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. , 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. 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. .

Relation to the spectral sequence of the filtered singular complex

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:

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 $X_{p-1}$-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 $X_p$;

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

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

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. 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. 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

Early references on (co)homology

The original references on chain homology/cochain cohomology and ordinary cohomology in the form of cellular cohomology:

• Andrei Kolmogoroff, Über die Dualität im Aufbau der kombinatorischen Topologie, Recueil Mathématique 1(43) (1936), 97–102. (mathnet)

A footnote on the first page reads as follows, giving attribution to Alexander 35a, 35b:

Die Resultate dieser Arbeit wurden für den Fall gewöhnlicher Komplexe vom Verfasser im Frühling und im Sommer 1934 erhalten und teilweise an der Internationalen Konferenz für Tensoranalysis (Moskau) im Mai 1934 vorgetragen. Die hier dargestellte allgemeinere Theorie bildete den Gegenstand eines Vortrages, den der Verfasser an der Internationalen Topologischen Konferenz (Moskau, September 1935) hielt; bei letzterer Gelegenheit erfuhr er, dass ein grosser Teil dieser Resultate im Falle von Komplexen indessen von Herrn Alexander erhalten worden ist. Vgl. die inzwischen erschienenen Noten von Herrn Alexander in den «Proceedings of the National Academy of Sciences U.S.A.», 21, (1935), 509—512. Herr Alexander trug über seine Resultate ebenfalls an der Moskauer Topologischen Konferenz vor. Verallgemeinerungen für abgeschlossene Mengen und die Konstruktion eines Homologieringes für Komplexe und abgeschlossene Mengen, über welche der Verfasser ebenso an der Tensorkonferenz 1934 vorgetragen hat, werden in einer weiteren Publikation dargestellt. Diese weitere Begriffsbildungen sind übrigens ebenfalls von Herrn Alexander gefunden und teilweise in den erwähnten Noten publiziert.

• Andrei Kolmogoroff, Homologiering des Komplexes und des lokal-bicompakten Raumes, Recueil Mathématique 1(43) (1936), 701–705. mathnet.

• J. W. Alexander, On the chains of a complex and their duals, Proc. Nat. Acad. Sei. USA, 21(1935), 509–511 (doi:10.1073/pnas.21.8.50)

• J. W. Alexander, On the ring of a compact metric space, Proc. Nat. Acad. Sci. USA, 21 (1935), 511–512 (doi:10.1073/pnas.21.8.511)

• J. W. Alexander, On the connectivity ring of an abstract space, Ann. of Math., 37 (1936), 698–708 (doi:10.2307/1968484, pdf)

The term “cohomology” was introduced by Hassler Whitney in

• Hassler Whitney, On matrices of integers and combinatorial topology. Duke Mathematical Journal 3:1 (1937), 35–45 (doi:10.1215/S0012-7094-37-00304-1)

See also

• Hassler Whitney, On products in a complex, Annals of Math. 39 (1938) 397–432 (doi:10.2307/1968795)

The notion of singular cohomology is due to

• Samuel Eilenberg, Singular homology theory, Annals of Mathematics 45:3 (1944) (doi:10.2307/1969185)

The general abstract perspective on cohomology (subsuming sheaf cohomology, hypercohomology, non-abelian cohomology and indications of Whitehead-generalized cohomology) was essentially established in:

• Kenneth Brown, Abstract homotopy theory and generalized sheaf cohomology (1973)

but probably known in one form or other before that.

General

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)

Formulation in homotopy type theory:

• Ulrik Buchholtz, Kuen-Bang Hou (Favonia), Cellular Cohomology in Homotopy Type Theory, Logical Methods in Computer Science, Volume 16, Issue 2 (June 1, 2020) (arXiv:1802.02191, lmcs:6518)