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 $S^{-1} \coloneqq \emptyset$ 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.

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 $CW$-complex $X_{n-1}$ of dimension $n-1$;

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

3. 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 $X_n \simeq \coprod_{j \in Cell(X)_n} D^n \coprod_{j \in Cell(X)_n S^{n-1}} X_n$

$$\array{ \coprod_{j \in Cell(X)_{n}} S^{n-1} &\stackrel{(f_j)}{\to}& X_{n-1} \\ \downarrow && \downarrow \\ \coprod_{j \in Cell(X)_{n}} D^{n} &\to& X_{n} } \,.$$

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

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

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

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

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 $H_\bullet^{CW}(X) \in Ch_\bullet$ 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 $X_{n-1} \hookrightarrow X$:

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

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

  $$\partial^{CW}_n \colon H_{n+1}(X_{n+1}, X_n) \stackrel{\partial_n}{\to} H_n(X_n) \stackrel{i_n}{\to} H_n(X_n, X_{n-1}) \,,$$

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

Proposition

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

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, $\partial_{n} \circ \partial_{n+1} = 0$.

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

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

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 $\partial^{CW}_n$ 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

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

and is computed as the degree of a continuous function

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

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

Relation to singular homology

Theorem

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

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

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 $C_\bullet(X)$ 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

$$F_p C_\bullet(X) \coloneqq C_\bullet(X_p)$$

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

$$0 \hookrightarrow F_0 C_\bullet(X) \hookrightarrow F_1 C_\bullet(X) \hookrightarrow F_2 C_\bullet(X) \hookrightarrow \cdots \hookrightarrow F_\infty C_\bullet(X) \coloneqq C_\bullet(X) \,.$$

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

Write $\{E^r_{p,q}(X)\}$ 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.

Proposition

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

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

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

Proof

(…)

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

Corollary

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

Proof

By the third item of prop. 3 the $(r = 2)$-page of the spectral sequence $\{E^r_{p,q}(X)\}$ 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

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

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

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

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

Software

There are convenient software implementations for large-scale computations of cellular homology: one may use LinBox, CHomP or Perseus.

References

A standard textbook account is from p. 139 on in

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

Lecture notes include

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