Cellular homology

Short description: Theory in algebraic topology

In mathematics, cellular homology in algebraic topology is a homology theory for the category of CW-complexes. It agrees with singular homology, and can provide an effective means of computing homology modules.

## Contents

* 1 Definition
* 2 Examples
* 2.1 The n-sphere
* 2.2 Genus g surface
* 2.3 Torus
* 2.4 Complex projective space
* 2.5 Real projective space
* 3 Functoriality
* 4 Other properties
* 5 Generalization
* 6 Euler characteristic
* 7 References
* 7.1 Notes
* 7.2 General References

## Definition

If X is a CW-complex with n-skeleton Xn, the cellular-homology modules are defined as the homology groups Hi of the cellular chain complex

⋯->Cn+1(Xn+1,Xn)->Cn(Xn,Xn−1)->Cn−1(Xn−1,Xn−2)->⋯,

where X−1 is taken to be the empty set.

The group

Cn(Xn,Xn−1)

is free abelian, with generators that can be identified with the n-cells of X. Let enα be an n-cell of X, and let χnα:∂enα≅𝕊n−1->Xn−1 be the attaching map. Then consider the composition

χnαβ:𝕊n−1⟶≅∂enα⟶χnαXn−1⟶qXn−1/(Xn−1∖en−1β)⟶≅𝕊n−1,

where the first map identifies 𝕊n−1 with ∂enα via the characteristic map Φnα of enα, the object en−1β is an (n−1)-cell of X, the third map q is the quotient map that collapses Xn−1∖en−1β to a point (thus wrapping en−1β into a sphere 𝕊n−1), and the last map identifies Xn−1/(Xn−1∖en−1β) with 𝕊n−1 via the characteristic map Φn−1β of en−1β.

The boundary map

∂n:Cn(Xn,Xn−1)->Cn−1(Xn−1,Xn−2)

is then given by the formula

∂n(enα)=∑βdeg⁡(χnαβ)en−1β,

where deg⁡(χnαβ) is the degree of χnαβ and the sum is taken over all (n−1)-cells of X, considered as generators of Cn−1(Xn−1,Xn−2).

## Examples

The following examples illustrate why computations done with cellular homology are often more efficient than those calculated by using singular homology alone.

### The n-sphere

The n-dimensional sphere Sn admits a CW structure with two cells, one 0-cell and one n-cell. Here the n-cell is attached by the constant mapping from Sn−1 to the 0-cell. Since the generators of the cellular chain groups Ck(Skn,Sk−1n) can be identified with the k-cells of Sn, we have that Ck(Skn,Sk−1n)=ℤ for k=0,n, and is otherwise trivial.

Hence for n>1, the resulting chain complex is

⋯⟶∂n+20⟶∂n+1ℤ⟶∂n0⟶∂n−1⋯⟶∂20⟶∂1ℤ⟶0,

but then as all the boundary maps are either to or from trivial groups, they must all be zero, meaning that the cellular homology groups are equal to

Hk(Sn)={ℤk=0,n{0}otherwise.

When n=1, it is possible to verify that the boundary map ∂1 is zero, meaning the above formula holds for all positive n.

### Genus g surface

Cellular homology can also be used to calculate the homology of the genus g surface Σg. The fundamental polygon of Σg is a 4n-gon which gives Σg a CW-structure with one 2-cell, 2n 1-cells, and one 0-cell. The 2-cell is attached along the boundary of the 4n-gon, which contains every 1-cell twice, once forwards and once backwards. This means the attaching map is zero, since the forwards and backwards directions of each 1-cell cancel out. Similarly, the attaching map for each 1-cell is also zero, since it is the constant mapping from S0 to the 0-cell. Therefore, the resulting chain complex is

⋯->0→∂3ℤ→∂2ℤ2g→∂1ℤ->0,

where all the boundary maps are zero. Therefore, this means the cellular homology of the genus g surface is given by

Hk(Σg)={ℤk=0,2ℤ2gk=1{0}otherwise.

Similarly, one can construct the genus g surface with a crosscap attached (a non-orientable genus g surface) as a CW complex with one 0-cell, g 1-cells {a1,⋯,ag}, and one 2-cell which is attached along the word a11⋯ag2. Therefore, the resulting chain complex is:

⋯->0→∂3ℤ→∂2ℤg→∂1ℤ->0,

where the boundary maps are ∂3=∂1=0 and ∂2(1)=2a1+2a2+⋯+2ag=2(a1+⋯+ag).

Its homology groups areHk(Ng)={ℤk=0ℤg−1⊕ℤ2k=1{0}otherwise.

### Torus

The n-torus (S1)n can be constructed as the CW complex with one 0-cell, n 1-cells, ..., and one n-cell. The chain complex is 0->ℤ(nn)->ℤ(nn−1)->⋯->ℤ(n1)->ℤ(n0)->0 and all the boundary maps are zero. This can be understood by explicitly constructing the cases for n=0,1,2,3, then see the pattern.

Thus, Hk((S1)n)≃ℤ(nk) .

### Complex projective space

If X has no adjacent-dimensional cells, (so if it has n-cells, it has no (n-1)-cells and (n+1)-cells), then HnCW(X) is the free abelian group generated by its n-cells, for each n.

The complex projective space ℂPn is obtained by gluing together a 0-cell, a 2-cell, ..., and a (2n)-cell, thus Hk(ℂPn)=ℤ for k=0,2,...,2n, and zero otherwise.

### Real projective space

The real projective space ℝPn admits a CW-structure with one k-cell ek for all k∈{0,1,…,n}. The attaching map for these k-cells is given by the 2-fold covering map φk:Sk−1->ℝPk−1. (Observe that the k-skeleton ℝPkn≅ℝPk for all k∈{0,1,…,n}.) Note that in this case, Ck(ℝPkn,ℝPk−1n)≅ℤ for all k∈{0,1,…,n}.

To compute the boundary map

∂k:Ck(ℝPkn,ℝPk−1n)->Ck−1(ℝPk−1n,ℝPk−2n),

we must find the degree of the map

χk:Sk−1⟶φkℝPk−1⟶qkℝPk−1/ℝPk−2≅Sk−1.

Now, note that φk−1(ℝPk−2)=Sk−2⊆Sk−1, and for each point x∈ℝPk−1∖ℝPk−2, we have that φ−1({x}) consists of two points, one in each connected component (open hemisphere) of Sk−1∖Sk−2. Thus, in order to find the degree of the map χk, it is sufficient to find the local degrees of χk on each of these open hemispheres. For ease of notation, we let Bk and B~k denote the connected components of Sk−1∖Sk−2. Then χk|Bk and χk|B~k are homeomorphisms, and χk|B~k=χk|Bk∘A, where A is the antipodal map. Now, the degree of the antipodal map on Sk−1 is (−1)k. Hence, without loss of generality, we have that the local degree of χk on Bk is 1 and the local degree of χk on B~k is (−1)k. Adding the local degrees, we have that

deg⁡(χk)=1+(−1)k={2if k is even,0if k is odd.

The boundary map ∂k is then given by deg⁡(χk).

We thus have that the CW-structure on ℝPn gives rise to the following chain complex:

0⟶ℤ⟶∂n⋯⟶2ℤ⟶0ℤ⟶2ℤ⟶0ℤ⟶0,

where ∂n=2 if n is even and ∂n=0 if n is odd. Hence, the cellular homology groups for ℝPn are the following:

Hk(ℝPn)={ℤif k=0 and k=n odd,ℤ/2ℤif 0<k<n odd,0otherwise.

## Functoriality

Cellular homology is a functor from the category of CW complexes with cellular maps to the category of abelian groups. A cellular map f:X->Y gives a map of pairs f:(Xn,Xn−1)->(Yn,Yn−1) for all n, and thus induces a map f*:Hn(Xn,Xn−1)->Hn(Yn,Yn−1) between the cellular chain groups of X and Y. That f* is a chain map follows from the naturality of the long exact sequence of a pair. Hence f* is a map between the cellular homology groups of X and Y.

The formula presented below allows one to compute the chain map f* in terms of the degrees of certain maps, similarly to the formula above for the boundary map in the cellular chain complex.[1] Let enα be an n-cell of X and enβ be an n-cell of Y.

Consider the composition

fβα:Sn⟶≅Dn/Sn−1⟶Φ¯nαXn/Xn−1⟶f¯Yn/(Yn∖enβ)⟶≅Sn,

where Φ¯nα is the quotient map obtained from the characteristic map of enα, and f¯ is the quotient map induced by the composition Xn⟶fYn->Yn/(Yn∖enβ). The last map comes from the characteristic map of enβ.

Then the chain map f*:Hn(Xn,Xn−1)->Hn(Yn,Yn−1) is determined by the formula

f*(enα)=∑βdeg⁡(fβα)enβ,

where the summation takes place over all n-cells of Y.

## Other properties

One sees from the cellular chain complex that the n-skeleton determines all lower-dimensional homology modules:

Hk(X)≅Hk(Xn)

for k<n.

An important consequence of this cellular perspective is that if a CW-complex has no cells in consecutive dimensions, then all of its homology modules are free. For example, the complex projective space ℂℙn has a cell structure with one cell in each even dimension; it follows that for 0≤k≤n,

H2k(ℂℙn;ℤ)≅ℤ

and

H2k+1(ℂℙn;ℤ)=0.

## Generalization

The Atiyah–Hirzebruch spectral sequence is the analogous method of computing the (co)homology of a CW-complex, for an arbitrary extraordinary (co)homology theory.

## Euler characteristic

For a cellular complex X, let Xj be its j-th skeleton, and cj be the number of j-cells, i.e., the rank of the free module Cj(Xj,Xj−1). The Euler characteristic of X is then defined by

χ(X)=∑j=0n(−1)jcj.

The Euler characteristic is a homotopy invariant. In fact, in terms of the Betti numbers of X,

χ(X)=∑j=0n(−1)jRank⁡(Hj(X)).

This can be justified as follows. Consider the long exact sequence of relative homology for the triple (Xn,Xn−1,∅):

⋯->Hi(Xn−1,∅)->Hi(Xn,∅)->Hi(Xn,Xn−1)->⋯.

Chasing exactness through the sequence gives

∑i=0n(−1)iRank⁡(Hi(Xn,∅))=∑i=0n(−1)iRank⁡(Hi(Xn,Xn−1))+∑i=0n(−1)iRank⁡(Hi(Xn−1,∅)).

The same calculation applies to the triples (Xn−1,Xn−2,∅), (Xn−2,Xn−3,∅), etc. By induction,

∑i=0n(−1)iRank⁡(Hi(Xn,∅))=∑j=0n∑i=0j(−1)iRank⁡(Hi(Xj,Xj−1))=∑j=0n(−1)jcj.

## References

### Notes

1. ↑ Switzer, Robert (1975). Algebraic Topology — Homotopy and Homology. Springer-Verlag. ISBN 3-540-42750-3. Proposition 10.13

### General References

* Albrecht Dold: Lectures on Algebraic Topology, Springer ISBN 3-540-58660-1.
* Allen Hatcher: Algebraic Topology, Cambridge University Press ISBN 978-0-521-79540-1. A free electronic version is available on the author's homepage.

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