for computing the latter.
For definiteness and to fix notation which we need in the following, we recall the definition of CW-complex. The actual definition of cellular homology is below.
For $n \in \mathbb{N}$ write
$S^n \hookrightarrow D^{n+1}$ for the continuous function that includes the $n$-sphere as the boundary of the $(n+1)$-disk.
Write furthermore $S^{-1} \coloneqq \emptyset$ for the 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.
A CW complex of dimension $(-1)$ is the empty topological space.
By induction, for $n \in \mathbb{N}$ a CW complex of dimension $n$ is a topological space $X_{n}$ obtained from
a $CW$-complex $X_{n-1}$ of dimension $n-1$;
an index set $Cell(X)_n \in Set$;
a set of continuous maps (the attaching maps) $\{ f_i \colon S^{n-1} \to X_{n-1}\}_{i \in Cell(X)_n}$
as the pushout $X_n \simeq \coprod_{j \in Cell(X)_n} D^n \coprod_{j \in Cell(X)_n S^{n-1}} X_n$
By this construction an $n$-dimensional CW-complex is canonical a filtered topological space with filter inclusion maps
the right vertical morphisms in these pushout diagrams.
A general 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
For the following a CW-complex is all this data: the chosen filtering with the chosen attaching maps.
We define “ordinary” cellular homology with coefficients in the group $\mathbb{Z}$ of integers. The analogous definition for other coefficients is immediate.
For $X$ a CW-complex, def. , its cellular chain complex $H_\bullet^{CW}(X) \in Ch_\bullet$ is the chain complex such that for $n \in \mathbb{N}$
the abelian group of chains is the relative singular homology group of $X_n \hookrightarrow X$ relative to $X_{n-1} \hookrightarrow X$:
the differential $\partial^{CW}_{n+1} \colon H_{n+1}^{CW}(X) \to H_n^{CW}(X)$ is the composition
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 $(X_n, \emptyset) \to (X_n, X_{n-1})$.
The composition $\partial^{CW}_{n} \circ \partial^{CW}_{n+1}$ of two differentials in def. is indeed zero, hence $H^{CW}_\bullet(X)$ is indeed a chain complex.
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, $\partial_{n} \circ \partial_{n+1} = 0$.
By the discussion at 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
a cellular $n$-chain is a singular $n$-chain required to sit in filtering degree $n$, hence in $X_n \hookrightarrow X$;
a 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 $X_{n-2} \hookrightarrow X$.
For every $n \in \mathbb{N}$ we have an isomorphism
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 $X_n$.
This is discussed at Relative homology - Homology of CW-complexes.
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
and is computed as the degree of a continuous function
where the inclusion $X_n - D^n \hookrightarrow X_n$ corresponds to the attaching map for the $i^{th}$ disk $D^n$.
For $X$ a CW-complex, its cellular homology $H^{CW}_\bullet(X)$ agrees with its singular homology $H_\bullet(X)$:
This appears for instance as (Hatcher, theorem 2.35). A proof is below as the proof of cor. .
The structure of a CW-complex on a topological space $X$, def. naturally induces on its singular simplicial complex $C_\bullet(X)$ the structure of a filtered chain complex:
For $X_0 \hookrightarrow X_1 \hookrightarrow \cdots \hookrightarrow X$ a CW complex, and $p \in \mathbb{N}$, write
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
(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.
We identify various of the pages of this spectral sequences with structures in singular homology theory.
$r = 0$ – $E^0_{p,q}(X) \simeq C_{p+q}(X_p)/C_{p+q}(X_{p-1})$ is the group of $X_{p-1}$-relative (p+q)-chains in $X_p$;
$r = 1$ – $E^1_{p,q}(X) \simeq H_{p+q}(X_p, X_{p-1})$ is the $X_{p-1}$-relative singular homology of $X_p$;
$r = 2$ – $E^2_{p,q}(X) \simeq \left\{ \array{ H_p^{CW}(X) & for\; q = 0 \\ 0 & otherwise } \right.$
$r = \infty$ – $E^\infty_{p,q}(X) \simeq F_p H_{p+q}(X) / F_{p-1} H_{p+q}(X)$.
(…)
This now directly implies the isomorphism between the cellular homology and the singular homology of a CW-complex $X$:
By the third item of prop. 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
for all $p,q$. By the third and fourth item of prop. this is equivalently
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 CW complex – singular homology.
There are convenient software implementations for large-scale computations of cellular homology: one may use LinBox, CHomP or Perseus.
A standard textbook account is from p. 139 on in
Lecture notes include
Last revised on February 22, 2019 at 12:22:51. See the history of this page for a list of all contributions to it.