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cellular homology

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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 nn \in \mathbb{N} write

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

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

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

Write furthermore S 1S^{-1} \coloneqq \emptyset for the empty topological space and think of S 1D 0*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)(-1) is the empty topological space.

By induction, for nn \in \mathbb{N} a CW complex of dimension nn is a topological space X nX_{n} obtained from

  1. a CWCW-complex X n1X_{n-1} of dimension n1n-1;

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

  3. a set of continuous maps (the attaching maps) {f i:S n1X n1} iCell(X) n\{ f_i \colon S^{n-1} \to X_{n-1}\}_{i \in Cell(X)_n}

as the pushout X n jCell(X) nD n jCell(X) nS n1X nX_n \simeq \coprod_{j \in Cell(X)_n} D^n \coprod_{j \in Cell(X)_n S^{n-1}} X_n

jCell(X) nS n1 (f j) X n1 jCell(X) nD n 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 nn-dimensional CW-complex is canonical a filtered topological space with filter inclusion maps

X 0X 1X n1X 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 XX is a topological space given as the sequential colimit over a tower diagram each of whose morphisms is such a filter inclusion

X 0X 1X. \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 XX a CW-complex, def. 1, its cellular chain complex H CW(X)Ch H_\bullet^{CW}(X) \in Ch_\bullet is the chain complex such that for nn \in \mathbb{N}

  • the abelian group of chains is the relative singular homology group of X nXX_n \hookrightarrow X relative to X n1XX_{n-1} \hookrightarrow X:

    H n CW(X)H n(X n,X n1), H_n^{CW}(X) \coloneqq H_n(X_n, X_{n-1}) \,,
  • the differential n+1 CW:H n+1 CW(X)H n CW(X)\partial^{CW}_{n+1} \colon H_{n+1}^{CW}(X) \to H_n^{CW}(X) is the composition

    n CW:H n+1(X n+1,X n) nH n(X n)i nH n(X n,X n1), \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 n\partial_n is the boundary map of the singular chain complex and where i ni_n is the morphism on relative homology induced from the canonical inclusion of pairs (X n,)(X n,X n1)(X_n, \emptyset) \to (X_n, X_{n-1}).

Proposition

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

Proof

On representative singular chains the morphism i ni_n acts as the identity and hence n CW n+1 CW\partial^{CW}_{n} \circ \partial^{CW}_{n+1} acts as the double singular boundary, n n+1=0\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 n1)H_n(X_n, X_{n-1}) is isomorphic to the the reduced homology H˜ n(X n/X n1)\tilde H_n(X_n/X_{n-1}) of X n/X n1X_n/X_{n-1}.

This implies in particular that

  • a cellular nn-chain is a singular nn-chain required to sit in filtering degree nn, hence in X nXX_n \hookrightarrow X;

  • a cellular nn-cycle is a singular nn-chain whose singular boundary is not necessarily 0, but is contained in filtering degree (n2)(n-2), hence in X n2XX_{n-2} \hookrightarrow X.

Properties

Cellular chains

Proposition

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

H n CW(X)H n(X n,X n1)(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 nn-chains with the free abelian group whose set of basis elements is the set of nn-disks attached to X n1X_{n-1} to yield X nX_n.

This is discussed at Relative homology - Homology of CW-complexes.

Remark

Thus, each cellular differential n CW\partial^{CW}_n can be described as a matrix MM with integer entries M ijM_{i j}. Here an index jj refers to the attaching map f j:S nX nf_j \colon S^n \to X_n for the j thj^{th} disk D n+1D^{n+1}. The integer entry M ijM_{i j} corresponds to a map

H n+1(D n+1,S n)H n(S n)H n(D n,S n1)H n(S n)\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 nf jX nX n/(X nD n)D n/S n1S nS^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 nD nX nX_n - D^n \hookrightarrow X_n corresponds to the attaching map for the i thi^{th} disk D nD^n.

Relation to singular homology

Theorem

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

H CW(X)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 XX, def. 1 naturally induces on its singular simplicial complex C (X)C_\bullet(X) the structure of a filtered chain complex:

Definition

For X 0X 1XX_0 \hookrightarrow X_1 \hookrightarrow \cdots \hookrightarrow X a CW complex, and pp \in \mathbb{N}, write

F pC (X)C (X p) F_p C_\bullet(X) \coloneqq C_\bullet(X_p)

for the singular chain complex of X pXX_p \hookrightarrow X. The given topological subspace inclusions X pX p+1X_p \hookrightarrow X_{p+1} induce chain map inclusions F pC (X)F p+1C (X)F_p C_\bullet(X) \hookrightarrow F_{p+1} C_\bullet(X) and these equip the singular chain complex C (X)C_\bullet(X) of XX with the structure of a bounded filtered chain complex

0F 0C (X)F 1C (X)F 2C (X)F 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 XX is of finite dimension dimXdim X then this is a bounded filtration.)

Write {E p,q r(X)}\{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=0r = 0E p,q 0(X)C p+q(X p)/C p+q(X p1)E^0_{p,q}(X) \simeq C_{p+q}(X_p)/C_{p+q}(X_{p-1}) is the group of X p1X_{p-1}-relatvive (p+q)-chains in X pX_p;

  • r=1r = 1E p,q 1(X)H p+q(X p,X p1)E^1_{p,q}(X) \simeq H_{p+q}(X_p, X_{p-1}) is the X p1X_{p-1}-relative singular homology of X pX_p;

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

  • r=r = \inftyE p,q (X)F pH p+q(X)/F p1H p+q(X)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 XX:

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

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

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

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

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

Finally observe that G pH p(X)H p(X)G_p H_p(X) \simeq H_p(X) by the definition of the filtering on the homology as F pH p(X)∶−image(H p(X p)H p(X))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

Lecture notes include

  • Lisa Jeffrey, Homology of CW-complexes and Cellular homology (pdf)
Revised on June 4, 2014 11:39:56 by Anonymous Coward (128.6.168.245)