nLab
cellular homology

Idea

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.

Definition

CW-Complex

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

• $S^n\in$ Top for the standad $n$-sphere;

• $D^n\in$ Top for the standard $n$-disk;

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

Write furthermore $S^{-1}≔\varnothing$ 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.

Definition

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

1. a $\mathrm{CW}$-complex $X_{n-1}$ of dimension $n-1$;

2. an index set $\mathrm{Cell}(X{)}_n\in \mathrm{Set}$;

3. a set of continuous maps (the attaching maps) $\{f_i:S^{n-1}\to X_{n-1}{\}}_{i\in \mathrm{Cell}(X{)}_n}$

as the pushout $X_n\simeq {\coprod }_{j\in \mathrm{Cell}(X{)}_n}{D}^n{\coprod }_{j\in \mathrm{Cell}(X{)}_n{S}^{n-1}}{X}_n$

$$\begin{array}{ccc}\coprod _{j\in \mathrm{Cell}(X{)}_n}{S}^{n-1}& \stackrel{(f_j)}{\to }& X_{n-1}\\ \downarrow & & \downarrow \\ \coprod _{j\in \mathrm{Cell}(X{)}_n}{D}^n& \to & X_n\end{array}.$$\array{ \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} } \,.

By this construction an $n$-dimensional CW-complex is canonical a filtered topological space with filter inclusion maps

$$\varnothing \hookrightarrow X_0\hookrightarrow X_1\hookrightarrow \cdots \hookrightarrow X_{n-1}\hookrightarrow X_n$$\emptyset \hookrightarrow X_0 \hookrightarrow X_1 \hookrightarrow \cdots \hookrightarrow X_{n-1} \hookrightarrow X_n

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

$$\varnothing \hookrightarrow X_0\hookrightarrow X_1\hookrightarrow \cdots \hookrightarrow X.$$\emptyset \hookrightarrow X_0 \hookrightarrow X_1 \hookrightarrow \cdots \hookrightarrow X \,.

For the following a CW-complex is all this data: the chosen filtering with the chosen attaching maps.

Cellular homology

We define “ordinary” cellular homology with coefficients in the group $\mathbb{Z}$ of integers. The analogous definition for other coefficients is immediate.

Definition

For $X$ a CW-complex, def. 1, its cellular chain complex $H_{•}^{\mathrm{CW}}(X)\in {\mathrm{Ch}}_{•}$ 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 $X_{n-1}\hookrightarrow X$:

  $$H_n^{\mathrm{CW}}(X)≔H_n(X_n,X_{n-1}),$$H_n^{CW}(X) \coloneqq H_n(X_n, X_{n-1}) \,,

• the differential ${\partial }_{n+1}^{\mathrm{CW}}:H_{n+1}^{\mathrm{CW}}(X)\to H_n^{\mathrm{CW}}(X)$ is the composition

  $${\partial }_n^{\mathrm{CW}}:H_{n+1}(X_{n+1},X_n)\stackrel{{\partial }_n}{\to }{H}_n(X_n)\stackrel{i_n}{\to }{H}_n(X_n,X_{n-1}),$$\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}) \,,

  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 $(X_n,\varnothing )\to (X_n,X_{n-1})$.

Proposition

The composition ${\partial }_n^{\mathrm{CW}}\circ {\partial }_{n+1}^{\mathrm{CW}}$ of two differentials in def. 2 is indeed zero, hence $H_{•}^{\mathrm{CW}}(X)$ is indeed a chain complex.

Proof

On representative singular chains the morphism $i_n$ acts as the identity and hence ${\partial }_n^{\mathrm{CW}}\circ {\partial }_{n+1}^{\mathrm{CW}}$ acts as the double singular boundary, ${\partial }_n\circ {\partial }_{n+1}=0$.

Remark

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 $X_{n-2}\hookrightarrow X$.

Properties

Cellular chains

Proposition

For every $n\in \mathbb{N}$ we have an isomorphism

$$H_n^{\mathrm{CW}}(X)≔H_n(X_n,X_{n-1})\simeq \mathbb{Z}(\mathrm{Cell}(X{)}_n)$$H^{CW}_n(X) \coloneqq H_n(X_n, X_{n-1}) \simeq \mathbb{Z}(Cell(X)_n)

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.

Remark

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

$$\mathbb{Z}\cong H_{n+1}(D^{n+1},S^n)\to H_n(S^n)\to H_n(D^n,S^{n-1})\cong H_n(S^n)\cong \mathbb{Z}$$\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}

and is computed as the degree of a continuous function

$$S^n\stackrel{f_j}{\to }{X}_n\to X_n/(X_n-D^n)\cong D^n/S^{n-1}\cong S^n$$S^n \stackrel{f_j}{\to} X_n \to X_n/(X_n - D^n) \cong D^n/S^{n-1} \cong S^n

where the inclusion $X_n-D^n\hookrightarrow X_n$ corresponds to the attaching map for the $i^{\mathrm{th}}$ disk $D^n$.

Relation to singular homology

Theorem

For $X$ a CW-complex, its cellular homology $H_{•}^{\mathrm{CW}}(X)$ agrees with its singular homology $H_{•}(X)$:

$$H_{•}^{\mathrm{CW}}(X)\simeq H_{•}(X).$$H^{CW}_\bullet(X) \simeq H_\bullet(X) \,.

This appears for instance as (Hatcher, theorem 2.35). A proof is below as the proof of cor. 1.

Relation to the spectral sequence of the filtered singular complex

The structure of a CW-complex on a topological space $X$, def. 1 naturally induces on its singular simplicial complex $C_{•}(X)$ the structure of a filtered chain complex:

Definition

For $X_0\hookrightarrow X_1\hookrightarrow \cdots \hookrightarrow X$ a CW complex, and $p\in \mathbb{N}$, write

$$F_p{C}_{•}(X)≔C_{•}(X_p)$$F_p C_\bullet(X) \coloneqq C_\bullet(X_p)

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 $F_p{C}_{•}(X)\hookrightarrow F_{p+1}{C}_{•}(X)$ and these equip the singular chain complex $C_{•}(X)$ of $X$ with the structure of a bounded filtered chain complex

$$0\hookrightarrow F_0{C}_{•}(X)\hookrightarrow F_1{C}_{•}(X)\hookrightarrow F_2{C}_{•}(X)\hookrightarrow \cdots \hookrightarrow F_{\mathrm{\infty }}{C}_{•}(X)≔C_{•}(X).$$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) \,.

(If $X$ is of finite dimension $\dim X$ then this is a bounded filtration.)

Write $\{E_{p,q}^r(X)\}$ 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.

Proposition

• $r=0$ – $E_{p,q}^0(X)\simeq C_{p+q}(X_p)/C_{p+q}(X_{p-1})$ is the group of $X_{p-1}$-relatvive (p+q)-chains in $X_p$;

• $r=1$ – $E_{p,q}^1(X)\simeq H_{p+q}(X_p,X_{p-1})$ is the $X_{p-1}$-relative singular homology of $X_p$;

• $r=2$ – $E_{p,q}^2(X)\simeq \{\begin{array}{cc}{H}_p^{\mathrm{CW}}(X)& \mathrm{for}q=0\\ 0& \mathrm{otherwise}\end{array}$

• $r=\mathrm{\infty }$ – $E_{p,q}^{\mathrm{\infty }}(X)\simeq F_p{H}_{p+q}(X)/F_{p-1}{H}_{p+q}(X)$.

Proof

(…)

This now directly implies the isomorphism between the cellular homology and the singular homology of a CW-complex $X$:

Corollary

$$H_{•}^{\mathrm{CW}}(X)\simeq H_{•}(X)$$H^{CW}_\bullet(X) \simeq H_\bullet(X)

Proof

By the third item of prop. 3 the $(r=2)$-page of the spectral sequence $\{E_{p,q}^r(X)\}$ is concentrated in the $(q=0)$-row. This implies that all differentials for $r>2$ vanish, since their domain and codomain groups necessarily have different values of $q$. Accordingly we have

$$E_{p,q}^{\mathrm{\infty }}(X)\simeq E_{p,q}^2(X)$$E^\infty_{p,q}(X) \simeq E^2_{p,q}(X)

for all $p,q$. By the third and fourth item of prop. 3 this is equivalently

$$G_p{H}_p(X)\simeq H_p^{\mathrm{CW}}(X).$$G_p H_{p}(X) \simeq H^{CW}_p(X) \,.

Finally observe that $G_p{H}_p(X)\simeq H_p(X)$ by the definition of the filtering on the homology as $F_p{H}_p(X):-\mathrm{image}(H_p(X_p)\to H_p(X))$ and by standard properties of singular homology of CW complexes discusses at CW complex – singular homology.

References

A standard textbook account is from p. 139 on in

• Alan Hatcher, Algebraic topology, Cambridge Univ. Press 2002,

Lecture notes include

• Lisa Jeffrey, Homology of CW-complexes and Cellular homology (pdf)