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Romin Abdolahzadi 11.29.14

1 CW Complexes

CW complexes are spaces that are built out of successive gluings of Euclidean

closed n-discs along their boundaries. Formally speaking, define Dn to be

Dn = {x ∈ Rn | ||x|| ≤ 1}

This will be the standard n-disc that we will always be working with. To

phrase the description of cell complexes differently, we will give an inductive

description. Start with a set X 0 of 0-discs, or points. Now attach to this space

a collection of 1-discs, or line segments. In other words, if D10, . . . , D1

α is the

collection of 1-discs to be attached, then set X 1 to be the disjoint union

X 1 = X 0 D10 D1

1 . . . D1α

But note that if we proceeded in this manner, then our spaces would be rather

uninteresting as any such space would just be a disjoint union of n-discs. For

instance, we would never be able to retrieve the circle S 1. Morally we would

want to build S 1 as, say, a 1-disc attached to a discrete set of two points acting

as endpoints. One would therefore be gluing the boundary of the single 1-disc

to the endpoints. So gluing is need. Thus, “attach a collection of 1-discs” really

should entail, for every n-disc Dnγ in the collection, a continuous map

φnγ : ∂D

nγ = S

n−1γ → X

n−1

This map is called an attaching map; it tells you how to glue the disc onto

the previous space along its boundaries. Hence, the real definition should be

X 1 = [X 1 D10 D1

1 . . . D1α]/[∀x ∈ ∂Dn

γ : x ∼ φnγ (x)]

via the quotient topology. Continuing in this way yields a space called a CW-

complex X where each nth part in the definition is called the n-skeleton X n.

Since the interiors enγ of each n-disc Dnγ is not being glued anywhere, we can

write X n as a set via the expression

X n = X n−1

γ

enγ

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We call the open interior discs enγ of Dn

γ the n-cells of X . Thus what is really

happening is that a CW complex is a space that is built by successively attaching

n-cells. To see some pictures of CW complexes, look at this answer by Ronnie

Brown: here.

Definition 1.1. Start with a finite set X 0 of points endowed with the discrete

topology. Suppose that you already have X n−1. Start with a collection of

n-discs Dn0 , . . . , Dn

α. To each such disc pick a continuous map

φnγ : ∂Dn

γ = S n−1γ → X n−1

Define

X n = [X n−1 Dn0 . . . Dn

α

]/[∀x ∈ ∂Dn

γ

: x ∼ φn

γ

(x)]

via the quotient topology. Let X =

n X n, and endow it with the topology

such that A ⊂ X is open iff A ∩ X n is open for each n. Equivalently A ⊂ X

is closed iff A ∩ X n is closed for each n. Then X is called a CW complex,

and any space that can be homeomorphically retrieved in this inductive fashion

is also called a CW complex. Each φnγ is called an attaching map of X , and

each interior enγ := ∂ Dn

γ is called an n-cell of X . If X = X n for some n, then

X is said to be finite dimensional and of dimension n.

We will use the CW complexes to motivate and speak of homology.

2 Idea of Homology

One means of solving and understanding mathematical phenomenon involves

developing invariants that allow us to distinguish between the different math-

ematical objects. Homotopy and homology are such invariants for spaces. We

have already seen the notion of a fundamental group functor π1. Furthermore,

we know that π1(S 1) = Z. But π1(S n) = 0 for n ≥ 2. Thus it seems to

be the case that the fundamental group does a good job detecting holes in 1-

dimensional objects such as S 1, but not holes in n-dimensional objects such as

S n for n ≥ 2. One can complain that maybe this is because the fundamental

group encodes the maps I → X up to homotopy, and the unit interval I is 1-

dimensional. This suggests to emulate the definition of the fundamental group

for maps I n → X for higher n, up to homotopy. Indeed, this is how one derives

the nth-homotopy group functors πn for each n. And indeed, these πn do indeed

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http://www.math.stackexchange.com/questions/476582/characteristic-map-of-a-n-cell-in-a-cw-complex

http://www.math.stackexchange.com/questions/476582/characteristic-map-of-a-n-cell-in-a-cw-complex

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detect holes in n-dimensional objects as the following shows:

πi(S n) = 0 for i ∈ 1, . . . , n − 1

πi(S n) = Z for i = n

But there are a couple problems with homotopy groups.

- πn(S 2) = 0 for infinitely many n

- homotopy is hard to compute

Hence if we were given an arbitrary space X that was secretly 2-dimensional

(but we did not know it to be so), then it might be the case that the higher

homotopy groups πn tell us that there are higher dimensional holes when infact there aren’t. Secondly, homotopy is hard to compute. This is supposedly

because we do not have the nice machinery such as excision (to be studied

later) that allows us to more readily compute. Another reason that it is hard

to compute is because the homotopy groups πn for n ≥ 2 are not directly

computable from a cell-structure.

On the other hand,

H n(S n) = 0 for i ∈ {1, . . . , n − 1}

H n(S n) = Z for i = n

H n(S n

) = 0 for i > n

and

- homology is easier to compute

- homology is directed related to the cell structure

Let us look at these claims through some examples of CW complexes. We

will compute the homology for the spaces X 1, X 2, X 3, X 4 located on page 99 of

Hatcher.

[Attempt to understand abelianization of chains ]

[Ideally I will get a better understanding later ]Before we begin we want to make some informal observations. We let cycles

be informally defined to be embeddings of n-spheres into the space. Call these

embeddings n-loops. Sometimes we can compose these embeddings as is done

with the case of ordinary loops for n = 1 to form the fundamental group.

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http://www.math.cornell.edu/~hatcher/AT/ATchapters.html

http://www.math.cornell.edu/~hatcher/AT/ATchapters.html

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And sometimes we cannot, as issues of basepoint arise. Nevertheless, we want

an algebraic structure which will capture the notion of composition of n-loops

irrespective of basepoints. Furthermore, we want to be able to think of the same

n-loop without reference to the basepoint at which it is formed.

Assume that our spaces are path-connected. One thing to note is that if α1

and α2 are the same n-loops realized at different basepoints, then there exists

an automorphism of πn that sends α1 to α2. Furthermore, this automorphism

is by a conjugation via a path between the basepoints. So, in some sense, α1

and α2 commute via a path. This might suggest some sort of abelianization

(vague from Hatcher... understand this...).

The abelianization of πn corresponds to the n-cycles (I don’t exactly know

how; confer last vagueness comment). One can retrieve the n-cycles as follows.

Take the free abelian group generated by the (n − 1)-cells? Cycles arise as

kernels of some maps between these free abelian chain groups? Something like

this, I think? I give up... Will come back to this later!

2.1 Computing homology for simple CW complexes

We first work it out for X 1. Define

C 0(X 1) = frAb(x, y) 0-chains of 0-cells

C 1(X 1) = frAb(a,b,c,d) 1-chains of 1-cells

There exists a chain of maps

0 ∂ 2−→ C 1(X 1)

∂ 1−→ C 0(X 0) ∂ 0−→ 0

wherein ∂ 1 is defined by a, b, c, d → y −x. These ∂ i are called boundary maps.

The reason for this is presumably because their kernels are generated by cycles

that are boundaries of holes, as we will see below. When possible, define the

n-th homology group of X 1 to be

H n(X 1) := ker∂ n/im∂ n+1

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Thus

H 0(X 1) = ker∂ 0/im∂ 1 = C 0(X 1)/im∂ 1 = frAb(x, y)/frAb(y − x) ∼= frAb(y) ∼= Z

H 1(X 1) = ker∂ 1/im∂ 2 = ker∂ 1 = frAb(a − b, b − c, c − d) ∼= Z3

H 2(X 1) = H 3(X 1) = . . . = 0

Wherein the computation of ker∂ 1 comes by noting that

ka + lb + mc + nd = 0 ⇐⇒ k + l + m + n = 0

Furthermore, the isomorphism with Z3 also requires one to see that a − b, b −

c, c − d form a Z-basis for the kernel. These generators correspond to the holes

ab−1, bc−1, cd−1 in X 1. Hence, the 1st homology group H 1 of X 1 detects the2-dimensional holes of X 1. (But is this hole really 2-dimensional? It looks so.

How come there isn’t anything for 1-dimensional holes? For instance, what if I

take a small line segment out of the plane?) And furthermore, H 2, H 3, H 4, . . .

accurately tell us that there are no higher dimensional holes. On the other hand,

H 0(X 1) = Z which would desirably be saying that there are 1-dimensional holes.

In fact, we will see that H 0(X ) = Z for every path-connected space X so that

the 0th homology is a pathology of the theory. We get rid of the 0th homology

by considering reduced homology later.

Next we compute homology for X 2 wherein we have filled in the hole ab−1.

This time cleaner. Again, define



C 2(X 2) = frAb(A) 2-chains of 2-cells

This yields a chain of maps

0 ∂ 3−→ C 2(X 2)

∂ 2−→ C 1(X 2) ∂ 1−→ C 0(X 2)

∂ 0−→ 0

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And


H 1(X 2) = ker∂ 1/im∂ 2 = frAb(a − b, b − c, c − d)/frAb(a − b) = frAb(b − c, c − d) ∼= Z2

H 2(X 2) = ker∂ 2/im∂ 3 = 0/0 = 0

H 3(X 2) = H 4(X 2) = H 5(X 2) = . . . = 0

By filling in the hole corresponding to ab−1 with a 2-cell, we accurately see

the reflection in the homology as H 1(X 2) = Z2 = H 1(X 1)/Z. Now the Z2

corresponds to the two remaining holes bc−1, cd−1. Again, there is a pathological

0th homology, and the other homology groups are trivial as expected.

Next we compute the homology for X 3 wherein we have attached a 2-cell B

to the missing boundary of A. Again, define



C 2(X 3) = frAb(A, B) 2-chains of 2-cells

This again yields a chain of maps

0 ∂ 3−→ C 2(X 3)

∂ 2−→ C 1(X 3) ∂ 1−→ C 0(X 3)

∂ 0−→ 0

But now



H 2(X 3) = ker∂ 2/im∂ 3 = frAb(A − B)/0 = frAb(A − B) ∼= Z

H 3(X 3) = H 4(X 3) = H 5(X 3) = . . . = 0

By gluing the 2-cells A, B as was done, a 3-dimensional hole was formed. And,

as homology is awesome, the 2nd homology group H 2 detects this only hole.

Furthermore, the other homology groups (other than the 0th) do their jobs as

well.

Finally we compute the homology for X 4 wherein we have filled in the only

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3-dimensional hole by attaching a 3-cell C to it. Define



C 2(X 4) = frAb(A, B) 2-chains of 2-cells

C 3(X 4) = frAb(C ) 3-chains of 3-cells

This again yields a chain of maps

0 ∂ 4−→ C 3(X 4)

∂ 3−→ C 2(X 4) ∂ 2−→ C 1(X 4)

∂ 1−→ C 0(X 4) ∂ 0−→ 0

But now



H 2(X 4) = ker∂ 2/im∂ 3 = frAb(A − B)/frAb(A − B) = 0

H 3(X 4) = H 4(X 3) = H 5(X 3) = . . . = 0

Thus, filling in the 3-dimensional hole of X 3 with a 3-cell killed the 3rd-homology,

as desired, as there are no longer any 3-dimensional holes. Homology works! At

least for these simple examples...

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