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LECTURE 1: DEFINITION OF SINGULAR HOMOLOGY

As a motivation for the notion of homology let us consider the topological space X which isobtained by gluing a solid triangle to a ‘non-solid’ triangle as indicated in the following picture.The vertices and some paths (with orientations) are named as indicated in the graphic.

•x0α

•x1

β•x2

γ

ε•x3

δ

Let us agree that we define the boundary of such a path by the formal difference ‘target - source’.So, the boundary ∂(β) of β is given by ∂(β) = x2− x1. In this terminology, the geometric propertythat a path is closed translates into the algebraic relation that its boundary vanishes. Moreover,let us define a chain of paths to be a formal sum of paths. In our example, we have the chainsc1 = α+ β + γ and c2 = β + ε+ δ−1. Both c1 and c2 are examples of closed paths (this translatesinto the algebraic fact that the sum of the boundaries of the paths vanishes). However, from ageometrical perspective, both chains behave very differently: c1 is the boundary of a solid triangle(and is hence closed for trivial reasons) while c2 is not of that form. Thus, the chain c2 detectssome ‘interesting geometry’.

The basic idea of homology is to systematically measure closed chains of paths (which might beinteresting) and divide out by the ‘geometrically boring ones’. Moreover, we would like to extendthis to higher dimensions. Let us now begin with a precise development of the theory.

Definition 1. Let n ≥ 0 be a natural number. The (geometric) n-simplex ∆n ⊆ Rn+1 is theconvex hull of the standard basis vectors of Rn+1 endowed with the subspace topology.

Let us denote these standard basis vectors by e0, . . . , en. Every point v ∈ ∆n can uniquely bewritten as a convex linear combination of the ei, i.e., there is a unique expression

v = Σni=0tiei, ti ≥ 0, t0 + . . .+ tn = 1.

The coordinates ti are the barycentric coordinates of the point v. Thus, to be completelyspecific, we have

∆n = {(t0, t1, . . . , tn) ∈ Rn+1 | ti ≥ 0, t0 + . . .+ tn = 1}.

Recall that a convex linear map is a map which sends convex linear combinations to convex linearcombinations. It follows that a convex linear map

α : ∆n → ∆m

is uniquely determined by its values on ei ∈ ∆n for i = 0, . . . , n. In the important case that α sendsvertices to vertices, i.e., if we have α(ei) = ea(i) for a certain map of sets a : {0, . . . , n} → {0, . . . ,m},

1

2 LECTURE 1: DEFINITION OF SINGULAR HOMOLOGY

then we obtain

α(Σni=0tiei) = Σni=0tiea(i) = Σmj=0sjej with sj = Σa(i)=jti.

As a special case we have the face maps

di : ∆n−1 → ∆n, (t0, . . . , tn−1) 7→ (t0, . . . , ti−1, 0, ti, . . . , tn), 0 ≤ i ≤ n.

This map is determined by the unique injective, monotone map {0, . . . , n− 1} → {0, . . . , n} whichdoes not hit i. The image di(∆n−1) ⊆ ∆n is called the i-th face of ∆n. For example, the possibleface maps ∆0 → ∆1 are the inclusions of the ‘target’ or the ‘source’, namely

t = d0 : ∆0 → ∆1 and s = d1 : ∆0 → ∆1.

In the next dimension, we have three face maps

di : ∆1 → ∆2, i = 0, . . . , 2

given by the inclusions of the three ‘sides’ of the topological boundary of ∆2 ⊆ R3. We stronglyrecommend the reader to draw the corresponding pictures.

For iterated face maps there is the following key relation (it is a special case of the cosimplicialidentities) which ‘lies at the heart of many kinds of homology theories’.

Lemma 2. For every n ≥ 2 and every 0 ≤ j < i ≤ n the following iterated face maps coincide

di ◦ dj = dj ◦ di−1 : ∆n−2 → ∆n.

Proof. It is immediate to verify that both maps are given by

(t0, t1, . . . , tn−2) 7→ (t0, . . . , tj−1, 0, tj , . . . , ti−1, 0, ti, . . . , tn−2).

In fact, both iterated face maps are determined by the unique monotone injection

{0, . . . , n− 2} → {0, . . . , n}

which hits neither i nor j. �

The idea of singular homology consists of studying an arbitrary space by considering formal sumsof maps defined on simplices of a fixed dimension.

Definition 3. Let X be a topological space.

(1) A singular n-simplex in X is a continuous map σ : ∆n → X.(2) The singular n-chain group Cn(X) is the free abelian group generated by the singular n-

simplices in X. Its elements are called singular n-chains in X.

Let us recall the notion of a free abelian group generated by a set. As a motivation for theconcept we include the following reminder.

Reminder 4. Let V be a finite-dimensional vector space with basis b1, . . . , bn ∈ V and let W be afurther vector space over the same field. Then a linear map f : V → W is uniquely determined bythe values f(b1), . . . , f(bn) ∈W.

Definition 5. Let S be a set. A free abelian group generated by S is a pair (F (S), iS)consisting of an abelian group F (S) and a map of sets iS : S → F (S) which satisfies the followinguniversal property: Given a further pair (A, j : S → A) with A an abelian group and j a map ofsets then there is a unique group homomorphism f : F (S)→ A such that f ◦ iS = j.

LECTURE 1: DEFINITION OF SINGULAR HOMOLOGY 3

More diagrammatically, this universal property can be depicted as follows (the reason why we splitthe diagram into two parts will become more apparent later: the left part is a diagram of sets, whilethe right part is a diagram of abelian groups):

S

=

iS //

∀j --

F (S)

f

��

F (S)

∃!f��

A A

The next lemma establishes the existence and essential uniqueness of free abelian groups gener-ated by a set. This motivates us to think of the free abelian group generated by a set as ‘the’ bestapproximation of a set by an abelian group.

Lemma 6. (1) Let S be a set. The free abelian group (F (S), iS) exists and is unique up to aunique isomorphism compatible with the maps from S. In more detail, given a further freeabelian group (F (S)′, i′S) then there is a unique isomorphism of groups φ : F (S) → F (S)′

which satisfies the relation φ ◦ iS = i′S:

S

=

iS //

i′S **

F (S)

φ

��

F (S)

∃! φ��

F (S)′ F (S)′

(2) The assignment S 7→ F (S) sending a set S to a free abelian group generated by S can beextended to a free abelian group functor from the category Set of sets to the category Ab ofabelian groups:

F : Set→ Ab

Proof. Exercise. �

The proof of this lemma will show that every element of F (S) can be written as a finite sum ofelements in S, i.e, for z ∈ F (S) we have

z = n1s1 + . . .+ nksk, ni ∈ Z, si ∈ S, i = 1, . . . , k.

Moreover, this expression is unique up to a permutation of the summands if we insist that the niare different from 0 and that the si are pairwise different. In particular, this applies to the singularchain group Cn(X) associated to a topological space X. Thus, a singular n-chain can be writtenas a formal sum of singular n-simplices in X.

Let f : X → Y be a map of spaces (unless stated differently all maps between spaces will beassumed to be continuous). Given a singular n-simplex σ : ∆n → X then f ◦ σ : ∆n → Y is asingular n-simplex in Y. The linear extension of this assignment (whose existence is guaranteed bythe last lemma) defines a group homomorphism:

Cn(f) = f∗ : Cn(X)→ Cn(Y )

Corollary 7. The assignments X 7→ Cn(X) and f 7→ f∗ define a functor, the singular n-chaingroup functor Cn, from the category Top of topological spaces to the category Ab of abelian groups:

Cn : Top→ Ab


The next aim is to relate the singular chain group functors of the various dimensions. For thispurpose, recall that we have the i-th face map di : ∆n−1 → ∆n for 0 ≤ i ≤ n. Given a singularn-simplex σ : ∆n → X in a space X we obtain a singular (n− 1)-simplex di(σ) in X by setting:

di(σ) = σ ◦ di : ∆n−1 di→ ∆n σ→ X, 0 ≤ i ≤ n

By linear extension this gives rise to a group homomorphism

di : Cn(X)→ Cn−1(X), 0 ≤ i ≤ n,

which will also be called the i-th face map. The key definition of the entire business is the followingone.

Definition 8. Let X be a topological space.

(1) The n-th singular boundary operator ∂ is given by

∂ =

n∑i=0

(−1)idi : Cn(X)→ Cn−1(X).

(2) The kernel Zn(X) of the boundary operator ∂ : Cn(X)→ Cn−1(X), i.e., the abelian group

Zn(X) = ker(∂ : Cn(X)→ Cn−1(X)),

is the group of singular n-cycles in X. An element of Zn(X) is sometimes also referredto as a closed singular n-chain.

(3) The image Bn(X) of the boundary operator ∂ : Cn+1(X)→ Cn(X), i.e., the abelian group

Bn(X) = im(∂ : Cn(X)→ Cn−1(X)),

is the group of singular n-boundaries in X.

Thus, by forming the alternating sum of the face maps we obtain a map between the groups invarious dimensions and this gives rise to two subgroups

Bn(X), Zn(X) ⊆ Cn(X), n ≥ 0.

In the special case of n = 0 we define Z0(X) = C0(X), i.e., every 0-chain is by definition also a0-cycle. A key property of these boundary maps is given in the next proposition. Once one getsused to the calculation in its proof, one remarks that the proposition is an immediate consequenceof the cosimplicial identity in Lemma 2.

Proposition 9. Given a topological space X then the singular boundary maps define a differentialon {C•(X)}, i.e., we have the relations

∂ ◦ ∂ = 0: Cn(X)→ Cn−2(X), n ≥ 2.

LECTURE 1: DEFINITION OF SINGULAR HOMOLOGY 5

Proof. This follows from the following algebraic manipulation in which Lemma 2 plays a key roleand where the last step is given by a shift of the inner summation index.

∂ ◦ ∂ =

n−1∑j=0

n∑i=0

(−1)i+jdj ◦ di

=

n−1∑j=0

j∑i=0

(−1)i+jdj ◦ di +

n−1∑j=0

n∑i=j+1

(−1)i+jdj ◦ di

!=

n−1∑j=0

j∑i=0

(−1)i+jdj ◦ di +

n−1∑j=0

n∑i=j+1

(−1)i+jdi−1 ◦ dj

=

n−1∑j=0

j∑i=0

(−1)i+jdj ◦ di +

n−1∑j=0

n−1∑i=j

(−1)i+j+1di ◦ dj

If we now interchange the roles of i and j in the –say– second sum we remark that the sums canceleach other as intended. �

Definition 10. The singular chain complex of a topological space X is the pair (C∗(X), ∂)consisting of the singular chain groups together with the singular boundary operators:

. . .∂→ Cn+1(X)

∂→ Cn(X)∂→ Cn−1(X)

∂→ . . .∂→ C1(X)

∂→ C0(X)

We will usually abuse notation and simply write C(X) or C∗(X) for the singular chain complex.The above proposition is very important. It implies that in each dimension n we have inclusions ofsubgroups

Bn(X) ⊆ Zn(X) ⊆ Cn(X).

Moreover, since all groups occurring here are abelian, the subgroups are normal subgroups so thatthe following definition makes sense.

Definition 11. Let X be a topological space. The n-th singular homology group Hn(X) of Xis the abelian group defined by

Hn(X) = Zn(X)/Bn(X).

This definition completes the program motivated by our initial example. Given a topologicalspace we can associate to it an abelian group which is obtained by taking the singular cycles ina fixed dimension (which might be geometrically interesting) and by dividing out those which aregeometrically uninteresting. The result of this gives us by definition the singular homology of thespace in that fixed dimension.

Given two singular n-cycles z1, z2 ∈ Zn(X) which represent the same homology class, i.e., theirdifference is a boundary, are called homologous. This will be denoted by:

z1 ∼ z2 :⇐⇒ z1 − z2 ∈ Bn(X)

For example a cycle z is a boundary if and only if z ∼ 0.

Example 12. (1) Let ∗ denote the space consisting of one point only. Then we have H0(∗) ∼= Zand Hn(∗) ∼= 0 for n ≥ 1. (Exercise.)


(2) The formation of singular homology is additive in the following sense. Let X be a topologicalspace and let Xα, α ∈ I, be its path components, then there is a (natural) isomorphism⊕

α∈IHn(Xα)

∼=→ Hn(X), n ≥ 0.

Thus, for many calculations it suffices to restrict attention to path-connected spaces. However,certain statements might be nicer if we allow for more general spaces (see for example the nextproposition). We will prove the claimed additivity in a later lecture. Although one could alreadyeasily make precise the definition of the above map we prefer to first establish the functoriality ofsingular homology. Of course, the reader is invited to convince her- or himself that such a relationshould be true.

We close this lecture by the following low-dimensional identification. Associated to a connectedspace X we have the following (natural) augmentation map ε. Recall that C0(X) is freely gener-ated by the singular 0-simplices in X. Thus, an element of this group is just a formal sum of pointsin X. Sending each point of X to 1 ∈ Z and then forming the linear extension gives rise to theaugmentation map

ε : C0(X)→ Z :

k∑i=1

nixi 7→k∑i=1

ni.

By our convention in dimension 0 we have C0(X) = Z0(X). Now, it is easy to check that theaugmentation map vanishes on all 0-boundaries. Thus, the universal property of the quotient ofabelian groups (see the exercises) implies that we get a unique induced group homomorphism ε∗ asindicated in:

B0(X)

=

i //

0 ..

Z0(X)

=ε

��

q// H0(X)

ε∗ppZ

Proposition 13. (1) Let X be a path-connected topological space, then the augmentation in-duces a (natural) isomorphism ε∗ : H0(X)→ Z.

(2) Let X be a topological space, then we have a (natural) isomorphism H0(X) ∼= Zπ0(X).

In the second statement of this proposition we use the notation ZS for the free abelian groupgenerated by the set S. Moreover, π0(X) denotes the set of path components of the space X. Theproof of this proposition will be given in the next lecture.

Remark 14. In algebraic topology it is very convenient to make systematical use of the languageof category theory. Since we did not wish to overwhelm the reader by too much of this languageat the very beginning we decided to slowly develop the corresponding terminology as the coursegoes on. In particular, the notion of a natural transformation between functors will only be madeprecise at a later stage although they already showed up in this lecture. As a compromise we wrote‘(natural) morphism’ or ‘(natural) isomorphism’ in the corresponding situations.

LECTURE 2: LOW-DIMENSIONAL IDENTIFICATIONS

The aim of this lecture is to show that the zeroth singular homology group H0(X) can beconstructed from π0(X) in a purely algebraic way. Similarly, for a connected pointed space (X,x0)the first singular homology group H1(X) can be obtained from π1(X,x0) by algebraic means only.

We begin with the case of H0(X). Let us recall from the last lecture that associated to aconnected space X we have the following (natural) augmentation map ϵ:

ϵ : C0(X) → Z :k∑

i=1

nixi 7→k∑

i=1

ni

By our convention, we have C0(X) = Z0(X). Since the augmentation map vanishes on all 0-boundaries there is a unique induced group homomorphism ϵ∗ as indicated in:

B0(X)

=

i //

0 ..

Z0(X)

=ϵ

��

q// H0(X)

ϵ∗ppZ

Proposition 1. (1) Let X be a path-connected topological space. Then the augmentation in-duces a (natural) isomorphism ϵ∗ : H0(X) → Z.

(2) Let X be a topological space. Then we have a (natural) isomorphism H0(X) ∼= Zπ0(X).

Proof. (1): Let us show that ϵ∗ is surjective. Given an integer n ∈ Z, we can choose an arbitrarypoint x ∈ X and consider the 0-cycle z = nx. We then have ϵ∗([z]) = ϵ(z) = n.We now show that ϵ∗ is injective. So let us assume that the homology class [z] represented by

z =∑k

j=1 njxj lies in the kernel of ϵ∗, i.e., that we have∑k

j=1 nj = 0. Since X is connected we

can find a point x0 ∈ X and paths σj : ∆1 → X such that σj(0) = x0 and σj(1) = xj . Let us form

the singular 1-chain σ =∑k

j=1 njσj . The following calculation shows that the homology class [z] istrivial:

∂(σ) =k∑

j=1

nj(xj − x0) =k∑

j=1

njxj −k∑

j=1

njx0 =k∑

j=1

njxj = z

(2): The proof of this part is very similar and uses the additivity of singular homology. It is left asan exercise to the reader. �

We now continue with the relation between π1 and H1. Let us begin by recalling some basicterminology concerning the manipulation of paths in a space. Two paths γ0, γ1 : ∆

1 → X are calledcomposable if they satisfy γ0(1) = γ1(0). If we have two such composable paths γ0 and γ1, then wedenote their concatenation by

γ0 ∗ γ1 : ∆1 → X.

This is the path obtained by first running through γ0 and then through γ1, and both at a doublespeed. The inverse path γ−1 of a path γ is defined by

γ−1 : ∆1 → X : t 7→ γ(1− t).1

2 LECTURE 2: LOW-DIMENSIONAL IDENTIFICATIONS

Let us next construct the so-called Hurewicz homomorphism π1(X,x0) → H1(X) associated toa pointed space (X,x0). So, let us consider a homotopy class α = [γ] ∈ π1(X,x0) represented by apointed loop γ : S1 → X, 1 7→ x0. The quotient map

e : ∆1 → S1 = ∆1/∂∆1

allows us to associate a singular 1-chain γ ◦ e ∈ C1(X) to such a loop γ:

γ ◦ e : ∆1 e→ S1 γ→ X

Since γ is a loop it is immediate that γ◦e is a 1-cycle and hence represents a homology class. Thus, wecould try to associate the homology class [γ◦e] ∈ H1(X) to the homotopy class α = [γ] ∈ π1(X,x0).In order to see that this assignment is well-defined we have to check the following: if two loops γ0and γ1 are homotopic relative to the base point, then the singular 1-cycles γ0 ◦ e and γ1 ◦ e arehomologous, i.e., their difference is a boundary. But a 2-chain realizing this can be constructedfrom such a pointed homotopy H : ∆1 ×∆1 → X. In fact, the homotopy satisfies the relations

H(0,−) = γ0 ◦ e, H(1,−) = γ1 ◦ e, and H(t, 0) = H(t, 1) = x0, t ∈ ∆1.

If we set w(t) = H(t, t) then the above relations of the homotopy can be graphically depicted by

κx0 //

γ0◦eOO

κx0

//

w

??��γ1◦eOO

where κx0 denotes the constant map with value x0. The restrictions of the homotopy H to theupper left and the lower right 2-simplex give us maps σ1 : ∆

2 → X and σ2 : ∆2 → X respectively.

We can conclude by calculating the boundary of σH = σ1 − σ2 ∈ C2(X):

∂(σH) = ∂σ1 − ∂σ2

= (κx0− w + γ0 ◦ e)− (γ1 ◦ e− w + κx0

)

= γ0 ◦ e− γ1 ◦ e

Thus, we obtain γ0 ◦ e ∼ γ1 ◦ e as intended.

Proposition 2. If (X,x0) is a pointed topological space then the assignment

h : π1(X,x0) → H1(X) : [γ] 7→ [γ ◦ e]

defines a (natural) homomorphism of groups, the Hurewicz homomorphism of (X,x0).

Proof. By the above discussion the map of sets h : π1(X,x0) → H1(X) is well-defined. Let usnow check that it is a group homomorphism. The homotopy class of the constant loop κS1,x0

at x0 is the neutral element of π1(X,x0). It’s image under h is the homology class of the 1-cycleκS1,x0

◦ e = κ∆1,x0where the latter denotes the constant path at x0. But the boundary of the

constant 2-simplex κ∆2,x0: ∆2 → X is given by

∂(κ∆2,x0) = κ∆1,x0

− κ∆1,x0+ κ∆1,x0

= κ∆1,x0.

We thus deduce that h(1) = 0 as intended.For the compatibility with the group structures let us consider two homotopy classes α1 = [γ1]

and α2 = [γ2] in π1(X,x0). Note that the paths γ1 ◦ e, γ2 ◦ e, and (γ1 ∗ γ2) ◦ e can be used to define

LECTURE 2: LOW-DIMENSIONAL IDENTIFICATIONS 3

a map from the (geometric) boundary of ∆2 to X. This is indicated in the next diagram (strictlyspeaking we have to use some linear reparametrizations of the paths but we will ignore this issue):

γ2◦e

��??

????

??γ1◦e

??��(γ1∗γ2)◦e

//

We can extend this to a continuous map σ : ∆2 → X which is constant along the ‘vertical lines’.But this singular 2-simplex σ implies the intended relation h(α1α2) = h(α1) + h(α2). �

As a preparation for the main theorem of this lecture, let us collect a few convenient facts.

Lemma 3. Let X be a topological space.

(1) A constant path κ : ∆1 → X is a boundary.(2) If two paths γ0, γ1 : ∆

1 → X are homotopic relative to the boundary then γ0 ∼ γ1.(3) If γ0, γ1 : ∆

1 → X are composable then γ0 + γ1 − γ0 ∗ γ1 is a boundary.(4) If γ0 : ∆

1 → X is a path then γ0 + γ−10 is a boundary.

Proof. (1): We proved this already when we showed that the Hurewicz homomorphism h preservesneutral elements.(2): This was already proved when we checked that the Hurewicz homomorphism is well-defined.(3): We established the corresponding result for loops when we showed that h is multiplicative.But that proof never used that we considered loops as opposed to more general paths.(4): This is a combination of the previous results:

γ0 + γ−10

iii)∼ γ0 ∗ γ−10

ii)∼ κγ0(0)i)∼ 0 �

The Hurewicz homomorphism is a group homomorphism whose target is an abelian group whilethe source itself is not necessarily abelian. Let us shortly abstract from this specific situation andconsider a group G, an abelian group A, and a group homomorphism f : G → A. The group G willbe written multiplicatively while A will be written additively. Then, for two elements g1 and g2of G we obtain:

f(g1g2) = f(g1) + f(g2) = f(g2) + f(g1) = f(g2g1)

Hence, all elements of the form g1g2g−11 g−1

2 ∈ G are sent to the neutral element 0 ∈ A. Such anelement is called a commutator and we will denote by [G,G] the subgroup of G generated by thecommutators:

[G,G] = ⟨{g1g2g−11 g−1

2 | g1, g2 ∈ G}⟩ ⊆ G

Lemma 4. In the above notation we have the following facts:

(1) The subset [G,G] is a normal subgroup of G and the quotient group Gab = G/[G,G] isabelian. The subgroup [G,G] is the commutator subgroup of G and the quotient groupGab = G/[G,G] is called the abelianization of G.

(2) The pair (Gab, q) consisting of the abelianization Gab and the canonical group homomor-phism q : G → Gab has the following universal property: Given a further pair (A, r) con-sisting of an abelian group A and a group homomorphism r : G → A then there is uniquegroup homomorphism g : Gab → A such that g ◦ q = r.


4 LECTURE 2: LOW-DIMENSIONAL IDENTIFICATIONS

More diagrammatically, this universal property can be visualized as follows:

G

=

q//

∀r --

Gab

g

�� Gab

∃!g��

A A

Note again that the two parts of this diagram take place in different categories: the diagram on theleft lives in the category of groups while the one on the right is a diagram in the category of abeliangroups. Thus, we can think of the abelianization as ‘the’ best approximation of an arbitrary groupby an abelian group.

Let us now return to the context of the Hurewicz homomorphism

h : π1(X,x0) → H1(X)

associated to a pointed space (X,x0). The above lemma implies that h factors uniquely through a

homomorphism h : π1(X,x0)ab → H1(X), i.e., we have

h : π1(X,x0) → π1(X,x0)ab h→ H1(X).

Theorem 5. Let (X,x0) be a path-connected pointed space. Then the (natural) group homomor-

phism h : π1(X,x0)ab → H1(X) induced by the Hurewicz homomorphism is an isomorphism.

Proof. We have seen that h induces a well-defined group homomorphim h, sinceH1(X) is an abeliangroup. To construct a map in the opposite direction, choose first for each x ∈ X a path τx from thebasepoint x0 to x. Then, to every path α in X from x to y we can associate a loop φ(α) at x0 bydefining φ(α) = τx ∗ α ∗ τ−1

y . This induces a homomorphism φ : C1(X) → π1(X,x0)ab. Moreover,

φ induces a group homomorphism

φ : H1(X) −→ π1(X,x0)ab

because it vanishes on the image of C2(X)∂→ C1(X). Indeed, if σ ∈ C2(X), then we can define a

homotopy from d2σ ∗ d0σ to d1σ:

x

y

z

x0

τzτx

τy

d2σd0σ

d1σ

σ

0

2

1

LECTURE 2: LOW-DIMENSIONAL IDENTIFICATIONS 5

so thatτx ∗ d2σ ∗ τ−1

y ∗ τy ∗ d0σ ∗ τ−1z ≃ τx ∗ d2σ ∗ d0σ ∗ τ−1

z ≃ τx ∗ d1σ ∗ τ−1z .

That is, φ(d2σ) · φ(d0σ) = φ(d1σ) in π1(X,x0), so φ(∂σ) = 0 in π1(X,x0)ab.

Now we will prove that h and φ are mutually inverses. Remember that we choose τx to be apath from x0 to x. Let’s agree to choose τx to be the constant path κx0 if x = x0. Then clearly,for a loop α based at x0 we have that

(φ ◦ h)(α) = κx0∗ α ∗ κx0

≃ α.

On the other hand, let α ∈ C1(X). Then we have that (h ◦φ)(α) = h(τx ∗α ∗ τ−1y ) = τx +α− τy =

α+τx−τy. Let’s take a class [β] in H1(X) represented by β =∑

niαi ∈ C1(X) with ∂β = 0. Then

(h ◦ φ)(β) =∑

niαi +∑

ni(τxi − τyi) = β +∑

ni(τxi − τyi) = β,

the latter because ∂β =∑

ni(xi − yi) = 0. Thus (h ◦ φ)([β]) = [β] also, which concludes theproof. �

The theorem allows us to do some first calculations for several spaces, if you already know theirfundamental groups. Let RPn denote the real projective space of dimension n and let Tn denotethe n-dimensional torus, i.e., Tn is an n-fold product of 1-spheres S1.

Corollary 6. The Hurewicz homomorphism induces the following identifications:

(1) H1(S1) ∼= Z,

(2) H1(Sn) ∼= 0, n ≥ 2,

(3) H1(RPn) ∼= Z/2Z, n ≥ 2,(4) H1(Tn) ∼= Zn.

LECTURE 3: RELATIVE SINGULAR HOMOLOGY

In this lecture we want to cover some basic concepts from homological algebra. These prove tobe very helpful in our discussion of singular homology. The following definition abstracts the keyingredients which were necessary to introduce the singular homology groups of a topological space.

Definition 1. A chain complex (of abelian groups) C consists of abelian groups Cn, n ≥ 0,together with group homomorphisms ∂ : Cn → Cn−1, n ≥ 1, such that

∂ ◦ ∂ = 0: Cn → Cn−2, n ≥ 2.

The homomorphisms ∂ are called boundary homomorphisms or differentials.

Thus, a chain complex of abelian groups can be depicted as

. . .∂→ Cn+1

∂→ Cn∂→ . . .

∂→ C1∂→ C0.

The elements of Cn are said to be of degree n and are called n-chains of C. Given such a chaincomplex C, we call

Zn = Zn(C) = ker(∂ : Cn → Cn−1) ⊆ Cnthe subgroup of n-cycles and

Bn = Bn(C) = im(∂ : Cn+1 → Cn)

the subgroup of n-boundaries. By convention, we set Z0 = C0, i.e., we define all 0-chains to be0-cycles. The fundamental relation ∂ ◦ ∂ = 0 implies that we have an inclusion Bn ⊆ Zn for alln ≥ 0. The n-th homology group Hn = Hn(C) of a chain complex C is the quotient group

Hn(C) = Zn(C)/Bn(C).

Elements of Hn(C) are cosets zn+Bn(C) which satisfy zn ∈ Zn(C). Such an element is also denotedby [zn] and is called the homology class of degree n represented by zn.

Example 2. Associated to a topological space X ∈ Top we earlier constructed the singular chaincomplex C(X) = C∗(X). In degree n it is given by the free abelian group generated by the singularn-simplices σ : ∆n → X in X. The singular boundary operator Cn(X)→ Cn−1(X) is induced by theface maps ∆n−1 → ∆n. By definition, the homology groups of this chain complex are the singularhomology groups of our given space.

Using the universal property of free abelian groups generated by a set we saw already that theassignment X 7→ Cn(X) is functorial. But also the singular boundary operators behave nicely withrespect to maps of spaces. Let us consider topological spaces X and Y and a continuous mapf : X → Y between them. Given a singular n-simplex σ : ∆n → X then the associativity of thecomposition law for maps of spaces implies the relation

(f ◦ σ) ◦ di = f ◦ (σ ◦ di) : ∆n−1 → Y, n ≥ 1, 0 ≤ i ≤ n.1

2 LECTURE 3: RELATIVE SINGULAR HOMOLOGY

By linearity, this implies that the square

Cn(X)di //

Cn(f)

��

Cn−1(X)

Cn−1(f)

��

Cn(Y )di

// Cn−1(Y )

commutes. Since the singular boundary operators Cn(X)→ Cn−1(X) and Cn(Y )→ Cn−1(Y ) areobtained from these face maps by forming alternating sums this implies that the various (Cn(f))n≥0assemble into a chain map C(X)→ C(Y ) in the following sense.

Definition 3. Let C andD be chain complexes of abelian groups. A chain map f : C → D consistsof group homomorphisms fn : Cn → Dn, n ≥ 0, which commute with the boundary operators inthe sense that

∂ ◦ fn = fn−1 ◦ ∂ : Cn → Dn−1, n ≥ 1.

Thus, if we depict chain complexes by diagrams as above, then a morphism of chain complexesgives us a ‘commutative ladder’ as described by the next diagram:

. . . ∂ // Cn+1∂ //

fn+1

��

Cn∂ //

fn

��

. . . ∂ // C1

f1

��

∂ // C0

f0

��

C

f

��. . .

∂// Dn+1

∂// Dn

∂// . . .

∂// D1

∂// D0 D

Given chain maps f : C → D and g : D → E then it is immediate to see that the degreewisecompositions gn ◦ fn : Cn → En assemble into a chain map g ◦ f : C → E. It is similarly obviousthat the identity maps Cn → Cn assemble to a chain map id: C → C. Moreover, it follows fromthe definition that a chain map sends cycles to cycles and boundaries to boundaries.

Lemma 4. (1) Chain complexes of abelian groups together with chain maps assemble into acategory which is denoted by Ch = Ch(Z).

(2) The formation of cycles, boundaries, and homology in a fixed dimension is functorial, i.e.,for all n ≥ 0 the assignments

C 7→ Zn(C), C 7→ Bn(C), and C 7→ Hn(C)

extend to functors Zn, Bn, Hn : Ch→ Ab.

Proof. This is straightforward. Let us only mention that, given a chain map f : C → D, then theinduced map in homology is defined by

f∗ = Hn(f) : Hn(C)→ Hn(D) : [cn] 7→ [fn(cn)].

We leave it to the reader to check the details. �

Corollary 5. The singular chain group functors Cn : Top→ Ab and the singular boundary operatorstogether define a singular chain complex functor C : Top → Ch. In particular, there is a singularhomology functor Hn : Top→ Ab defined as the composition

Hn : TopC→ Ch

Hn→ Ab.

LECTURE 3: RELATIVE SINGULAR HOMOLOGY 3

A formal consequence of having a functor Hn : Top→ Ab is that singular homology groups formtopological invariants, i.e., homeomorphic spaces have isomorphic singular homology groups (moreprecisely, any such homeomorphism induces an isomorphism in singular homology). There is amuch stronger statement though: even spaces which are only homotopy equivalent have canonicallyisomorphic homology groups. We refer to this statement by saying that ‘singular homology ishomotopy-invariant ’. A proof of this important result will be given later in the course.

Let us be given a (not necessarily bounded) sequence of abelian groups An together with grouphomomorphisms ∂ : An → An−1 for all n. Thus, we have a diagram of abelian groups as follows:

. . .∂→ An+1

∂→ An∂→ An−1

∂→ . . .

Such a sequence is a chain complex if we always have ∂◦∂ = 0. In light of this more general definition,we sometimes refer to objects considered in Definition 1 as non-negative or non-negatively gradedchain complexes. We say that such a sequence is exact at An if we have the equality of subgroups

im(∂ : An+1 → An) = ker(∂ : An → An−1) ⊆ An.Such a sequence is called an exact sequence if it is exact at all An. Note that such an exactsequence is, in particular, a chain complex since the relation ∂ ◦ ∂ = 0 is equivalent to the fact thatwe have inclusions of subgroups

im(∂ : An+1 → An) ⊆ ker(∂ : An → An−1)

for all n. A chain complex is exact if we also have inclusions in the opposite directions, namely

im(∂ : An+1 → An) ⊇ ker(∂ : An → An−1)

for all n. Note also that this is equivalent to the vanishing of all homology groups.A particularly important special case is given by so-called short exact sequences of abelian

groups. By definition, this is an exact sequence of abelian groups of the form

0→ A′i→ A

p→ A′′ → 0.

The following lemma makes precise what kind of structure is encoded by such a short exact sequence.

Lemma 6. (1) The sequence of abelian groups 0 → A′i→ A is exact at A′ if and only if i is

injective.

(2) The sequence of abelian groups Ap→ A′′ → 0 is exact at A′′ if and only if p is surjective.

(3) The sequence of abelian groups 0→ A′i→ A

p→ A′′ → 0 is exact if and only if i is injective, pis surjective, and we have im(i) = ker(p).

(4) The sequence of abelian groups 0→ Af→ B → 0 is exact if and only if f is an isomorphism.

Thus, a short exact sequence basically only encodes an inclusion of a subgroup together with itsquotient map. Nevertheless, this notion proves to be very useful. It is straightforward to extendit to chain complexes. A short exact sequence of chain complexes is a diagram of chaincomplexes and chain maps

0→ C ′i→ C

p→ C ′′ → 0

which induces a short exact sequence of abelian groups

0→ C ′nin→ Cn

pn→ C ′′n → 0

in each degree. As in the case of abelian groups, also short exact sequences of chain complexesbasically only encode the inclusion of a subcomplex together with the corresponding projection to


the quotient complex. Let us make these notions precise. Given a chain complex C, a subcomplexof C is given by a family of subgroups C ′n ⊆ Cn such that the boundary operator ∂ : Cn → Cn−1restricts to a homomorphism C ′n → C ′n−1 for all n. In this situation the family of subgroups C ′ncan be uniquely turned into a chain complex C ′ such that the inclusions of the subgroups C ′n ⊆ Cnassemble into a morphism of chain complexes i : C ′ → C (do this as an exercise!). In particular,given a chain map f : C → D, the kernels

ker(fn : Cn → Dn) ⊆ Cnassemble into a chain complex ker(f), the kernel of f . As in the case of abelian groups, the kernelof a chain map comes with a canonical map ker(f) → C which satisfies an appropriate universalproperty.

Exercise 7. Define the image im(f) ∈ Ch of a chain map f : C → D. Show that any chain mapf : C → D factors as a composition

f : C → im(f)→ D.

We already discussed subobjects of chain complexes, namely subcomplexes. We now turn to thedual concept which is given by quotient complexes.

Lemma 8. Let i : C ′ → C be the inclusion of a subcomplex and let pn : Cn → C ′′n = Cn/C′n be the

(levelwise) quotient map. Then there is a unique way to turn the (C ′′n)n≥0 into a chain complex C ′′

such that the (pn)n≥0 assemble into a chain map p : C → C ′′. Moreover, it follows that the sequence

0→ C ′i→ C

p→ C ′′ → 0 is exact.

Proof. This is left to the reader as an exercise. �

The chain complex C ′′ constructed in the lemma is the quotient complex associated to theinclusion i : C ′ → C. Similarly, using pointwise definitions, once can construct the cokernel cok(f)of a chain map f : C → D. This is a chain complex which comes with a chain map D → cok(f)and this pair satisfies the usual universal property.

As a punchline of this lengthy discussion you should take away that basic constructions likekernels, cokernels, subobjects and quotient objects can be extended from abelian groups to chaincomplexes by pointwise definitions. And these extended definitions still ‘behave as expected’. How-ever, it is tremendously important to note that the formation of homology is not compatible withthese constructions. Let us be more specific about this and consider a short exact sequence

0→ C ′ → C → C ′′ → 0

of chain complexes. Since homology is functorial we obtain induced maps

Hn(C ′)→ Hn(C)→ Hn(C ′′)

and one might wonder if there are short exact sequences

0→ Hn(C ′)→ Hn(C)→ Hn(C ′′)→ 0

in each dimension n. In general, this turns out to be an unreasonable demand but there is thefollowing proposition.

Proposition 9. Let 0 → C ′i→ C

p→ C ′′ → 0 be a short exact sequence of chain complexes ofabelian groups. Then there is a (natural) connecting homomorphism

δn : Hn(C ′′)→ Hn−1(C ′)


such that the following sequence is exact:

. . .→ Hn+1(C ′′)δn+1→ Hn(C ′)

i∗→ Hn(C)p∗→ Hn(C ′′)

δn→ Hn−1(C ′)→ . . .

Proof. Let us begin with the construction of the connecting homomorphism δn for a given n. Forthis purpose let us draw the part of the short exact sequence which is relevant in that situation:

Cnpn //

∂n

��

C ′′n

C ′n−1 in−1

// Cn−1

Let us consider a homology class ω ∈ Hn(C ′′) and let us represent it by an n-cycle z′′n ∈ Zn(C ′′).By the surjectivity of pn we can find an n-chain cn ∈ Cn such that pn(cn) = z′′n. For the image∂n(cn) of cn under the boundary operator we calculate pn−1(∂n(cn)) = ∂n(pn(cn)) = ∂n(z′′n) = 0since z′′n is a cycle. Thus, the fact that we have a short exact sequence in level n − 1 implies thatthere is a unique z′n−1 ∈ C ′n−1 such that in−1(z′n−1) = ∂n(cn). This (n − 1)-chain z′n−1 is, in fact,a cycle as the following calculation combined with the injectivity of in−2 implies:

in−2(∂n−1(z′n−1)) = ∂n−1(in−1(z′n−1)) = ∂n−1(∂n(cn)) = 0

Thus, z′n−1 represents a homology class [z′n−1] ∈ Hn−1(C ′). We define the connecting homomor-phism δn as follows:

δn : Hn(C ′′)→ Hn−1(C ′) : [z′′n] 7→ [z′n−1]

In the construction of the connecting homomorphism some choices were made. We leave it to thereader to check that our definition is well-defined and that δn is a group homomorphism.

Let us now turn to the exactness issues. However, we will only give the proof of the exactnessat Hn−1(C ′). In order to establish the inclusion im(δn) ⊆ ker(i∗ : Hn−1(C ′∗) → Hn−1(C∗) we willstill use the notation from the construction of δn. But his inclusion is immediate since

i∗(δn[z′′n]) = i∗[z′n−1] = [in−1(z′n−1)] = [∂n(cn)] = 0.

Let us assume conversely that we have a homology class ω′ ∈ Hn−1(C ′) such that i∗(ω′) = 0. Thus,

if we represent ω′ by z′n−1 we have in−1(z′n−1) = ∂n(cn) for some cn ∈ Cn. The image z′′n = pn(cn)of cn under pn is a cycle since:

∂n(z′′n) = ∂n(pn(cn)) = pn−1(∂n(cn)) = pn−1(in−1(z′n−1)) = 0

Hence, we can form the homology class ω′′ = [z′′n] ∈ Hn(C ′′) and it follows from the constructionof the connecting homomorphism and the fact that it is well-defined that we have ω′ = δn(ω′′).

The proofs of the exactness at Hn(C) and Hn(C ′′) are similar and are left to the reader as anexercise. �

This long exact sequence is referred to as the long exact homology sequence induced bya short exact sequence of chain complexes. It is a very powerful tool – both for theoretical andcomputational purposes. In the case of nonnegative chain complexes, this long exact sequence endson

. . .→ H2(C ′′)δ2→ H1(C ′)→ H1(C)→ H1(C ′′)

δ1→ H0(C ′)→ H0(C)→ H0(C ′′)→ 0.

The final aim in this lecture is to apply Proposition 9 in a topological context. More precisely,let us consider a pair of spaces (X,A), i.e., a topological space X together with a subspace A ⊆ X.Let us denote the inclusion of the subspace by i : A → X. It is immediate that the induced map


of singular chain complexes i∗ : C(A)→ C(X) is levelwise injective and we can hence consider thequotient complex C(X,A) = C(X)/C(A). (To be completely precise we should form the quotientby the subcomplex i∗(C(A)) ⊆ C(X), but we allow ourselves to blur the distinction between theisomorphic complexes C(A) and i∗(C(A)).)

Definition 10. The relative singular chain complex C(X,A) of a pair of spaces (X,A) is thequotient complex

C(X,A) = C(X)/C(A).

The homology of C(X,A) is the relative singular homology of the pair (X,A), and it will bedenoted by

Hn(X,A) = Hn(C(X,A)), n ≥ 0.

Thus, if we have a pair of spaces (X,A), then there is by definition a short exact sequence ofchain complexes

0→ C(A)→ C(X)→ C(X,A)→ 0.

An application of Proposition 9 implies immediately the following result.

Corollary 11. Let (X,A) be a pair of spaces. Then there are (natural) connecting homomorphisms

δn : Hn(X,A)→ Hn−1(A), n ≥ 1,

such that the following sequence is exact

. . .→ H2(X,A)δ2→ H1(A)→ H1(X)→ H1(X,A)

δ1→ H0(A)→ H0(X)→ H0(X,A)→ 0.

This is the long exact homology sequence associated to the pair of spaces (X,A).

Example 12. Let X be a space, i : A→ X the inclusion of a subspace, and let x0 ∈ X be a point.

(1) The homology of the empty space is trivial in all dimensions. Thus, the inclusion j : ∅ → Xinduces isomorphisms

Hn(X)∼=→ Hn(X, ∅), n ≥ 0.

(2) The inclusion i induces isomorphisms in homology i∗ : Hn(A)∼=→ Hn(X), n ≥ 0, if and only

if all relative homology groups Hn(X,A), n ≥ 0, vanish, i.e.,

Hn(X,A) ∼= 0.

In particular, the homology groups Hn(X,X) vanish for all n ≥ 0.(3) If the maps Hn(A) → Hn(X) are injective for n ≥ 0 or if the maps Hn(X) → Hn(X,A)

are surjective for all n ≥ 1, then we have short exact sequences

0→ Hn(A)→ Hn(X)→ Hn(X,A)→ 0, n ≥ 0.

In fact, in both cases the connecting homomorphisms are trivial and the result follows. Thisapplies, in particular, to the case of the inclusion of a retract.

(4) Let us consider the inclusion k : {x0} → X. Our earlier calculation Hn(x0) ∼= 0 for alln ≥ 1 together with the long exact homology sequence for the pair (X,x0) implies that wehave (natural) isomorphisms Hn(X) → Hn(X,x0), n ≥ 2. Observe that H0(x0) → H0(X)is injective since it is just the Z-linear extension of the inclusion of the path-componentof x0 in π0(X). Hence the connecting homomorphism H1(X,x0) → H0(x0) is trivial andwe obtain a short exact sequence 0 → H1(x0) → H1(X) → H1(X,x0) → 0. We concludefrom this discussion that there are natural isomorphisms

Hn(X)∼=→ Hn(X,x0), n ≥ 1.


More generally, if A has trivial homology in positive degrees and if the map π0(A)→ π0(X)is injective, then we have (natural) isomorphisms Hn(X)→ Hn(X,A) for all n ≥ 1.

Let us close this lecture with a short comment on the notions introduced here. With our mainexample of the singular chain complex associated to a space in mind, we decided to restrict attentionto chain complexes of abelian groups. However, the concepts of chain complexes, chain maps,homology, and exactness make perfectly well sense in other contexts. For example, given a field k,we could consider the category Ch(k) of chain complexes of vector spaces over k. The only differenceis that in this case all maps in sight are supposed to be k-linear. More generally, given a commutativering R, we can consider the category Ch(R) of chain complexes of R-modules. From this perspective,we considered the case of the ring R = Z since the categories of abelian groups and Z-modules areisomorphic.

LECTURE 4: SINGULAR HOMOLOGY OF CONTRACTIBLE SPACES

In the last lecture we introduced relative singular homology of a pair of spaces. Let us quicklyrecall the construction. Given a space X and a subspace A together with its inclusion i : A → Xthen we have the induced short exact sequence of singular chain complexes:

0→ C(A)→ C(X)→ C(X,A) = C(X)/C(A)→ 0

The quotient complex C(X,A) is the relative singular chain complex and its homology is the relativesingular homology of the pair :

Hn(X,A) = Hn(C(X,A))

By construction of the quotient complex chains in it can be represented by singular chains c in X.Moreover, such a chain c ∈ Cn(X) represents a cycle in the relative chain complex if its usualsingular boundary ∂(c) lies in the image of Cn−1(A) → Cn−1(X). In this situation we say that cis a cycle mod A or a cycle relative to A. Thus, relative singular homology classes, in general, cannot be represented by cycles in X but always by cycles relative to A. We leave it to the reader todefine the category Top2 of pairs of spaces and to remark that relative singular chain complexesand relative singular homology groups define functors C : Top2 → Ch(Z) and Hn : Top2 → Abrespectively (Exercise!).

The aim of this and the next lecture is to show that (relative) singular homology is homotopy-invariant. Recall that it is an immediate consequence of the functoriality of singular homology thathomeomorphic spaces have naturally isomorphic homology groups. We want to show next that thisalso holds true for homotopy equivalent spaces. In fact, this will be a consequence of the moregeneral result that homotopic maps induce the same maps on singular homology.

We begin by splitting of an algebraic definition which ‘mimics’ the notion of a homotopy at thelevel of chain complexes. With our main application in mind we restrict attention to non-negativechain complexes. Recall that given two chain complexes C, D of abelian groups, then a chainmap f : C → D consists of a family of group homomorphisms fn : Cn → Dn which commute withthe differentials.

Definition 1. Let C, D ∈ Ch(Z) be chain complexes and let f, g : C → D be chain maps. A chainhomotopy s from f to g, denoted s : f ' g, consists of group homomorphisms sn : Cn → Dn+1 forall n ≥ 0 such that:

∂ ◦ sn + sn−1 ◦ ∂ = gn − fn, n ≥ 1, and ∂ ◦ s0 = g0 − f0Two chain maps f and g are chain homotopic, denoted f ' g, if there is such a chain homotopy.

Thus, in the situation of the definition we have the following diagram in which the vertical homo-morphisms are given by gn − fn for the corresponding value of n:

. . . ∂ // Cn+1∂ //

��

Cn∂ //

��

sn

||yyyyyyyyCn−1

∂ //

��

sn−1

||yyyyyyyy. . . ∂ // C1

��

∂ // C0

��

s0

~~||||

||||

. . .∂

// Dn+1∂

// Dn∂

// Dn−1∂

// . . .∂

// D1∂

// D0

1

2 LECTURE 4: SINGULAR HOMOLOGY OF CONTRACTIBLE SPACES

One reason why we are interested in this concept is given by Lemma 3. But let us first collectsome elementary facts about chain homotopies (the precise formulations are left to the reader).

Lemma 2. The chain homotopy relation defines an equivalence relation on the set of chain mapsbetween two chain complexes. Moreover, it is compatible with composition, addition, and the for-mation of additive inverses.


Lemma 3. Let C, D ∈ Ch(Z) and let us consider chain maps f, g : C → D. If f and g are chainhomotopic then the induced maps in homology are equal, i.e., we have:

Hn(f) = Hn(g) : Hn(C)→ Hn(D), n ≥ 0

Proof. Let s : f ' g be a chain homotopy and let us consider a homology class ω = [zn] ∈ Hn(C).For n ≥ 1 we calculate

Hn(g)(ω) = [gn(zn)] = [fn(zn) + ∂sn(zn) + sn−1∂(zn)] = [fn(zn)] = Hn(f)(ω)

The third equality uses that zn is a cycle and that a homology class is not changed if we add aboundary. In degree zero it is even slightly simpler since we can conclude the proof by:

H0(g)(ω) = [g0(z0)] = [f0(z0) + ∂s0(z0)] = [f0(z0)] = H0(f)(ω) �

Thus, in order to establish the homotopy invariance of singular homology we have to showthat homotopic maps between topological spaces induce chain homotopic maps between the corre-sponding singular chain complexes. However, this will require some preparation and will only becompleted in the next lecture.

Let us begin by recalling the quotient of a space by a subspace. We first construct the underlyingset of the quotient space. Any pair of spaces (X,A) with inclusion map i : A → X induces anequivalence relation on X, namely the equivalence relation ∼A generated by i(a) ∼A i(a′) for alla, a′ ∈ A. Let us denote the set of equivalence classes with respect to ∼A by X/A:

X/A = X/ ∼AThere is the natural quotient map q : X → X/A which sends an element x ∈ X to its equivalenceclass [x] ∈ X/A. The equivalence class of i(a) for an arbitrary a ∈ A will be denoted by ∗. Weendow X/A with the finest topology such that the quotient map q : X → X/A is continuous. Inother words, a subset U ⊂ X/A is open by definition if and only if q−1(U) is open in X. It isimmediate from the construction that the composition q ◦ i : A → X → X/A is constant withvalue ∗. This quotient space construction has the following universal property.

Lemma 4. Let (X,A) be a pair of spaces and let (X/A, q) be the quotient space X/A together withits quotient map q : X → X/A. Given a further such pair (Y, r) consisting of a topological space Yand a continuous map r : X → Y such that the composition r ◦ i : A→ Y is constant then there isa unique continuous map r′ : X/A→ Y such that r′ ◦ q = r:

Ai //

∗′ **

Xq

//

r

��

X/A

∃!r′ssY


LECTURE 4: SINGULAR HOMOLOGY OF CONTRACTIBLE SPACES 3

We now apply this to the cone construction of a topological space. Let X be a topological spacethen the product I ×X where I = [0, 1] is the cylinder of X. The cone CX of X is obtained fromthe cylinder by collapsing the top:

I ×X p→ CX = I ×X/{1} ×XElements of this space are equivalence classes [t, x], t ∈ I, x ∈ X and we have [1, x] = [1, x′] forall x, x′ ∈ X. The point given by [1, x] is called the apex of the cone and will be denoted by ∗. Weobtain an inclusion of X in the cone by composing the inclusion X → I ×X : x 7→ (0, x) with thequotient map to CX:

i : X → I ×X p→ CX : x 7→ [0, x]

Lemma 5. Let f : X → Y be a map of topological spaces. Then f is homotopic to a constant mapif and only if f extends over the cone CX in the sense that there is a map K : CX → Y such thatthe following diagram commutes:

Xf

//

i

��

Y

CX

K

=={{{{{{{{

Proof. Given a homotopy H : I ×X → Y with H0 = f and H1 = κy the constant map at y ∈ Yobviously factors over CX to give the desired map K. Conversely, if we have such a map K thenwe can precompose it with p : I × X → CX to obtain a homotopy H between f and a constantmap. That these two assignments are inverse bijections is just a special case of the last lemma. �

More precisely, the proof shows that there is a bijection between homotopies to constant maps andextensions over the cone. In the special case of X = ∆n we have a homeomorphism C∆n ∼= ∆n+1

such that the apex of the cone corresponds to e0 ∈ ∆n+1. A precise formula for this homeomorphismreads as follows where we use the short hand notation t′ = t1 + . . . + tn+1 and where ti are thebarycentric coordinates:

φ : ∆n+1 ∼=→ C∆n : (t0, . . . , tn+1) 7→{

[t0, t1/t′, . . . , tn+1/t

′] , t′ 6= 0∗ , t′ = 0

Moreover, this homeomorphism has the nice property that it identifies the face map

d0 : ∆n → ∆n+1 : (t0, . . . , tn) 7→ (0, t0, . . . , tn)

with the inclusion i : ∆n → C∆n, i.e., we have φ ◦ d0 = i : ∆n → C∆n. Thus, we have a bijectionbetween homotopies of σ : ∆n → Y to constant maps and extensions of σ to maps s(σ) : ∆n+1 → Ysuch that s(σ) ◦ d0 = σ.

Proposition 6. Let X be a contractible space, then all homology groups Hn(X) vanish for n ≥ 1.

Proof. Let X be contractible, i.e., we can find a point x0 ∈ X and a homotopy H : I × X → Xwith H0 = idX and H1 = κx0

, the constant map at x0. Now, let σ : ∆n → X be a basis element ofthe singular chain group Cn(X). Precomposition of H with idI ×σ yields a further such homotopyas indicated in the first row of the following diagram:

I ×∆n id×σ//

��

I ×X H // X

C∆n ∼= ∆n+1 sn(σ)

::

4 LECTURE 4: SINGULAR HOMOLOGY OF CONTRACTIBLE SPACES

Now, the composition H ◦ (id × σ) gives us a homotopy to a constant map so that it correspondsto a unique map sn(σ) : ∆n+1 → X. Additive extension gives us a group homomorphism:

sn : Cn(X)→ Cn+1(X)

By construction, it follows that we have the following relations for the faces of sn(σ):

d0(snσ) = σ, n ≥ 0, and di(sn(σ)) = sn−1(di−1(σ)), i = 1, . . . , n+ 1, n ≥ 1

In the remaining case n = 0 and i = 1 we have d1(s0σ) = x0 ∈ C0(X), where we use the standardconvention that x0 also denotes the map ∆0 → X sending the unique point in ∆0 to x0. But theserelations allow us to make the following calculation for n ≥ 1:

∂sn(σ) =

n+1∑i=0

(−1)idi(sn(σ))

= d0(sn(σ)) +

n+1∑i=1

(−1)idi(snσ)

= σ +

n+1∑i=1

(−1)isn−1(di−1σ)

= σ − sn−1(∂σ)

By linear extension we thus deduce for n ≥ 1 the relation ∂ ◦ sn + sn−1 ◦ ∂ = id. In the remainingdegree n = 0 where an 0-simplex is just given by a point x : ∆0 → X we have:

∂s0(x) = d0(s0(x))− d1(s0(x)) = x− x0Thus, we have the relation ∂ ◦ s0 = id − ε0 where ε0 : C0(X) → C0(X) sends each basis elementto x0. As an upshot of these calculations we thus constructed a chain homotopy s : id→ ε where εis the chain map (!) which is zero in all positive degrees and ε0 in degree 0. Thus, by Lemma 3 wehave id = 0: Hn(X)→ Hn(X) for n ≥ 1 which can only be the case if these groups vanish. �

This proposition implies that the homology of a large family of spaces vanishes in positivedimensions. This applies to points, vector spaces, discs, simplices, products of such spaces etc. Insome sense we are happy with this result since the motivational idea of homology was that we wantto have invariants which measure the ‘geometric complexity’ of spaces. And it is good to know thatthe invariants are trivial in these examples.

Furthermore, this result (applied to products of simplices) is essential in our approach to thehomotopy-invariance of singular homology. Given two spaces X, Y we want to relate the singularchain complexes C(X), C(Y ), C(X×Y ) and this relation is to be natural in the spaces. Recall thatgiven maps f : X → X ′ and g : Y → Y ′ then there is the product map (f, g) : X × Y → X ′ × Y ′which sends (x, y) to (f(x), g(y)).

Theorem 7. Given topological spaces X, Y then there are bilinear maps

× : Cp(X)× Cq(Y )→ Cp+q(X × Y ) : (c, d) 7→ c× d

for all p, q ≥ 0 called cross product maps with the following properties:i) For x ∈ X, y ∈ Y, σ : ∆p → X, and τ : ∆q → Y we have:

x× τ : ∆q ∼= ∆0 ×∆q (x,τ)→ X × Y and σ × y : ∆p ∼= ∆p ×∆0 (σ,y)→ X × Y

LECTURE 4: SINGULAR HOMOLOGY OF CONTRACTIBLE SPACES 5

ii) The cross product is natural in X and Y , i.e., for maps f : X → X ′ and g : Y → Y ′ we have:

(f, g)∗(c× d) = f∗(c)× g∗(d) in Cp+q(X′ × Y ′)

iii) The boundary ∂ is a derivation with respect to × in the sense that for c ∈ Cp(X) and d ∈ Cq(Y )we have:

∂(c× d) = ∂(c)× d + (−1)pc× ∂(d) in Cp+q−1(X × Y )

We will begin the next lecture with a proof of this theorem. A crucial step in this proof will usethat spaces of the form ∆p ×∆q have trivial homology groups in positive dimensions.

LECTURE 5: HOMOTOPY INVARIANCE OF SINGULAR HOMOLOGY

The aim of this lecture is to show that singular homology is homotopy-invariant. In the lastlecture we showed that the singular homology of a contractible space is trivial in positive dimensions.This result (applied to products of simplices) is essential to our approach to the homotopy-invarianceof singular homology.

Given X, Y ∈ Top we want to relate the singular chain complexes C(X), C(Y ), and C(X × Y )to each other and this relation is to be natural in the spaces. Recall that given maps f : X → X ′

and g : Y → Y ′ then there is the product map (f, g) : X × Y → X ′ × Y ′ which sends (x, y) to(f(x), g(y)).

Theorem 1. Associated to topological spaces X and Y there are bilinear maps

× : Cp(X)× Cq(Y )→ Cp+q(X × Y ) : (c, d) 7→ c× d

for all p, q ≥ 0, called cross product maps, with the following properties:i) For x ∈ X, y ∈ Y, σ : ∆p → X, and τ : ∆q → Y we have:

x× τ : ∆q ∼= ∆0 ×∆q (x,τ)→ X × Y and σ × y : ∆p ∼= ∆p ×∆0 (σ,y)→ X × Y

ii) The cross product is natural in X and Y , i.e., for maps f : X → X ′ and g : Y → Y ′ we have:

(f, g)∗(c× d) = f∗(c)× g∗(d) in Cp+q(X′ × Y ′)

iii) The boundary ∂ is a derivation with respect to × in the sense that for c ∈ Cp(X) and d ∈ Cq(Y )we have:

∂(c× d) = ∂(c)× d + (−1)pc× ∂(d) in Cp+q−1(X × Y )

Proof. The proof will be given by induction on n = p + q. By condition i) there is no choice ifp = 0 or q = 0. So let us assume we already constructed a cross product map for all p′+ q′ ≤ n− 1and let us consider p + q = n with p > 0, q > 0. Let us begin with the special case whereX = ∆p and Y = ∆q. In this case there are the special singular simplices σ = idp : ∆p → ∆p andτ = idq : ∆q → ∆q. Whatever idp× idq ∈ Cp+q(∆p×∆q) will be, condition iii) forces its boundaryto be:

∂(idp × idq) = ∂(idp)× idq + (−1)pidp × ∂(idq)

Using this relation again and our induction assumption we can calculate the boundary of theexpression on the right-hand-side to be zero, i.e.,:

∂(idp)× idq + (−1)pidp × ∂(idq) ∈ Zp+q−1(∆p ×∆q)

Since p+q ≥ 2 we can use that Hp+q−1(∆p×∆q) ∼= 0 to deduce that this cycle must be a boundaryof some chain in Cp+q(∆

p×∆q). Let us choose an arbitrary such chain and use this as the definitionof idp×idq ∈ Cp+q(∆p×∆q). We have now defined the cross product in the universal example whichforces the definition on all other simplices: Given a pair of simplices σ : ∆p → X and τ : ∆q → Ythe definition of their cross product is forced by property ii):

σ × τ = σ∗(idp)× τ∗(idq)!= (σ, τ)∗(idp × idq)

1

2 LECTURE 5: HOMOTOPY INVARIANCE OF SINGULAR HOMOLOGY

A bilinear extension concludes the definition of × : Cp(X)× Cq(Y )→ Cp+q(X × Y ). We still haveto verify that property iii) is satisfied for arbitrary basis elements σ and τ . But this is done by thefollowing calculation:

∂(σ × τ) =(∂ ◦ (σ, τ)∗

)(idp × idq)

=((σ, τ)∗ ◦ ∂

)(idp × idq)

= (σ, τ)∗(∂(idp)× idq + (−1)pidp × ∂(idq)

)= σ∗(∂(idp))× τ∗(idq) + (−1)pσ∗(idp)× τ∗(∂(idq))

= ∂(σ∗(idp))× τ∗(idq) + (−1)pσ∗(idp)× ∂(τ∗(idq))

= ∂(σ)× τ + (−1)pσ × ∂(τ)

Thus we have checked that the map × : Cp(X) × Cq(Y ) → Cp+q(X × Y ) satisfies the relations ii)and iii) which finishes the induction step. �

There were many choices made during the construction of the cross product. Nevertheless, it canbe shown that the collection of all such maps is essentially unique in a certain precise sense. Sincewe do not have the algebraic toolkit necessary to attack this statement we will not be pursuing thisany further.

Recall that a pair of spaces (X,A) is given by a space X together with a subspace A. Given twosuch pairs (X,A) and (Y,B) their product is defined to be (X,A)×(Y,B) = (X×Y,A×Y ∪X×B).It is straightforward to generalize the construction of the cross product to the context of pairs ofspaces. Namely, given (X,A), (Y,B) ∈ Top2 we obtain an induced cross product

× : Cp(X,A)× Cq(Y,B)→ Cp+q((X,A)× (Y,B)) = Cp+q(X × Y,A× Y ∪X ×B)

with similar formal properties as in the absolute case. Let us only give the definition of this map.For this purpose let c′′ ∈ Cp(X,A) be represented by c ∈ Cp(X) so that each other representativeof c′′ is of the form c + c′ for some c′ ∈ Cp(A) (we allow ourselves to be a bit sloppy in that wedrop the inclusions from notation!). Similarly, all representatives of d′′ ∈ Cq(Y,B) can be writtenas d + d′. The idea is to define c′′ × d′′ to be the relative chain represented by c × d. To see thatthis is well-defined it suffices to make the following calculation:

(c+ c′)× (d+ d′) = c× d + c× d′ + c′ × d + c′ × d′

But the sum of the last three chains lives in Cp+q(A× Y ∪X ×B) so that both (c+ c′)× (d+ d′)and c× d define the same relative chain in Cp+q((X,A)× (Y,B)). For later reference we collect thefollowing consequence of our last result.

Corollary 2. Given two pairs of spaces (X,A) and (Y,B) then the cross product at the level ofsingular chain groups induces a bilinear map:

× : Hp(X,A)×Hq(Y,B)→ Hp+q((X,A)× (Y,B))

This homomorphism is natural in (X,A) and (Y,B) and is called the (homology) cross product.

Proof. Let us consider relative homology classes ω ∈ Hp(X,A) and ω′ ∈ Hq(Y,B). We can repre-sent ω by a chain z in X which is a cycle relative to A, i.e., ∂(z) ∈ Cp−1(A). Similarly, ω′ can berepresented by a z′ ∈ Cq(Y ) with ∂(z′) ∈ Cq−1(B). Now, the boundary of z × z′ lies in:

∂(z × z′) = ∂(z)× z′ + (−1)pz × ∂(z′) ∈ Cp+q−1(A× Y ∪X ×B)

LECTURE 5: HOMOTOPY INVARIANCE OF SINGULAR HOMOLOGY 3

In other words, z × z′ defines a relative cycle in (X,A) × (Y,B). Let us show that the relativehomology class represented by z × z′ is well-defined. If we take a different representing relativecycle z + ∂(c) for ω then we calculate:

(z + ∂(c))× z′ = z × z′ + ∂(c)× z′ = z × z′ + ∂(c× z′) − (−1)p+1c× ∂(z′)

But since c × ∂(z′) ∈ Cp+q(X × B) this expression represents the same relative homology class.A similar calculation shows that the class is independent of the chosen representing relative cyclefor ω′. Thus the assignment

× : Hp(X,A)×Hq(Y,B)→ Hp+q((X,A)× (Y,B)) : ([z], [z′]) 7→ [z × z′]is well-defined. The bilinearity and naturality follow immediately from the corresponding propertiesof the cross product at the level of singular chains. �

With this preparation we can now establish the homotopy invariance of singular homology.

Theorem 3. Let X be a space and let I = [0, 1] be the interval. The inclusions ij : X → I × Xwhich send x to (j, x), j = 0, 1, induce chain homotopic maps (i0)∗ ' (i1)∗ : C(X) → C(I × X).Similarly, if (X,A) ∈ Top2 then the induced chain maps (i0)∗, (i1)∗ : C(X,A) → C(I ×X, I × A)are chain homotopic.

Proof. We give the proof in the absolute case. Let ι : ∆1 → I be the affine isomorphism with e0 7→ 0and e1 7→ 1. This singular 1-simplex has boundary ∂(ι) = ε1−ε0 where we wrote εj : ∆0 → I for the0-simplex corresponding to j ∈ I. We can use the cross product to obtain group homomorphisms:

sn = ι×− : Cn(X)→ Cn+1(I ×X) : c 7→ ι× cA calculation of the boundary of sn(c) gives:

∂(sn(c)) = ∂(ι× c) = ∂(ι)× c − ι× ∂(c) = ε1 × c − ε0 × c − sn−1(∂(c))

In the special case of n = 0 the right-hand-side simply reads as ε1× c − ε0× c. The ‘normalization’of the cross products shows that (ij)∗ = εj × − : Cn(X) → Cn(I ×X). Thus, these equations tellus that the homomorphisms sn = ι × − assemble to a chain homotopy s : (i0)∗ ' (i1)∗. Thus theinduced maps in homology are the same. �

Recall that a homotopy of maps f, g : (X,A)→ (Y,B) mod A is a map H : I×X → Y such thatH0 = f, H1 = g, and H(I × A) ⊂ B. If we agree that I × (X,A) is the pair (I ×X, I × A) thensuch a homotopy is, in particular, a map H : I × (X,A)→ (Y,B).

Corollary 4. Let f, g : X → Y be homotopic maps then the maps f∗, g∗ : C(X)→ C(Y ) are chainhomotopic. In particular, homotopic maps induce the same map in singular homology. Similarly, iff, g : (X,A)→ (Y,B) are maps which are homotopic relative to A then f∗, g∗ : C(X,A)→ C(Y,B)are chain homotopic. Thus, relative singular homology is homotopy-invariant.

Proof. We again content ourselves to give a proof in the absolute case. Let H : I × X → Y be ahomotopy from f to g, i.e., we have f = H ◦ i0 : X → I×X → Y and g = H ◦ i1 : X → I×X → Y .It suffices to do the following short calculation to conclude the proof:

f∗ = H∗ ◦ i0∗ ' H∗ ◦ i1∗ = g∗

Here we used that the chain homotopy relation behaves nicely with compositions (see the exercisessheet). �

Corollary 5. A homotopy equivalence f : X → Y induces isomorphisms f∗ : Hn(X)→ Hn(Y ).

4 LECTURE 5: HOMOTOPY INVARIANCE OF SINGULAR HOMOLOGY

Proof. By definition of a homotopy equivalence we can find a map g : Y → X and homotopiesg ◦ f ' idX and f ◦ g ' idY . But the last corollary then implies that we have:

g∗ ◦ f∗ = id and f∗ ◦ g∗ = id �

We want to finish this lecture by summarizing our main results so far. But before that letus quickly mention the naturality of the connecting homomorphisms associated to a short exactsequence. Let us sketch the construction of this homomorphism from Lecture 3. In the context of

a short exact sequence 0 → C ′i→ C

p→ C ′′ → 0 of chain complexes we defined a homomorphismδn : Hn(C ′′)→ Hn−1(C ′) by considering the following diagram:

Cnpn //

∂n

��

C ′′n

C ′n−1 in−1

// Cn

If ω ∈ Hn(C ′′) is represented by z′′n ∈ Zn(C ′′) then we showed that the expression ‘i−1n−1◦∂n◦p−1n (z′′n)’makes sense and defines a well-defined homology class δn(ω) ∈ Hn−1(C ′). We now want to showthat this homomorphism is natural with respect to morphisms of short exact sequences.

Proposition 6. Let us consider the following diagram of chain complexes in which the rows areshort exact sequences:

0 // C ′i //

f ′

��

C

f

��

p// C ′′ //

f ′′

��

0

0 // D′j// D

q// D′′ // 0

The connecting homomorphism is natural in the short exact sequence in the sense that for all n ≥ 1the following diagram commutes:

Hn(C ′′)δn //

f ′′∗��

Hn−1(C ′)

f ′∗��

Hn(D′′)δn

// Hn−1(D′)

Proof. The proof is left as an exercise and follows more or less directly from the fact that theconnecting homomorphism is well-defined. �

In the situation of the proposition it is immediate that the long exact sequences in homologyassociated to the respective short exact sequences of chain complexes assemble to the followingcommutative ‘ladder’:

. . . // Hn+1(C ′′)δn+1

//

f ′′∗��

Hn(C ′)

f ′∗��

i∗ // Hn(C)

f∗

��

p∗ // Hn(C ′′)

f ′′∗��

// . . .

. . . // Hn+1(D′′)δn+1

// Hn(D′)j∗// Hn(D)

q∗// Hn(D′′) // . . .

LECTURE 5: HOMOTOPY INVARIANCE OF SINGULAR HOMOLOGY 5

Corollary 7. Let f : (X,A)→ (Y,B) be a map of pairs of topological spaces. Then the long exactsequences in homology of the respective pairs fit together to give the following commutative diagram:

. . . // Hn+1(X,A)δn+1

//

f∗

��

Hn(A)

f∗

��

// Hn(X)

f∗

��

// Hn(X,A)

f∗

��

// . . .

. . . // Hn+1(Y,B)δn+1

// Hn(B) // Hn(Y ) // Hn(Y,B) // . . .

Proof. This is immediate from the above algebraic fact. In fact, it suffices to observe that f inducesthree chain maps C(A)→ C(B), C(X)→ C(Y ), and C(X,A)→ C(Y,B). These chain maps takentogether define a morphism between the short exact sequences of chain complexes which are usedto define the respective relative singular chain complexes. �

Let us summarize the most important results which we obtained during the last five lectures.We defined the singular homology groups of a space and also of a pair of spaces. We showed thatsingular homology is functorial in the pair and that the connecting homomorphisms assemble to acertain natural transformation. Moreover, this datum consisting of the singular homology functorsand the connecting homomorphisms has certain key properties:

• homotopy invariance: Corollary 4• long exact sequence in homology associated to a pair: Lecture 3, Corollary 11• homology of a point is concentrated in degree zero: Lecture 1, Example 12(2); exercises

sheet• additivity: Lecture 1, Example 13ii); exercises sheet

There is one additional important property, namely the so-called excision property of singularhomology. The reason why we emphasize these properties of singular homology is the following: Itcan be shown that the singular homology theory is essentially characterized by these five properties!In the next lecture we will state the remaining property. This is the last important ingredientnecessary for many interesting applications. Some of them will be discussed in the next lecture sothat we are sufficiently motivated to attack the proof of the excision property one week later.

LECTURE 6: EXCISION PROPERTY AND MAYER-VIETORIS SEQUENCE

In this lecture we will state the important excision property of singular homology which isone of the key features of singular homology allowing for calculations. While the proof of thisexcision property will only be given in the next lecture, we will here focus on some consequencesand applications. In particular, we will deduce the important Mayer-Vietoris sequence and thencalculate the homology groups of all spheres. A convenient reformulation is obtained in terms ofreduced homology groups.

Here is the important excision theorem.

Theorem 1. Let U ⊂ A ⊂ X be subspaces such that the closure U of U lies in the interior A◦ of A.Then the inclusion (X\U,A\U)→ (X,A) induces isomorphisms on relative homology groups:

Hn(X\U,A\U)∼=→ Hn(X,A), n ≥ 0

This theorem can be equivalently reformulated as follows.

Theorem 2. Let X be a space and let us consider subspaces X1, X2 ⊆ X such that X◦1 ∪X◦2 = X.Then the inclusion (X1, X1∩X2)→ (X,X2) induces isomorphisms on all relative homology groups:

Hn(X1, X1 ∩X2)∼=→ Hn(X,X2), n ≥ 0

Lemma 3. Theorem 1 and Theorem 2 are equivalent.

Proof. Exercise. (Hint: consider the assignments A = X2 and U = X\X1.) �

An important consequence of the second formulation of the theorem is given by the so-calledMayer-Vietoris sequence. This result is obtained by specializing the following algebraic fact to acertain topological situation.

Lemma 4. (Algebraic Mayer-Vietoris sequence) Let us consider the following commutative diagramof abelian groups in which the rows are exact and all the f ′′n are isomorphisms:

. . . // C ′′n+1

δn+1//

f ′′n+1

��

C ′n

f ′n��

in // Cn

fn

��

pn // C ′′n

f ′′n��

δn // C ′n−1//

f ′n−1

��

. . .

. . . // D′′n+1δ′n+1

// D′n jn// Dn qn

// D′′nδ′n

// D′n−1// . . .

Then there is an exact sequence in which ∆n = δn ◦ f ′′−1n ◦ qn : Dn → C ′n−1 :

. . . // C ′n(in,f

′n)// Cn ⊕D′n

fn−jn // Dn∆n // C ′n−1

// . . .

Proof. Let us give a proof of the exactness at C ′n−1. The relation (in−1, f′n−1)◦∆n = 0 is immediate

since we calculate for both coordinates:

in−1 ◦∆n = in−1 ◦ δn ◦ f ′′−1n ◦ qn = 0 and f ′n−1 ◦∆n = f ′n−1 ◦ δn ◦ f ′′−1

n ◦ qn = δ′n ◦ qn = 01

2 LECTURE 6: EXCISION PROPERTY AND MAYER-VIETORIS SEQUENCE

Conversely, let us assume that we have an element c′n−1 such that in−1(c′n−1) = 0 = f ′n−1(c′n−1).Exactness of the upper row at C ′n−1implies that there is an element c′′n ∈ C ′′n with δn(c′′n) = c′n−1.But

0 = f ′n−1(c′n−1) = f ′n−1(δn(c′′n)) = δ′n(f ′′n (c′′n))

shows that f ′′n (c′′n) lies in the kernel of δ′n. Finally, exactness of the lower row at D′′n implies thatthere is a dn ∈ Dn such that qn(dn) = f ′′n (c′′n). But for this dn we calculate

∆n(dn) = δn ◦ f ′′−1n ◦ qn(dn) = δn ◦ f ′′−1

n ◦ f ′′n (c′′n) = δn(c′′n) = c′n−1

as intended. Thus c′n−1 lies in the image of ∆n. The remaining two cases are left as an exerciseand can be established by similar diagram chases. �

The choice of the sign in the lemma was arbitrary. There are further choices and any of thesewould be equally good and lead to a similar statement. The only constraint was to reexpress thecommutativity of the square induced by the inclusion as the vanishing of the composition of twohomomorphisms with the given domains and targets.

We want to apply this to the following topological situation. Let X = X◦1 ∪X◦2 for two subspacesX1, X2 ⊂ X and let us consider the inclusion ι : (X1, X1∩X2)→ (X,X2) obtained from the followingcommutative square

X1 ∩X2j1 //

j2

��

X1

i1

��

X2i2

// X.

The naturality of the long exact sequence in homology with respect to morphisms of pairs impliesthat we have the following commutative ladder with exact rows:

. . . // Hn+1(X1, X1 ∩X2)δn+1

//

ι∗∼=��

Hn(X1 ∩X2)

j2∗

��

j1∗ // Hn(X1)

i1∗

��

// Hn(X1, X1 ∩X2)

ι∗∼=��

// . . .

. . . // Hn+1(X,X2)δ′n+1

// Hn(X2)i2∗

// Hn(X) // Hn(X,X2) // . . .

Since we are in the situation of Theorem 2 we know that all induced maps ι∗ are isomorphisms.Thus, if we denote by ∆n : Hn(X)→ Hn−1(X1 ∩X2) the homomorphism

∆n : Hn(X)→ Hn(X,X2)ι−1∗→ Hn(X1, X1 ∩X2)→ Hn−1(X1 ∩X2),

then the algebraic Mayer-Vietoris sequence (Lemma 4) specializes to the following result.

Theorem 5. In the above situation we have an exact sequence, the Mayer-Vietoris sequence:

. . . // Hn(X1 ∩X2)(j1∗,j2∗)// Hn(X1)⊕Hn(X2)

i1∗−i2∗// Hn(X)∆n // Hn−1(X1 ∩X2) // . . .

This theorem allows for inductive calculations of homology groups. We will illustrate this by thecalculation of the homology groups of spheres. Besides being interesting for its own sake, this willprovide the basis for a large class of examples to be studied in a later lecture.

LECTURE 6: EXCISION PROPERTY AND MAYER-VIETORIS SEQUENCE 3

Proposition 6. The singular homology groups of the spheres are as follows:

Hn(S0) ∼={

Z⊕ Z , n = 00 , otherwise

and Hn(Sm) ∼={

Z , n = 0, m0 , otherwise

if m > 0

Proof. Since S0 is just a disjoint union of two points we already calculated its homology groups.Moreover, since all the remaining spheres are path connected we know already all the zeroth ho-mology groups. Let us calculate the homology group of Sm for m ≥ 1. In these cases let us denoteby X1 ⊂ X = Sm the subspace obtained by removing the ‘north pole’ NP ∈ Sm. Similarly, letX2 ⊂ Sm be obtained by removing the ‘south pole’ SP ∈ Sm. We are in the situation of the Mayer-Vietoris sequence since the interiors of X1 and X2 cover X. Moreover, the intersection X1 ∩X2 ishomotopy equivalent to Sm−1 in these cases and the subspaces X1, X2 are contractible. Thus, thehomotopy invariance of singular homology implies that H∗(Xi) is trivial in positive dimensions andthat H∗(X1 ∩X2) ∼= H∗(S

m−1).Let us begin by applying the Mayer-Vietoris sequence in the case of m = 1. We know already

that H1(S1) ∼= Z since the same is true for the fundamental group of S1. For Hn(S1), n ≥ 2, therelevant part of the Mayer-Vietoris sequence is given by

. . .→ Hn(X1)⊕Hn(X2)→ Hn(S1)→ Hn−1(∗ t ∗)→ . . . .

But since both Hn(X1)⊕Hn(X2) and Hn−1(∗ t ∗) are zero the same is true for Hn(S1), n ≥ 2.We now proceed by induction: consider m ≥ 2 and assume that the calculations of the homology

of Sk, k ≤ m− 1, are already done. For the calculation of H1(Sm) we consider the following partof the Mayer-Vietoris sequence:

. . .→ H1(X1)⊕H1(X2)→ H1(Sm)→ H0(Sm−1)→ H0(X1)⊕H0(X2)

The path connectedness of Sm−1, X1, and X2 shows us that the last map in this sequence isisomorphic to Z → Z ⊕ Z : c 7→ (c, c). Since this map is injective and the Xi are contractible weconclude that H1(Sm) ∼= 0. For n ≥ 2 the interesting part of the Mayer-Vietoris sequence is:

. . .→ Hn(X1)⊕Hn(X2)→ Hn(Sm)→ Hn−1(Sm−1)→ Hn−1(X1)⊕Hn−1(X2)→ . . .

But the contractibility of the Xi implies that the outer groups are trivial so that we obtain anisomorphism Hn(Sm) ∼= Hn−1(Sm−1) in this range. The inductive assumption allows us to concludethe proof. �

These calculations show us that for spheres the homology groups are trivial ‘above the dimension’,which turns out to be an important feature of singular homology theory. Let us mention thefollowing immediate consequences.

Corollary 7. (1) For m 6= n the spheres Sm and Sn are not homotopy equivalent.(2) For m ≥ 0 the sphere Sm is not contractible.

Proof. This follows immediately from Proposition 6 and the homotopy invariance of singular ho-mology. �

All vector spaces of the form Rn are contractible so that they are all homotopy equivalent. How-ever, we can now use our above calculations to show that these vector spaces are not homeomorphicunless they have the same dimensions.

Corollary 8. (Invariance of dimension) For m 6= n the spaces Rm and Rn are not homeomorphic.

4 LECTURE 6: EXCISION PROPERTY AND MAYER-VIETORIS SEQUENCE

Proof. Let us assume we are given a homeomorphism f : Rm → Rn and let us choose an arbitraryx0 ∈ Rm. We obtain an induced homeomorphism f : Rm\{x0} ∼= Rn\{f(x0)}. If either of thedimensions happens to be zero then this must also be the case for the other one. So, let us assumethat both of them are different from zero. Then the domain and the target of this restrictedhomeomorphism is homotopy equivalent to a sphere of the respective dimension, and we thus get ahomotopy equivalence Sm−1 ' Sn−1. By Corollary 7, this can only be the case if m = n. �

The corresponding result in the ‘differentiable category’ is much easier. A diffeomorphismf : Rm → Rn induces an isomorphism at the level of tangent spaces dfx0 : Tx0Rm ∼= Tf(x0)Rn.Thus, the dimensions of these vector spaces have to coincide so that we get m = n.

There are many further classical applications. For the time being, we will content ourselves withthe following one. Let us denote by Dn ⊂ Rn the closed ball of radius 1 centered at the origin.

Proposition 9. (Brouwer fixed point theorem) Every continuous map f : Dn → Dn has a fixedpoint.

Proof. Let us assume that f has no fixed points at all. For each x ∈ Dn there is thus a uniqueray from f(x) through x. Each such ray has a unique point of intersection with the boundary Sn−1

of Dn which we denote by r(x). We leave it as an exercise to the reader to check that the assignmentx 7→ r(x) defines a continuous retraction r : Dn → Sn−1. Moreover, we leave it to the reader toshow that this contradicts the calculations in Proposition 6. �

In many cases it is convenient to consider a minor variant of singular homology given by reducedsingular homology. By definition, the k-th reduced homology group Hk(X) of a space X is the

k-th homology group of the following augmented chain complex C(X):

. . .→ C2(X)∂→ C1(X)

∂→ C0(X)ε→ Z

This chain complex differs from the usual singular chain complex by the fact that in degree −1there is an additional copy of the integers. The map

ε : C0(X)→ Z :

k∑i=1

nixi 7→k∑i=1

ni

is the augmentation map which already played a role in the identification of H0. We leave it as anexercise to show that from this definition we obtain isomorphisms:

Hk(X) ∼={Hk(X) , k > 0

Z⊕ H0(X) , k = 0

Thus, in positive dimensions the notions coincide while in dimension zero the reduced homologygroup is obtained from the unreduced one by splitting off a copy of the integers. Let us give twoexamples which follow immediately from our earlier calculations.

Example 10. (1) If X is a contractible space, then Hk(X) ∼= 0 for all k ≥ 0.

(2) For the spheres we have Hn(Sn) ∼= Z and Hk(Sn) ∼= 0 for all k 6= n.

Exercise 11. For a pointed space (X,x0) there is an isomorphism Hi(X) ∼= Hi(X,x0) which isnatural with respect to pointed maps, i.e., maps sending base points to base points.

Exercise 12. Let (X,A) be a pair of spaces. Then there is a long exact sequence

· · · → H2(X,A)→ H1(A)→ H1(X)→ H1(X,A)→ H0(A)→ H0(X)→ H0(X,A)→ 0.

LECTURE 6: EXCISION PROPERTY AND MAYER-VIETORIS SEQUENCE 5

Corollary 13. For each n ≥ 1 we have isomorphisms

Hk+1(Dn+1, Sn) ∼= Hk(Sn) ∼= Hk(Sn, ∗) ∼={

Z , k = n0 , otherwise.

Proof. We already know that this description is correct for the homology of the sphere so it remainsto discuss the case of (Dn+1, Sn). But this follows immediately from the long exact sequence inreduced homology. �

A generator of any of the groups Hn(Dn, Sn−1) ∼= Z is called a fundamental class or orienta-tion class. Note that these classes are only well-defined up to a sign. Sometimes it is convenientto make explicit, compatible choices for these orientation classes in all dimensions, and we will bea bit more specific when we discuss the degree maps Sn → Sn.

Related to this notion are the local homology groups of manifolds. Let M be a topologicalmanifold of dimension n and let x0 ∈ M. Then the k-th local homology group of M at x0 isHk(M,M − {x0}). By definition of a manifold, we can find an open neighborhood x0 ∈ V and ahomeomorphism V ∼= Rn sending x0 to 0 ∈ Rn. An application of excision implies that we haveisomorphisms

Hk(V, V − {x0})∼=→ Hk(M,M − {x0}).

Using the homeomorphism we obtain a further isomorphism Hk(V, V − {x0}) ∼= Hk(Rn,Rn − {0})and the reader will easily show that this group is isomorphic to Hk−1(Sn−1). Thus, the we have

Hk(M,M − {x0}) ∼={

Z , k = n0 , otherwise,

and any generatorω = ωx0

∈ Hn(M,M − {x0})is a local orientation class of M at x0. Also these local orientation classes are only well-definedup to a sign, and one can show that a manifold M is orientable if and only if local orientationclasses ωx0 , x0 ∈M, can be chosen in a compatible way (which we do not want to make precise).

LECTURE 7: PROOF OF EXCISION PROPERTY OF SINGULAR

HOMOLOGY

In this lecture we will give a proof of the excision property of singular homology. For convenience,let us quickly recall the statement.

Theorem 1. Let U ⊂ A ⊂ X be subspaces such that the closure U of U lies in the interior A◦ of A.Then the inclusion (X\U,A\U)→ (X,A) induces isomorphisms on relative homology groups:

Hn(X\U,A\U)∼=→ Hn(X,A), n ≥ 0

The proof of this theorem uses the notion of small chains in X. In the situation of the theorem,let us call a generator α : ∆n → X small if:

α(∆n) ⊆ A or α(∆n) ∩ U = ∅The second condition is obviously equivalent to α(∆n) ⊆ X\U . This notion is extended to singularn-chains in the obvious way: such a chain is small if it is a linear combination of small generators.Let us denote the subgroup of small singular n-chains by:

C ′n(X) ⊆ Cn(X)

This clearly defines a subcomplex of C(X) since the boundary of a small chain is again small. Letus define C ′n(X,A) by the following short exact sequence of abelian groups:

0→ Cn(A)→ C ′n(X)→ C ′n(X,A)→ 0

The inclusion Cn(A) ⊆ C ′n(X) is part of a morphism of chain complexes. It is thus immediatethat the C ′n(X,A) can be uniquely assembled into a chain complex such that we have a short exactsequence of chain complexes:

0→ C(A)→ C ′(X)→ C ′(X,A)→ 0

There are now two quotient complexes in sight: the relative singular chain complex C(X\U,A\U)and C ′(X,A). The two defining short exact sequences are related in the following way:

0 // Cn(A\U)

��

// Cn(X\U)

��

// Cn(X\U,A\U)

��

// 0

0 // Cn(A) // C ′n(X) // C ′n(X,A) // 0

The two solid vertical arrows are the obvious ones and since the square commutes we get an inducedmorphism of chain complexes C(X\U,A\U)→ C ′(X,A).

Lemma 2. The induced chain map C(X\U,A\U) → C ′(X,A) is an isomorphism of chain com-plexes.

Proof. We have to check that we have an isomorphism in each degree. The injectivity is left asan easy exercise. For the surjectivity each element [α] ∈ C ′n(X,A) can be represented by a smallchain α ∈ C ′n(X). But such an α can be decomposed as a sum α = β + γ where β ∈ Cn(A) andγ ∈ Cn(X\U). Thus, the element of Cn(X\U,A\U) represented by γ is sent to [α]. �

1

2 LECTURE 7: PROOF OF EXCISION PROPERTY OF SINGULAR HOMOLOGY

The proof of excision is now based on the following proposition.

Proposition 3. The inclusion of chain complexes C ′(X) → C(X) induces isomorphisms in ho-mology.

Let us assume this proposition for the moment and let us see how we can deduce Theorem 1from this.

Proof. (of Theorem 1 assuming Proposition 3)By definition of the chain complexes C ′(X,A) and C(X,A) there are the following two short exactsequences of chain complexes:

0 // C(A)

=

��

// C ′(X) //

��

C ′(X,A) //

��

0

0 // C(A) // C(X) // C(X,A) // 0

Since the square on the left commutes we get an induced map on the quotients as indicated by thedashed arrow. By the last proposition, the vertical map in the middle induces isomorphisms on allhomology groups. Moreover, this is obviously also the case for the vertical map on the left. Thenaturality of the long exact sequence induced in homology can now be used to conclude that alsothe chain map C ′(X,A) → C(X,A) induces isomorphisms in homology. Finally, the chain mapC(X\U,A\U)→ C(X,A) which we consider in the statement of the excision theorem factors as acomposition:

C(X\U,A\U)→ C ′(X,A)→ C(X,A)

By the above and by Lemma 2 we know that both maps induce isomorphisms in homology whichconcludes the proof. �

The main work is thus to establish Proposition 3. The proof of the proposition is based on theconstruction of two natural maps: a morphism of chain complexes

bsX : C(X)→ C(X)

for any space X (bs stands for barycentric subdivision) and a chain homotopy

RX : C•(X)→ C•+1(X)

between bsX and the identity. Thus, for any α ∈ Cn(X) we want to have:

∂RXn (α) +RX

n−1(∂α) = bsXn (α)− α

Note that the existence of the chain homotopy implies that bsX induces the identity in homology, i.e.,

[bsXn (α)] = [α] ∈ Hn(X)

for any α ∈ Cn(X) with ∂α = 0. It will follow from the construction that

(1) For any α ∈ Cn(X), if we apply bsX to α sufficiently many –say k– times, we get a smallchain, i.e.,

(bsXn )k(α) ∈ C ′n(X), k large enough.

(2) For any small α the chain RXn (α) is also small.

Exercise 4. Show that the existence of such maps would indeed imply Proposition 3.

LECTURE 7: PROOF OF EXCISION PROPERTY OF SINGULAR HOMOLOGY 3

Let us begin with a preliminary construction, the cone construction. Let K be a convex set(in some Rd, say) and let p ∈ K. For α : ∆n → K, let

Conep(α) : ∆n+1 → K

be the map:

Conep(α)(t0, . . . , tn+1) = t0p+ (1− t0)α(t′1, . . . , t′n+1)

Here, t′i = ti/(1− t0), t0 < 1. More geometrically, ∆n+1 is the convex hull of its zeroth vertex andthe n-simplex opposite to that vertex (which is spanned by the remaining n+ 1 vertices). We wantConep(α) to map the zeroth vertex to p and we want it to be α on the opposite n-simplex. This isforced by defining Conep(α) to be the convex linear extension in the t0-direction of these two maps.The linear extension of Conep to chains will be denoted by the same notation:

Conep : Cn(K)→ Cn+1(K)

Notice the important cone formula:

(1) ∂Conep(α) = α− Conep(∂α), α ∈ Cn(K)

This formula and these constructions should remind you of the way we prepared the proof of thehomotopy invariance of singular homology (see Lecture 5, Lemma 5 and Proposition 6).

We now turn to the actual construction of the maps bsX and RX . Note first that naturalitymeans that the following squares commute for any f : X → Y :

Cn(X)bsXn //

f∗

��

Cn(X)

f∗

��

Cn(X)RX

n //

f∗

��

Cn+1(X)

f∗

��

Cn(Y )bsYn

// Cn(Y ) Cn(Y )RY

n

// Cn+1(Y )

This naturality together with the linearity of these maps has the consequence that bs and R arecompletely determined by their effect on the identity maps ηn:

ηn = (id: ∆n → ∆n) ∈ Cn(∆n)

Indeed, given a generator α ∈ Cn(X), α : ∆n → X, we have:

(2) bsXn (α) = α∗(bs∆n

n (ηn)) and RXn (α) = α∗(R

∆n

n (ηn))

We begin by defining bsXn for all X by induction on n. For n = 0 let us put bs∆0

0 (η0) = η0. This

defines bsX0 (α) for all spaces X and all α ∈ C0(X). For the induction step, let us suppose that

bsXn (α) has already been defined for all X and all α. Define

bs∆n+1

n+1 (ηn+1) = Conez(bs∆n+1

n (∂ηn+1))

where z = zn+1 is the barycenter of ∆n+1. Naturality forces the definition of bsXn+1(α) for anygenerator α ∈ Cn+1(X) by the above formulas, which is linearly extended to arbitrary (n+1)-chains.

This concludes the definition of the barycentric subdivision operators bsXn : Cn(X)→ Cn(X).

Exercise 5. Draw some low-dimensional pictures to convince yourself that this is a good definitionfor a barycentric subdivision. Do the exercise!

Lemma 6. The maps bsXn : Cn(X)→ Cn(X), n ≥ 0, define a chain map C(X)→ C(X).

4 LECTURE 7: PROOF OF EXCISION PROPERTY OF SINGULAR HOMOLOGY

Proof. We prove ∂ ◦ bsXn (α) = bsXn−1 ◦ ∂(α) by induction on n. For n = 0 it is clear. Suppose theformula holds for all X and α, for a fixed n. Then for n+1 we have the following chain of identitieswhere the first three are given by definition and by the fact that α∗ is a chain map:

∂ ◦ bsXn+1(α) = ∂ ◦ α∗ ◦ bs∆n+1

n+1 (ηn+1)

= ∂ ◦ α∗ ◦ Conez(bs∆n+1

n (∂ηn+1))

= α∗ ◦ ∂ ◦ Conez(bs∆n+1

n (∂ηn+1))

= α∗ ◦ bs∆n+1

n (∂ηn+1) − α∗ ◦ Conez(∂ ◦ bs∆n+1

n (∂ηn+1))

= α∗ ◦ bs∆n+1

n (∂ηn+1)

= bsXn ◦ α∗(∂ηn+1)

= bsXn ◦ ∂(α)

The fourth and the fifth step are given by the cone formula (1) and the induction assumptionrespectively while the remaining steps again follow from the definition. �

The next step is to define the chain homotopies RXn : Cn(X) → Cn+1(X), again by induction

on n and in such a way that the homotopy formula will hold. In this construction we use theso-called method af acyclic models. Recall from Lecture 4, Proposition 6 that contractible spaceshave trivial homology groups in positive dimensions which applies, in particular, to simplices. Indimension 0 we set:

R∆0

0 (η0) = (∆1 → ∆0) ∈ C1(∆0)

Since the boundary of R∆0

0 (η0) is zero the homotopy formula is satisfied in this dimension. Thisdefines RX

0 for all spaces X (by means of (2)). For the inductive step, let us now suppose that RXn

has already been defined for all X, in such a way that the homotopy formula

∂ ◦RXn +RX

n−1 ◦ ∂ = bsXn − id

holds for all X. As we already know, to define RXn+1 for all X, it is enough to find an element

β = R∆n+1

n+1 (ηn+1) ∈ Cn+2(∆n+1).

This β should satisfy the formula ∂β +R∆n+1

n (∂ηn+1) = bs∆n+1

n+1 (ηn+1)− ηn+1, i.e.,

∂β = −R∆n+1

n (∂ηn+1) + bs∆n+1

n+1 (ηn+1)− ηn+1 in Cn+1(∆n+1).

To prove that such a β exists, it is enough to show that the right-hand-side is a cycle. We canthen use that Hn+1(∆n+1) ∼= 0 (since n is at least 0!) in order to conclude that this cycle has tobe a boundary, i.e., that such a β exists. The fact that the right-hand-side is a cycle follows from

the following calculation using the homotopy formula in dimension n and the fact that bs∆n+1

is achain map:

∂(−R∆n+1

n (∂ηn+1) + bs∆n+1

n+1 (ηn+1)− ηn+1

)=

(R∆n+1

n−1 (∂∂ηn+1)− bs∆n+1

n (∂ηn+1) + ∂ηn+1

)+ ∂bs∆n+1

n+1 (ηn+1)− ∂ηn+1

= 0

Thus, we can find such a β and this concludes the inductive construction of the natural chainhomotopy RX for all X.

LECTURE 7: PROOF OF EXCISION PROPERTY OF SINGULAR HOMOLOGY 5

As a final step, given an arbitrary generator α : ∆n → X it only remains to show that (bsXn )k(α)is small for k large enough and that RX

n sends small simplices to small simplices. Let us observe

first that (bs∆n

n )k(ηn) is a linear combination of affine maps ∆n → ∆n. Moreover, the diameterof the images of these affine maps becomes arbitrarily small as k increases. Thus, the smallness of(bsXn )k(α) will follow from the existence of a Lebesgue number (see the next lemma) applied to theopen cover of ∆n given by

α−1(A◦) and α−1(X − U).

(This is an open cover by our assumption in the exicision theorem: U ⊆ A◦.) The fact that RXn (α)

is small if α is small follows immediately from the fact that RXn (α) lies in the image of α∗. Thus,

this concludes the proof of Proposition 3 and hence of the excision property.

Lemma 7. Let (Y, d) be a compact metric space and let (Ui)i∈I be an open cover of Y . Then thereis a positive real number λ, called a Lebesgue number of the cover, such that every subset of Yof diameter less than λ is entirely contained in Ui for some i.

Proof. This is one of the exercises of the exercise sheet. �

It might be enlightening for the reader to again have a look at the proof of the homotopyinvariance in Lecture 5. That proof was based on the construction of the chain-level cross productwhich in turn was also given by the method of acyclic models. Thus, two of the key features ofsingular homology (homotopy invariance and excision) have been established by this method.

Moreover, the method of acyclic models itself uses in an essential way that the homology ofsimplices (or contractible spaces) is trivial in positive dimensions. To get this vanishing resultwe already used the cone construction in Lecture 4. Thus, judged from this perspective the coneconstruction is one of the essential ingredients at least in our treatment of singular homology theory.

LECTURE 8: CW COMPLEXES

In this section we introduce CW complexes, an important class of spaces which can be builtinductively by gluing ‘cells’. Here we will study basic notions and examples, some facts concerningthe point set topology of these spaces, and also give elementary constructions.

By the very definition, a CW complex is given by a space which admits a filtration such thateach next filtration step is obtained from the previous one by attaching cells. Let us begin byintroducing this process. Let en = {(x0, . . . , xn−1) ∈ Rn |

∑x2i ≤ 1} be a copy of the (closed) n-

ball. Its boundary ∂en = Sn−1 is the (n− 1)-sphere (for n = 0 we take ∂e0 = ∅). If X is any spaceand χ : ∂en → X a map, one can form the new space X ∪χ en defined as the pushout

∂en

i

��

χ// X

��

en // X ∪χ en.

More explicitly, X ∪χ en is the space obtained from the disjoint union X t en by identifying eachi(y) ∈ en with χ(y) ∈ X for all y ∈ ∂en, and equipping the resulting set with the quotient topology.The universal property of this quotient is as follows.

Exercise 1. (1) The maps X → X ∪χ en and en → X ∪χ en are continuous and make theabove square commutative. Moreover, the triple consisting of the space X ∪χ en and thesetwo maps is initial with respect to this property. In other words, for all triples (W, g, h)consisting of a topological space W and continuous maps g : X →W and h : en →W suchthat the outer square in the following diagram commutes

∂en

��

// X

�� g

��

en //

h //

X ∪χ en

∃!$$

W

there is a unique dashed arrow X ∪χ en →W such that the two triangles commute.(2) Define more generally the notion of a pushout for two arbitrary maps A→ X and A→ Y

of spaces with a common domain. Show that the pushout exists and is unique up to aunique homeomorphism in a way which is compatible with the structure maps.

(3) The notion of a pushout makes sense in every category but does not necessarily exist. Tofamiliarize yourself with the concept, show that the categories of sets and of abelian groupshave pushouts by giving an explicit construction.

We refer to the space X ∪χ en as being obtained from X by ‘attaching an n-cell’, and callχ : ∂en → X the attaching map, and en → X ∪χ en the characteristic map of the ‘cell’ en.Note that this characteristic map restricts to a homeomorphism of the interior of en to its image

1

2 LECTURE 8: CW COMPLEXES

in X ∪χ en, i.e., we have a relative homeomorphism (en, ∂en) → (X ∪χ en, X). The image of thishomeomorphism is called the open cell, and the image of en → X ∪χ en the closed cell of thisattachment.

Usually one attaches more than one cell, and writes eσ for the cell with ‘index σ’, sometimesleaving the dimension implicit. If χ : ∂eσ → X is the attaching map, it is handy to freeze theindex σ, and write χ∂σ for the attaching map, χσ for the characteristic map, and refer to eσ or itsimage as the cell (with index) σ.

Thus if we obtain Y from X by attaching a set Jn of n-cells, then, by considering Jn as a discretespace, we have a pushout diagram of the form

Jn × ∂en =⊔σ∈Jn ∂e

nσ

��

// X

��

Jn × en =⊔σ∈Jn e

nσ

// Y.

In particular, a subset of Y is open if and only if its preimages in X and each en are open, i.e., Ycarries the quotient topology.

Definition 2. Let X be a topological space. A CW decomposition of X is a sequence ofsubspaces

X(0) ⊆ X(1) ⊆ X(2) ⊆ . . . ⊆ X(n) ⊆ . . . , n ∈ N,such that the following three conditions are satisfied:

(1) The space X(0) is discrete.(2) The space X(n) is obtained from X(n−1) by attaching a (possibly) infinite number of n-cells{enσ}σ∈Jn via attaching maps χσ : ∂enσ → X(n−1).

(3) We have X =⋃X(n) with the weak topology (this means that a set U ⊆ X is open if and

only if U ∩X(n) is open in X(n) for all n ≥ 0).

A CW decomposition is called finite if there are only finitely many cells involved. A (finite) CWcomplex is a space X equipped with a (finite) CW decomposition. Given a CW decomposition ofa space X then the subspace X(n) is called the n-skeleton of X.

Remark 3. (1) Note that by the very definition a CW complex is a space together with anadditional structure given by the CW decomposition. Nevertheless, we will always onlywrite X for a topological space endowed with a CW decomposition.

(2) Condition (3) in Definition 2 is only needed for infinite complexes.(3) From the definition of the weak topology it also follows that closed subsets of X can be

detected by considering the intersections with all skeleta X(n).(4) The image of a characteristic map χσ : eσ → X is called a closed cell in X, and the image

of χσ : e◦σ → X an open cell. These need not be open in X! Every point of X belongs toX(0) or lies in a unique open cell.

(5) Each X(n) is a closed subspace of X(n+1), and hence of X. (The open (n+1)-cells are openin X(n+1) but not necessarily in X).

Example 4. (1) The interval I = [0, 1] has a CW decomposition with two 0-cells and one1-cell by identifying the boundary of the unique 1-cell with the two 0-cells as expected.

(2) The circle S1 has a CW decomposition with one 0-cell and one 1-cell and no other cells. Ofcourse, it also has a CW composition with two 0-cells and two 1-cells.

LECTURE 8: CW COMPLEXES 3

(3) More generally, if one identifies the boundary ∂en of the n-ball to a point, one obtains (aspace homeomorphic to) the n-sphere. Thus the n-sphere has a CW decomposition withone 0-cell and one n-cell, and no other cells. One can also build up the n-sphere by startingwith two points, then two half circles to form S1, then two hemispheres to form S2, andso on. Then Sn has a CW decomposition with exactly 2 i-cells for i = 0, . . . , n (draw apicture for n ≤ 2!). If we take the coordinates (x0, . . . , xn) with

∑x2i = 1 for Sn as before,

these two i-cells are

ei+ = {(x0, . . . , xi, 0 . . . , 0) ∈ Sn | xi ≥ 0}

and

ei− = {(x0, . . . , xi, 0 . . . , 0) ∈ Sn | xi ≤ 0}.(4) The real projective space RPn, the space of lines through the origin in Rn+1, can be

constructed as the quotient Sn/Z2 where Z2 = Z/2Z acts on the n-sphere by the antipodalmap; in other words, by the quotient of Sn obtained by identifying x and −x. This identi-fication maps the cell ei+ to ei−. Thus RPn has a CW decomposition with exactly one i-cellfor i = 0, . . . , n.

(5) The complex projective space CPn is the space of complex lines through the originin Cn+1. Such a line is determined by a point (z0, . . . , zn) 6= 0 on the line, and for anyscalar λ ∈ C − {0} the tuple (λz0, . . . , λzn) determines the same line for which we write[z0, . . . , zn]. The line can also be represented by a point z = (z0, . . . , zn) with |z| = 1, sothat z and λz represent the same line for all λ ∈ S1. Thus CPn = S2n+1/S1 is a space of(real) dimension 2n. There are inclusions

∗ = CP0 ⊆ CP1 ⊆ CP2 ⊆ . . .

where CPn−1 ⊆ CPn sends [z0, . . . , zn−1] to [z0, . . . , zn−1, 0]. An arbitrary point in CPn −CPn−1 can be uniquely represented by (z0, . . . , zn−1, t) where t > 0 is the real number√

1−∑zizi. This defines a map

e2n → CPn : z = (z0, . . . , zn−1) 7→ [z0, . . . , zn−1, t]

with t =√

1− ||z||. The boundary of e2n (where t = 0) is sent to CPn−1. In this way, CPn

is obtained from CPn−1 by attaching one 2n-cell. So CPn has a CW structure with one cellin each even dimension 0, 2, . . . , 2n.

(6) Every compact manifold is homotopy equivalent to a CW complex. One can even show thatevery topological space is weakly homotopy equivalent to a CW complex. (Both of thesestatements are theorems which we only include to indicate the generality of the notion.)

Exercise 5. (1) The torus T can be obtained from the square by identifying opposite sides.Use an adapted CW decomposition of the square to also turn the torus into a CW complex.

(2) Similarly we can obtain the Klein bottle from the unit square by identifying (0, t) ∼ (1, t)and (s, 0) ∼ (1− s, 1). Show that there is a similar CW decomposition of the Klein bottle.

(3) Can you come up with CW decompositions of the torus and the Klein bottle which havethe same number of cells in each dimension? In particular this shows the obvious fact thatthe number of cells does not determine the space.

Lemma 6. Let X be a CW complex and let U be a subset of X. Then a subset U ⊂ X is open ifand only if U ∩X(n) is open for each n if and only if χ−1σ (U) ⊆ enσ is open for each cell σ of X.


Proof. The equivalence of the first two statements holds true by definition of CW complexes. Itis immediate that the second condition implies the third one. We want to prove the converseimplication by induction so let us begin by observing that U ∩X(0) is open in X(0) since X(0) isdiscrete. For the inductive step, let us assume that U ∩X(n−1) is open in X(n−1) for some n ≥ 1.Recall that we then have a pushout diagram of the following form:

Jn × ∂en =⊔σ∈Jn ∂e

nσ

��

// X(n−1)

��

Jn × en =⊔σ∈Jn e

nσ

// X(n)

By assumption χ−1σ (U) ⊆ enσ is open for every σ ∈ Jn. But the above pushout diagram togetherwith the induction assumption then tells us that also U ∩ X(n) is open in X(n) concluding theproof. �

Thus, given a CW complex X with n-cells parametrized by index sets Jn, then taking all theattaching maps together we obtain a map

(χσ)n,σ :⊔n

Jn × en ∼=⊔n

⊔σ∈Jn

enσ → X.

The above lemma shows that X carries the quotient topology with respect to this map.

Corollary 7. Let X be a CW complex, Y a topological space, and g : X → Y a map of sets. Thenthe following are equivalent:

(1) The map g : X → Y is continuous.(2) The restriction g | : X(n) → Y is continuous for all n ≥ 0.(3) The map g ◦ χσ : enσ → Y is continuous for each cell enσ.

This corollary allows us to build continuous maps ‘cell by cell’. Thus, not only CW complexescan be built inductively by attaching cells but the same holds also true for maps defined on a CWcomplex. There is also a similar result for homotopies.

Exercise 8. Let X be a CW complex, Y a topological space, and H : X × I → Y a map of sets.Then H is continuous if and only if each composition

H ◦ (χσ ×idI) : enσ × I → X × I → Y

is continuous for each cell enσ of X.

Before turning to CW subcomplexes and an adapted class of morphisms, let us establish somemore fundamental properties of CW complexes.

Exercise 9. A CW complex is normal. Thus show that disjoint closed subsets have disjoint openneighborhoods and that points are closed.

In studying the topology of CW complexes, one often uses the following fact.

Proposition 10. Any compact subset of a CW complex is contained in finitely many open cells.

This proposition in fact immediately follows from the following statement, by choosing a pointin every open cell that intersects non-trivially the given compact subset.

LECTURE 8: CW COMPLEXES 5

Lemma 11. Let X be a CW complex and A ⊂ X a subspace. If A has at most one point in eachopen cell then A is closed in X and the subspace topology on A is discrete.

Proof. We check this by induction on n and for each A∩X(n) as a subspace of X(n) (the closure thenfollows by definition of the weak topology on X). For n = 0 there is nothing to prove since X(0) isdiscrete. Suppose the statement has been proved for A∩X(n−1) ⊆ X(n−1). Write A∩X(n) = BtCwhere B = A∩X(n−1) and C = A∩ (X(n)−X(n−1)). Then C is open in A because the open n-cellsare open in X(n), and for the same reason C is discrete. The set C is closed in X(n) because ifx ∈ C then x lies in the same open cell as any point c ∈ C close to x, hence x = c. So C isclosed and discrete in X(n). Also B is closed and discrete in X(n−1) by induction hypothesis, hencein X(n) because X(n−1) ⊆ X(n) is closed. Then B t C has the same properties, which completesthe induction step. �

Remark 12. This proposition allows us to explain the terminology ‘CW complex’. In the originaldefinition given by J.H.C. Whitehead, the following two properties played a more essential role:

(C): The closure of every cell lies in a finite subcomplex (‘closure finite’).(W): A subset is open if and only if it is open in the n-skeleton for all n (‘weak topology’).

We now turn to an adapted class of morphisms between CW complexes.

Definition 13. A map f : X → Y between CW complexes is cellular if it satisfies f(X(n)) ⊆ Y (n)

for all n. It is immediate that we have a category of CW complexes and cellular maps.

Thus, such a cellular map induces commutative diagrams of the form:

X(n)

i

��

f |// Y (n)

i

��

Xf

// Y

Let us give some examples of cellular maps. One can show that this notion is rather generic.

Example 14. (1) The vector space Rn maps injectively to Rn+1 by adding a zero as the lastcoordinate, i.e., we have a map

in : Rn → Rn+1 : (t1, . . . , tn) 7→ (t1, . . . , tn, 0).

These maps restrict to maps of spheres as follows

jn = in+1 | : Sn → Sn+1

and these maps are cellular with respect to the CW decompositions on the spheres withprecisely two cells in each dimension lower or equal to the dimension of the respective sphere(but not with respect to the other one).

(2) Since the inclusions in : Rn → Rn+1 are compatible with the actions by R×, we obtainedinduced maps j′n : RPn−1 → RPn which are easily seen to be cellular with respect to theCW decomposition of Example 4. The maps are also obtained from the maps jn of the lastexample by passing to the quotient by the Z/2Z-action and these quotient maps are also


cellular. Thus, we have a diagram of cellular maps:

S0 j0 //

q0��

S1 j1 //

q1��

S2 j2 //

q2��

. . .

RP 0

j′0

// RP 1

j′1

// RP 2

j′2

// . . .

Similarly, in the case of complex numbers, we have cellular maps:

CP 0 → CP 1 → CP 2 → . . .

(3) In Example 4 we introduced two CW decompositions on the n-sphere. Let us write Sn for

the one with two cells in each dimension d ≤ n while we write Sn for the one with precisely

one 0-cell and one n-cell. Then the identity map id: Sn → Sn is cellular, while this is not

the case for id : Sn → Sn if n ≥ 2.

LECTURE 9: BASIC HOMOLOGICAL ASPECTS OF CW COMPLEXES

We begin this lecture by a brief introduction to some basic aspects of CW complexes (subcom-plexes, quotient comlexes, and the subtleties concerning products). In the remainder of the lecturewe establish a few basic homological properties of CW complexes. This will make necessary a briefdiscussion of the compatibility of singular homology and directed colimits.

Recall from the previous lecture that CW complexes are spaces which can be inductively obtainedform discrete spaces by attaching cells. We now turn to subcomplexes of CW complexes.

Proposition 1. Let X be a CW complex and let Y ⊆ X be a closed subspace such that theintersection Y ∩ (X(n) −X(n−1)) is the union of open n-cells. The filtration

Y (0) ⊆ Y (1) ⊆ . . . ⊆ Y

given by Y (n) = Y ∩X(n) then defines a CW decomposition on Y . Moreover, the inclusion Y → Xis then a cellular map.

As a special case, this proposition suggests how to define pointed CW complexes and, moregenerally, pairs of CW complexes.

Definition 2. In the notation of the above proposition, we refer to Y as a CW subcomplex of Xand to (X,Y ) as a CW pair. A pointed CW complex (X,x0) is a CW complex X togetherwith a chosen base point x0 ∈ X(0).

In the obvious way, this gives us the category of pointed CW complexes and CW pairs whosedefinitions are left to the reader.

Example 3. (1) For an arbitrary CW complex X, we have CW pairs (X,X(n)) for all n andsimilarly (X(n), X(m)) for n ≥ m.

(2) We have CW pairs (Sn, Sm), (RPn,RPm) and similarly in the complex case for n ≥ m. Ifwe endow the unions

S∞ =⋃n

Sn, RP∞ =⋃n

RPn, and CP∞ =⋃n

CPn

with the weak topology then each of the three spaces carries canonically a CW structure.Moreover, we have CW pairs (S∞, Sn), (RP∞,RPn). and (CP∞,CPn) for all n.

Exercise 4. Let (X,Y ) be a CW pair. Then the quotient space X/Y can be turned in a CWcomplex such that the quotient map X → X/Y is cellular.

We will now establish a few more closure properties of CW complexes. Let us begin with a moredifficult one, namely the product. Recall that we observed that each CW complex is obtained froma disjoint union of cells by passing to a quotient space. Namely, for a CW complex X we have aquotient map: ⊔

n

Jn × en → X

1

2 LECTURE 9: BASIC HOMOLOGICAL ASPECTS OF CW COMPLEXES

Given two CW complexes X and Y one might now try to take two such presentations⊔n

Jn(X)× en → X and⊔m

Jm(Y )× em → Y

and use homeomorphisms en × em ∼= en+m to obtain a map⊔k

Jk(X × Y )× ek → X × Y

where Jk(X×Y ) = tk′+k′′=kJk′(X)×Jk′′(Y ). However, this map is, in general, not a quotient map.More conceptually, the problem is that the formation of products and quotients in the category ofspaces are not compatible in general. Nevertheless, under certain ‘finiteness conditions’ one canobtain a positive result.

Proposition 5. Let X,K be CW complexes such that K is finite. Then the product X × K isagain a CW complex with the above CW decomposition.

Using the previous proposition we can establish additional closure properties for the class of CWcomplexes.

Corollary 6. (1) The disjoint union of two CW complexes is again a CW complex such thatthe inclusions of the respective summands are cellular.

(2) Given a CW complex X then the cylinder X × I is again a CW complex. For each n-cell enσ of X we obtain three cells for X × I, namely two n-cells enσ × {0}, enσ × {1}, and an(n + 1)-cell enσ × e1. Moreover, the cylinder comes with cellular maps i0, i1 : X → X × Iand p : X × I → X.

(3) Given a CW complex X, then the unreduced suspension SX is again a CW complex. Infact, we know that the cylinder of X is a CW complex, and SX is obtained in two steps bypassing to the quotient of a subcomplex.

Proof. The first statement is immediate while the other ones follow from Proposition 5 and theexamples of the previous lecture. �

In the definition of a CW complex X, the first condition we imposed was that X(0) is to bea discrete space and then that the higher skeleta are obtained from the lower ones by attachingn-cells for n ≥ 1. We can also think of X(0) as being obtained from the empty space by attaching0-cells; in fact, using the convention that ∂e0 = ∅ we have a pushout:

X(0) × ∂e0 =⊔σ∈X0

∂e0σ

��

∼= // X(−1) = ∅

��

X(0) × e0 =⊔σ∈J0 e

0σ ∼=

// X(0)

This observation is more than only a rather picky remark since it motivates the following general-ization of the notion of CW complex.

Definition 7. Let (X,A) be a pair of spaces. Then X is a CW complex relative to A, if thereis a filtration of X,

A = X(−1) ⊆ X(0) ⊆ X(1) ⊆ . . . ⊆ X,such that the following two properties are satisfied:

(1) The space X(n) is obtained from X(n−1) by attaching n-cells for n ≥ 0.

LECTURE 9: BASIC HOMOLOGICAL ASPECTS OF CW COMPLEXES 3

(2) The space X is the union⋃n≥−1X

(n) endowed with the weak topology.

In this situation, the pair (X,A) is called a relative CW complex.

Example 8. (1) Let X be a CW complex and x0 ∈ X0. Then we have a relative CW complex(X,x0).

(2) More generally, every CW pair is a relative CW complex.

We now turn to basic homological aspects of CW complexes. Recall that we have a goodunderstanding of the homology of the spheres,

Hk(Sn, ∗) ∼={


The n-sphere is obtained from a point by attaching an n-cell. Since CW complexes are obtained in-ductively by attaching cells, a first step towards an understanding of the homology of CW complexeswould consist of understanding the effect of the attachment of a cell at the level at homology.

Lemma 9. Let X be obtained from A by attaching a n-cell along f : ∂en → A, X = A∪f en. Then

Hk(X,A) ∼={


Moreover, the attaching map applied to any orientation class of (en, ∂en) gives us a generator ofHn(X,A).

Proof. Let N ⊆ en be a small collar around the boundary of en, e.g., N = ∂en × (1− ε, 1] for somesmall ε > 0. Then by gradually shrinking this collar back to the boundary, we obtain a homotopyequivalences (X,A) ' (X,A∪N), and hence isomorphisms Hk(X,A) ∼= Hk(X,A∪N) for all k ≥ 0by homotopy invariance. But A ⊆ (A ∪ N)◦ so by excision the map induced by the inclusion inhomology is an isomorphism,

Hk(X −A,N −A) ∼= Hk(X,A ∪N), k ≥ 0.

Now X −A is an open ball of dimension n and N −A is a collar, so clearly by collapsing this collarto a sphere we obtain an additional homotopy equivalence

(X −A,N −A) ' (en, ∂en).

Thus, again by homotopy invariance, we obtain isomorphisms

Hk(X −A,N −A) ∼= Hk(en, ∂en)

for all k ≥ 0, and the first statement follows by the long exact sequence associated to this latterpair. We leave it to the reader to go through the construction and to check the statement aboutthe generators of Hn(X,A). �

Thus this lemma tells us that if a space is obtained by attaching an n-cell then there is a uniquecopy of the integer in the corresponding homology group, and a generator is given by the cell itself.Using this lemma, we can draw some consequences for the homology of CW complexes by ‘inductionon the number of the cells’. In this lecture we will only achieve a first step and continue with theprogram in the next lecture.

As the case of finite CW complexes requires less machinery, we treat that case independently.

Proposition 10. For any finite CW complex X, Hk(X(n), X(n−1)) ∼= 0 for all k 6= n.


Proof. The proof will be given by induction. The assertion is true if X has dimension 0. SupposeX = A∪f ei and the assertion is true for A. We claim it also holds for X. There are three cases tobe discussed:

(1) If n < i then (X(n), X(n−1)) = (A(n), A(n−1)) and there is nothing to prove.(2) If n > i then (X(n), X(n−1)) = (A(n) ∪χ ei, A(n−1) ∪χ ei) where χ denotes the attaching

map of the i-cell. If B is a closed ball inside the interior of ei, then the B is also containedin the interior of X(n−1). Thus by excision we deduce that the inclusion

(X(n) −B,X(n−1) −B)→ (X(n), X(n−1))

induces isomorphisms in homology. But by gradually expanding this ball to fill ei, weobtain a homotopy equivalence (X(n) − B,X(n−1) − B) ' (A(n), A(n−1)). The homotopyinvariance implies that the inclusion (A(n), A(n−1))→ (X(n), X(n−1)) induces isomorphismsis homology so that the induction assumption on A establishes this case.

(3) So the remaining case is where i = n in which case there are inclusions

A(n−1) = X(n−1) ⊆ A(n) ⊆ A(n) ∪χ ei = X(n).

For this sequence of spaces, the inclusions of pairs

(A(n), A(n−1)) ⊆ (X(n), A(n−1)) = (X(n), X(n−1)) ⊆ (X(n), A(n))

induces a long exact sequence in homology (the long exact sequence of a triple)

. . .→ Hk+1(X(n), A(n))∆→ Hk(A(n), A(n−1))→ Hk(X(n), X(n−1))→ Hk(X(n), A(n))

∆→ . . . .

We show that Hk(X(n), X(n−1)) ∼= 0 for all k 6= 0 by showing that in this range the twogroups next to it in the above long exact sequence vanish. But for k 6= n, the groupHk(A(n), A(n−1)) is already known to vanish by induction hypothesis. In order to obtainthe vanishing of Hk(X(n), A(n)) for k 6= 0 it suffices to note that X(n) is obtained from A(n)

by attaching an n-cell. Thus we can conclude since Hk(X(n), X(n−1)) sits between zeros inthe sequence, hence must itself be zero.

This concludes the inductive step and hence the proof since our CW complexes were assumed tobe finite. �

In order to obtain a similar result for not necessarily finite CW complexes, let us first include ashort detour on directed colimits. Let us begin by establishing some terminology.

Definition 11. (1) A partially ordered set (P,≤) is directed if it is non-empty and if forevery two elements i, j ∈ P there is an element k ∈ P such that i ≤ k and j ≤ k.

(2) A directed system in a category C over a directed poset P consists of a family of objectsCi, i ∈ P, and morphisms fij : Cj → Ci for every pair of elements i, j ∈ P , i ≥ j, whichsatisfy the relations

fii = idCi: Ci → Ci, i ∈ P, and fij ◦ fjk = fik, i ≥ j ≥ k.

(3) A directed colimit of such a directed system (Ci, fij) consists of an object C ∈ C togetherwith morphisms fi : Ci → C such that fj = fi ◦fij whenever i ≥ j. Moreover, this datum issupposed to be universal with respect to this property in the following sense: whenever there

LECTURE 9: BASIC HOMOLOGICAL ASPECTS OF CW COMPLEXES 5

is an object D together with morphisms gi : Ci → D which also satisfy gj = gi ◦ fij , i ≥ j,then there is a unique morphism g : C → D such that gi = g ◦ fi.

Cjfij

//

fj

gj

��

Cifi

��

gi

��

C

��

D

(4) A category C has directed colimits if there is a directed colimit for every directed systemin C.

(5) A morphism of directed systems (Ci, fij) → (C ′i, f′ij) consists of morphisms Ci → C ′i

which commute with the maps fij and f ′ij , i.e., such the following squares commute

Ci //

��

Cj

��

C ′i// C ′j .

Exercise 12. Show that given two directed colimits (C, fi) and (C ′, f ′i) of the same directed system(Ci, fij) then there is a unique isomorphism g : C → C ′ such that f ′i = g ◦ fi. This justifies that wetalk about the directed colimit and we write C = colimi∈PCi for it. Conclude that a morphism ofdirected systems induces a morphism of directed colimits (provided both directed colimits exist).

Lemma 13. The categories of abelian groups, of chain complexes of abelian groups, and of topo-logical spaces have directed colimits.

Proof. Let P be a directed partially ordered set and let (Ai, fij : Aj → Ai) be a directed system ofabelian groups (over P ). Then we can form the direct sum

⊕i∈P Ai which comes with the subgroup

R generated by

fij(x)− x, i, j ∈ P, x ∈ Ai.We define C to be the quotient of

⊕iAi/R. The natural inclusions Aj →

⊕Ai induce homomor-

phisms fj : Aj → C. Note that any c ∈ C can be represented as fj(aj) for suitable aj ∈ Aj . In fact,by definition of C, every element is a coset represented by a finite sum Σnk=1aik for some aik ∈ Aik .Since P is directed we can find an element j ∈ P such that ik ≤ j for all k = 1, . . . , n. Thus we canconsider the element Σnk=1fjik(aik) ∈ Aj and it is immediate that this element represents c ∈ C. Itfollows from the definition of R that the relations fij ◦ fj = fi are satisfied.

Now, let D be an abelian group coming with similar maps gi : Ai → D such that gi ◦ fij = gj .The induced map g = (gi) :

⊕Ai → B sends the generators of R to zero and hence factors uniquely

through C. This concludes the construction of the colimit of the directed system.The case of chain complexes follows more or less directly from this (using that the differentials

can be considered as defining a morphism of directed systems). The details about this case and thecategory of topological spaces will be discussed in the exercises. �

From now on given a directed system (Ai, fij)i∈P of abelian groups, chain complexes, or topo-logical spaces, we will write colimi∈P Ai for the directed colimit.


Exercise 14. Let (Ai, fij)i∈P be a directed system of abelian groups and let fj : Aj → colimi∈P Aibe the canonical map. Then an element xj ∈ Aj satisfies fj(aj) = 0 ∈ colimi∈P Ai if and only ifthere is an element k ∈ P, j < k, such that fkj(xj) = 0 ∈ Ak.

Thus, elements which are sent to zero in the directed colimit already vanish at some finite stage.Using this exercise we establish the following lemma.

Lemma 15. Let (A′i, f′ij) → (Ai, fij) → (A′′i , f

′′ij) be morphisms of directed systems of abelian

groups. If the sequences A′iιi→ Ai

πi→ A′′i are exact for all i ∈ P, then also the induced sequence

colimi∈P A′iι→ colimi∈P Ai

π→ colimi∈P A′′i

of homomorphisms of abelian groups is exact.

Proof. We use the explicit description of elements in a directed colimit of abelian groups to provethis result. Let us consider an element a′ ∈ colimi∈P P

′i . Then there is an index j ∈ P such that a′

can be represented by some a′j ∈ A′j , i.e., a′ = f ′j(a′j) where f ′j : A′j → colimi∈P A

′i is the canonical

map to the colimit (we will use similar notation for the canonical maps in the other two cases). Bydefinition of the induced maps between colimits we obtain

πι(a′) = πι(f ′j(a′j)) = f ′′j (πjιj(a

′j)) = f ′′j (0) = 0,

showing that the image of ι lies in the kernel of π.Conversely, let a ∈ colimi∈P Ai lie in the kernel of π, π(a) = 0. By definition of the induced

homomorphism we can find an element aj ∈ Aj representing the element a, fj(aj) = a, and suchthat πj(aj) becomes zero in the colimit colimi∈P A

′′i . But πj(aj) has then already to vanish at a

finite stage. More precisely, there is an index k ∈ P , j < k, such that f ′′kj(πj(aj)) = 0. But we also

have 0 = f ′′kj(πj(aj)) = πk(fkj(aj)). Using the exactness of the morphisms of directed systems in

the k-th level, we conclude that there is an element a′k ∈ A′k such that ιk(a′k) = fkj(aj). But forthe element a′ = f ′k(a′k) ∈ colimi∈P A

′i we then calculate

ι(a′) = ι(f ′k(a′k)) = fk(ιk(a′k)) = fk(fkj(aj)) = fj(aj) = a,

as intended. �

Corollary 16. Homology of chain complexes commutes with directed colimits. More precisely, forevery n and every directed system of chain complexes (Ci, fij) there is a natural isomorphism

colimi∈PHn(Ci) ∼= Hn(colimi∈PCi).

Proof. This follows from the previous lemma, and the reader is asked to fill in the details in theexercises. �

LECTURE 10: CELLULAR HOMOLOGY

In this lecture we continue the study of homological properties of CW complexes, culminating inthe definition of cellular homology for such complexes, and the proof that this alternative homologytheory is naturally isomorphic to singular homology and that it is useful in explicit calculations.

We begin by recalling some basics about (homological) orientations. Recall that Hn(Sn) ∼= Z. Anorientation of Sn is a choice of generator in Hn(Sn); so there are two orientations. The boundary∂∆n of the n-simplex is a model of Sn−1, and has a canonical orientation given by the order of itsvertices

v0, . . . , vn

where vi = (0, . . . , 0, 1, 0, . . . , 0) ∈ Rn+1 with 1 in the i-th place, i = 0, . . . , n. More precisely, the(n−1)-cycle

∑(−1)i∂i is a generator, where ∂i : ∆n−1 → ∆n is the face opposite to the i-th vertex,

∂i(x0, . . . , xn−1) = (x0, . . . , xi−1, 0, xi, . . . , xn−1).

An orientation of the n-cell en is a generator of Hn(en, ∂en) (note that Hn(en, ∂en) ∼= Z by thelong exact sequence of the pair and the contractibility of en). Each homeomorphism α : ∆n → en

determines an orientation, since the map α itself is a cycle and represents an element of Hn(en, ∂en).An oriented n-cell in a CW complex X is a pair (e, θ) consisting of an n-cell e in X and an

orientation θ of e. We write Corn (X) for the free abelian group generated by the oriented n-cells

of X. Let Ccelln (X) be the quotient of Cor

n (X) obtained by identifying (e, θ) and −(e, θ′) if θ and θ′

are the two possible orientations of e. So Ccelln (X) is isomorphic to the free abelian group on the

set of n-cells (but the isomorphism would require a choice of orientations).In the final lecture we will prove the following theorem.

Theorem 1. Let X be a CW complex. The abelian groups Ccell• (X) can be turned into a chain

complex, the homology of which is isomorphic to the singular homology Hn(X) of X.

Of course a complete statement of the theorem, and its proof, requires an explicit description ofthe (cellular) boundary operator

∂ : Ccelln (X)→ Ccell

n−1(X).

This description will be given in the next lecture and is based on the homological degree of mapsSn → Sn. But even as it stands, it is already clear that the theorem is useful in calculations. Forexample, for the complex projective space CPn we had a CW decomposition with one 2i-cell foreach 0 ≤ i ≤ n. Thus, for the homology we obtain

Hk(CPn) ∼={

Z , k = 2l, 0 ≤ l ≤ n,0 , otherwise.

Let us say that a CW complex has dimension bounded by n if it has no cells in dimension largerthan n. Using this terminology, the following is immediate for a CW complex X:

(1) If dim(X) ≤ n, then Hk(X) ∼= 0 for all k > n.(2) If X is dimension-wise finite, then all Hk(X) are finitely generated.

For our further study of the homology of CW complexes let us recall the following two results.1

2 LECTURE 10: CELLULAR HOMOLOGY

Lemma 2. Let X be obtained from A by attaching an n-cell along f : ∂en → A, X = A ∪f en.Then

Hk(X,A) ∼={


Moreover, the attaching map applied to any orientation class of (en, ∂en) gives us a generator ofHn(X,A).

Using this lemma, we can draw some consequences for the homology of CW complexes by ‘in-duction on the number of cells’. In the case of finitely many n-cells the following result was alreadyestablished in the previous lecture. We leave it to the reader to deduce the general case from thisusing filtered colimits.

Proposition 3. For any CW complex X, Hk(X(n), X(n−1)) ∼= 0 for all k 6= n.

We now continue establishing some interesting facts about the singular homology of CW com-plexes.

Proposition 4. For any CW-complex X and any n ≥ 0 we have Hi(X,X(n)) ∼= 0, for i ≤ n.

Proof. It suffices to prove this result for finite CW complexes X. The general case will follow byan argument using filtered colimits. The proof will be by induction over the number of cells in X.If X has dimension 0, the assertion is clear. Let us suppose that the proposition holds for A, andlet us consider X = A ∪ ek. Then by excision as in the previous lecture, if n ≥ k then

Hi(X,X(n)) ∼= Hi(A,A

(n)).

If k > n let us again consider the long exact sequence in singular homology associated to the triple

A(n) = X(n) ⊆→ A⊆→ A ∪ ek = X,

a part of which looks like:

. . .→ Hi(A,A(n))→ Hi(X,A

(n)) = Hi(X,X(n))→ Hi(X,A)→ . . .

By Lemma 2 the group Hi(X,A) is nonzero only for i = k > n. Moreover, by induction the groupHi(A,A

(n)) is zero for i ≤ n. So surely the group in the middle is zero for i ≤ n as intended. �

Let us now show that the range in which the singular homology of a CW complex is possiblynontrivial is bounded by its dimension.

Proposition 5. Let X be a CW complex of dimension ≤ n. Then Hi(X) ∼= 0 for i > n.

Proof. Again, we prove the result for finite CW complexes by induction on the number of cells.The case of an infinite CW complex will follow by a colimit-argument. If dim(X) = 0 then theproposition is clear. Suppose the proposition holds for A, and let X = A ∪ ek, so in particulark ≤ n. A typical part of the long exact homology sequence of the pair (X,A) looks like:

. . .→ Hi(A)→ Hi(X)→ Hi(X,A)→ . . .

Now the group Hi(X,A) is trivial for i 6= k (so surely for i > n) as is the group Hi(A) for i > n byinduction assumption. Thus also Hi(X) ∼= 0 for i > n concluding the proof. �

We can now prove a more explicit addendum to Proposition 3. If X is a CW complex, let uschoose for each n-cell a characteristic map f and a homeomorphism α as in

∆n α→ enf→ X.

LECTURE 10: CELLULAR HOMOLOGY 3

Then f ◦α is a cycle in Cn(X(n), X(n−1)), so we obtain a homology class [f ◦α] ∈ Hn(X(n), X(n−1)).Doing this for each n-cell gives a well-defined homomorphism

φn : Ccelln (X)→ Hn(X(n), X(n−1)).

It might be helpful to refamiliarize yourself with the proof of Proposition 3 (as given in the previouslecture) before reading the proof of the following proposition.

Proposition 6. For all CW complexes X and all n, the map φn : Ccelln (X)→ Hn(X(n), X(n−1)) is

an isomorphism.

Proof. Again, it suffices to prove this for finite CW complexes X, the case where X has dimensionzero is clear, and we consider only the induction step X = A ∪ ek. If k 6= n then

Hn(X(n), X(n−1)) ∼= Hn(A(n), A(n−1))

(as we saw in the proof of Proposition 3 in the previous lecture) and also Ccelln (A) = Ccell

n (X). Sowe only need to look at the case k = n. But here we have a commutative diagram of the followingform

0 // Hn(A(n), A(n−1)) // Hn(X(n), X(n−1)) // Hn(X(n), A(n)) // 0

0 // Ccelln (A) //

φn

OO

Ccelln (X) //

φn

OO

Z //

∼=

OO

0.

It is obvious that the last row is exact but also the first row is exact: for this consider the longexact sequence associated to the triple

A(n−1) = X(n−1) ⊆→ A(n) ⊆→ X(n)

and use that both groups Hn+1(X(n), X(n−1)) and Hn−1(A(n), A(n−1)) vanish. The fact that thisdiagram commutes follows from the explicit description of the isomorphism Z ∼= Hn(X(n), A(n)).But by our induction assumption the vertical map on the left is an isomorphism. Thus we candeduce by the 5-lemma that also the vertical map in the middle is an isomorphism, completing theinduction step. �

Thus, these relative homology groups are just free abelian groups generated by the variousindexing sets of the cell structure. We now want to show that these relative homology groupsthemselves assemble into a chain complex, and in the next lecture we show that the homology ofthis new complex again calculates the homology of the space. A priori this does not seem to bean efficient idea: we build a complex consisting of relative homology groups of a space in order tocalculate the homology groups of that same space. However, as we saw these relative homologygroups have an easy explicit description and we will see that this alternative way of calculating thehomology is very convenient. This is also due to the fact that the differential can be given in quiteexplicit geometric terms. If one has a good understanding of the attaching maps of a given CWcomplex, then this allows for the calculation of its homology.

Here, we give an abstract description of the differentials. The translation into more geometricterms will be given in the following lecture. Let us recall that associated to a triple of spaces(X,A,B) there is a connecting homomorphism

∆n : Hn(X,A)δ→ Hn−1(A)→ Hn−1(A,B),


were δ is the connecting homomorphism of the pair (X,A) and the undecorated morphism belongsto the long exact sequence of the pair (A,B).

Let X be a CW complex. For each n ≥ 1 there is the triple of spaces (X(n), X(n−1), X(n−2))(we use the standard convention X(−1) = ∅). Let us denote the connecting homomorphism of thistriple by

∂celln : Hn(X(n), X(n−1))→ Hn−1(X(n−1), X(n−2)).

A key property of these maps is given by the following lemma.

Lemma 7. For a CW complex X and n ≥ 2 we have

0 = ∂celln−1 ◦ ∂celln : Hn(X(n), X(n−1))→ Hn−2(X(n−2), X(n−3)).

Proof. This follows since the composition of these cellular boundary homomorphisms is given by

Hn(X(n), X(n−1))

δ

��

Hn−1(X(n−1)) //

0,,

Hn−1(X(n−1), X(n−2))

δ

��

Hn−2(X(n−2)) // Hn−2(X(n−2), X(n−3)).

But the composition of the second and the third morphism is trivial since these are two subsequentmorphisms belonging to the long exact homology sequence of the pair (X(n−1), X(n−2)). �

From now on we will use these relative homology groups as definitions of Ccell• (X), but keep in

mind that these are isomorphic to the groups described at the beginning of this lecture. Thus wemake the following definition.

Definition 8. The cellular chain complex Ccell• (X) of a CW complex X is given by the cellular

chain groups

Ccelln (X) = Hn(X(n), X(n−1)), n ≥ 0,

together with the cellular boundary homomorphisms

∂celln : Ccelln (X) = Hn(X(n), X(n−1))→ Ccell

n−1(X) = Hn−1(X(n−1), X(n−2)).

The cellular homology Hcelln (X) of X is given by

Hcelln (X) = Hn(Ccell(X)), n ≥ 0.

Note that cellular homology is functorial with respect to cellular maps of CW complexes. Thisfollows from the naturality of the connecting homomorphism of a triple since any cellular mapf : X → Y induces maps of triples

f : (X(n), X(n−1), X(n−2))→ (Y (n), Y (n−1), Y (n−2)).

Thus, cellular homology defines a functor on the category of CW complexes and cellular maps.Note that this definition of the cellular chain complex of a CW complex does not only depend

on the underlying space but also on the chosen CW structure. In fact, by definition the cellularchain groups are relative homology groups of subsequent filtration steps in the skeleton filtration.Thus, one might wonder whether the resulting cellular homology is an invariant of the underlying

LECTURE 10: CELLULAR HOMOLOGY 5

space only (in that it would be independent of the actual choice of a CW structure). Theorem 10tells us, in particular, that this is indeed the case.

We split off a preliminary lemma.

Lemma 9. Let X be a CW complex. The canonical map Hn+1(X(n+1), X(n)) → Hn+1(X,X(n))is surjective for every n ≥ 0.

Proof. For this it suffices to consider the long exact homology sequences associated to the triple(X,X(n+1), X(n)). The relevant part of it is given by

Hn+1(X(n+1), X(n))→ Hn+1(X,X(n))→ Hn+1(X,X(n+1)).

But by Proposition 4, the group Hn+1(X,X(n+1)) is trivial, concluding the proof. �

Theorem 10. (Singular and cellular homology are isomorphic.)Let X be a CW complex. Then there is an isomorphism Hn(X) ∼= Hcell

n (X), n ≥ 0, which is naturalwith respect to cellular maps.

Proof. Let us begin by identifying the cellular cycles, i.e., the kernel of the cellular boundaryoperator,

Zcelln (X) = ker(Hn(X(n), X(n−1))→ Hn−1(X(n−1), X(n−2))).

By definition, this boundary operator factors as

Hn(X(n), X(n−1))→ Hn−1(X(n−1))→ Hn−1(X(n−1), X(n−2)).

But the second map in this factorization is injective as one easily checks using the long exacthomology sequence of the pair (X(n−1), X(n−2)) together with the fact that Hn−1(X(n−2)) vanishes.This implies that Zcell

n (X) is simply the kernel ofHn(X(n), X(n−1))→ Hn−1(X(n−1)). If we considerthe long exact homology sequence of (X(n), X(n−1)), then the interesting part reads as

Hn(X(n−1))→ Hn(X(n))→ Hn(X(n), X(n−1))→ Hn−1(X(n−1)).

Using that Hn(X(n−1)) is trivial, we conclude that there is a canonical isomorphism

Hn(X(n))∼=→ Zcell

n (X),

and that this isomorphism is induced by the map Hn(X(n))→ Hn(X(n), X(n−1)).Let us now describe the cellular boundaries, i.e., the image of the cellular boundary operator,

Bcelln (X) = im(Hn+1(X(n+1), X(n))→ Hn(X(n), X(n−1))).

Again, by definition this map is Hn+1(X(n+1), X(n)) → Hn(X(n)) → Hn(X(n), X(n−1)). By thefirst part of this proof, we know that Hcell

n (X) is canonically isomorphic to the cokernel of the firstmap Hn+1(X(n+1), X(n)) → Hn(X(n)), the connecting homomorphism of the pair (X(n+1), X(n)).Recall that these connecting homomorphisms are natural with respect to maps of pairs, henceapplied to the map (X(n+1), X(n))→ (X,X(n)) this yields the following commutative diagram

Hn+1(X(n+1), X(n)) //

��

Hn(X(n))

=

��

Hn+1(X,X(n)) // Hn(X(n)) // Hn(X) // Hn(X,X(n)),

in which the lower row is part of the long exact sequence of the pair (X,X(n)). By Lemma 1, thevertical map on the left is surjective, and Hcell

n (X) is thus canonically isomorphic to the cokernel of


Hn+1(X,X(n)) → Hn(X(n)). But since Hn(X,X(n)) vanishes, the above exact sequence allows usto conclude that Hcell

n (X) is isomorphic to Hn(X). It follows from this proof that the isomorphismis compatible with cellular maps. �

LECTURE 11: THE DEGREE OF A MAP AND THE CELLULAR

BOUNDARIES

In this lecture we will study the (homological) degree of self-maps of spheres, a notion whichgeneralizes the usual degree of a polynomial. We will study many examples, establish basic proper-ties of the degree, and discuss some of the typical applications. We will also see how the boundaryoperator of the cellular chain complex of a space can be defined in terms of the degrees of self-mapsof the spheres.

1. Degrees of maps between spheres

Let us recall from Lecture 6 that for each n ≥ 1 we have isomorphisms

Hk(Dn, Sn−1) ∼= Hk(Sn) ∼= Hk(Sn, ∗) ∼={


Generators of the respective free abelian groups of rank one are fundamental classes or orientationclasses. Note that these are well-defined up to a sign and we will now make coherent choices whichwill then be denoted by

[Dn, Sn−1] ∈ Hn(Dn, Sn−1), [Sn] ∈ Hn(Sn) or [Sn] ∈ Hn(Sn, ∗)

respectively. Now, the n-sphere is obtained by gluing the ‘north’ hemisphere DnN and the ‘south’

hemisphere DnS along their common boundary (the ‘equator’). Both hemispheres are just copies of

the unit ball Dn and as such homeomorphic to ∆n. To be more specific, we take the homeomorphismσ : ∆n → Dn which is essentially given by a rescaling: Dn is homeomorphic by a translation and arescaling to a disc of dimension n centered at the barycenter of ∆n and with the radius chosen suchthat all vertices of ∆n lie on the boundary of that disc; given this disc then we choose σ to be justthe obvious homeomorphism given by rescaling. Using these homeomorphisms we obtain singularn-simplices

σN : ∆n σ→ Dn ∼= DnN⊂→ Sn and σS : ∆n σ→ Dn ∼= Dn

S⊂→ Sn,

where the undecorated homeomorphisms are obtained by projection into Rn × {0}. One can checkthat the formal difference zn = σS − σN ∈ Zn(Sn) is a cycle which actually represents a gen-

erator [Sn] ∈ Hn(Sn). Under the above isomorphisms this also defines the fundamental classes[Dn, Sn−1] ∈ Hn(Dn, Sn−1) and [Sn] ∈ Hn(Sn, ∗).

Recall from Lecture 2 the definition of the Hurewicz homomorphism: for every pointed space(X,x0) there is a natural group homomorphism h : π1(X,x0) → H1(X). Given a homotopy classα ∈ π1(X,x0) represented by a loop γ : S1 → X, h(α) ∈ H1(X) is the well-defined homology classrepresented by the cycle

γ ◦ e : ∆1 → ∆1/∂∆1 ∼= S1 → X

where e : ∆1 → ∆1/∂∆1 denotes the quotient map. The fundamental class [S1] which we con-structed above boils down to endowing S1 with the ‘usual’ counter-clockwise orientation. The

1

2 LECTURE 11: THE DEGREE OF A MAP AND THE CELLULAR BOUNDARIES

quotient map e : ∆1 → S1 can be chosen to be the concatenation e = σS ∗ σ−1N . Considering e as a

1-cycle in S1 and using Lemma 3 of Lecture 2 we see that e is homologous to z1:

e = σS ∗ σ−1N ∼ σS + σ−1

N ∼ σS − σN = z1

Thus, the Hurewicz homomorphism associated to a pointed space (X,x0) is given by

h : π1(X,x0)→ H1(X) : α 7→ α∗([S1])

where α∗ : H1(S1) → H1(X) denotes the induced map in homology. Of course, the homotopyinvariance of singular homology motivated us to write α∗ (no matter which representing loop wechoose we get the same map in homology!).

This description of the Hurewicz homomorphism suggests an extension to higher dimensions.Given a pointed homotopy class α of maps (Sn, ∗)→ (X,x0), we obtain the homology class

α∗([Sn]) ∈ Hn(X).

In order to give a precise definition of this higher dimensional Hurewicz homomorphism, one hasfirst to introduce higher dimensional analogues of the fundamental group. This is done in anystandard course on Homotopy Theory, and the thusly defined Hurewicz homomorphisms do playan important role.

Definition 1. Let f : Sn → Sn be a map. The unique integer deg(f) ∈ Z such that

f∗([Sn]) = deg(f) · [Sn] ∈ Hn(Sn)

is called the degree of f : Sn → Sn.

In this definition we used of course that Hn(Sn) ∼= Z so that every self-map of Hn(Sn) is givenby multiplication by an integer. Note that the definition of the degree is independent of the actualchoice of fundamental classes: a different choice would amount to replacing [Sn] by −[Sn] and hencewould give rise to the same value for deg(f).

Lemma 2. (1) If f, g : Sn → Sn are homotopic, then deg(f) = deg(g).(2) For maps g, f : Sn → Sn we have deg(g ◦ f) = deg(g) deg(f) and deg(idSn) = 1.

Proof. This is immediate from the definition. �

The second statement of this lemma is referred to by saying that the degree is multiplicative.

Example 3. (1) Let f : S1 → S1 : (x0, x1) = (−x0, x1) be the reflection in the axis x0 = 0.Then deg(f) = −1. One way to see this is as follows. Let σW , σE : ∆1 → S1 be pathsfrom the ‘south pole’ to the ‘north pole’ in the clockwise and the counterclockwise senserespectively. Obviously σW − σE is a cycle and it can be checked (using a minor variant ofLemma 4 of Lecture 2: see the exercises) that

[σW − σE ] = −[S1] ∈ H1(S1).

In particular, we can use [σW − σE ] to calculate degrees. Since f∗([σW − σE ]) = [σE − σW ]we deduce deg(f) = −1. It follows that a map S1 → S1 given by a reflection in an arbitraryline through the origin has degree −1.

(2) Let a : S1 → S1 : (x0, x1) 7→ (−x0,−x1) be the antipodal map. Then deg(a) = 1. Thisfollows from the previous example and the multiplicativity of the degree since the antipodalmap is the composition of two reflections.

(3) Let τ : S1 → S1 be given by τ(x0, x1) = (x1, x0). Then deg(τ) = −1. Indeed, τ is areflection in a line.

LECTURE 11: THE DEGREE OF A MAP AND THE CELLULAR BOUNDARIES 3

In order to extend these examples to higher dimensions, let us recall that we can construct Sn+1

as the suspension of Sn,

Sn+1 ∼= S(Sn).

If we write Sn = {(x0, . . . , xn) ∈ Rn+1 |∑x2i = 1} then Sn+1 = S(Sn) is homeomorphic to the

quotient of Sn× [−1, 1] obtained by identifying all (x0, . . . , xn, 1) to a point N , the north pole, andall (x0, . . . , xn,−1) to a point S, the south pole. The homeomorphism is induced by the map

Sn × [−1, 1]→ Sn+1 : ((x0, . . . , xn), t) 7→ (x′0, . . . , x′n, t)

where x′i =√

1− t2 · xi. Under this isomorphism, it is clear that the suspension of f : Sn → Sn inthe standard coordinates for Sn+1 is

Sf : Sn+1 → Sn+1 : (x0, . . . , xn+1) 7→ (rf(r−1x0, . . . , r−1xn), xn+1)

where r =√

1− x2n+1. In particular, Sf restricts to f on the ‘meridian’ xn+1 = 0.

Proposition 4. For any f : Sn → Sn and the associated Sf : Sn+1 → Sn+1, we have an equalityof degrees deg(Sf) = deg(f).

Proof. Write Sn+1 = S(Sn) = A ∪ B for the northern and southern hemispheres A and B (givenby xn+1 ≥ 0, resp. xn+1 ≤ 0). Then A ∩ B = Sn is the meridian. Since A and B are contractible,the Mayer-Vietoris sequence (for slight extensions to open neighborhoods of A and B which arehomotopy equivalent to A and B) gives:

Hn+1(Sn+1)∆∼=

//

(Sf)∗

��

Hn(Sn)

f∗

��

Hn+1(Sn+1)∆

∼= // Hn(Sn)

The square commutes by naturality of the Mayer-Vietoris sequence. From this, the statement followsby tracing the generator [Sn+1] through this diagram. In more detail, since ∆ is an isomorphism,we have ∆([Sn+1]) = ε · [Sn] with ε ∈ {−1,+1}, and hence

(f∗ ◦∆)([Sn+1]) = f∗(ε · [Sn]) = ε · f∗([Sn]) = ε · deg(f) · [Sn].

Similarly, if we trace [Sn+1] through the lower left corner, we calculate

(∆ ◦ (Sf)∗)([Sn+1]) = ∆(deg(Sf) · [Sn+1]) = deg(Sf) ·∆([Sn+1]) = deg(Sf) · ε · [Sn].

Comparing these two expressions concludes the proof. �

This proof is a further instance of a calculation showing that naturality of certain long exactsequences is not just a technical issue but actually useful in calculations. With this preparation wenow obtain the following higher-dimensional versions of Example 3.

Proposition 5. (1) The degree of the map f : Sn → Sn : (x0, x1, . . . , xn) 7→ (−x0, x1, . . . , xn)is −1. More generally, the degree of the any reflection at an arbitrary hyperplane throughthe origin is −1.

(2) Let f : Sn → Sn be given by f(x0, x1, . . . , xn) = (ε0x0, ε1x1, . . . , εnxn) for some signs εi.Then deg(f) = ε0 · . . . · εn. In particular, if f = a is the antipodal map (thus all εi are −1)then deg(a) = (−1)n+1.

(3) Let f : Sn → Sn be given by f(x0, x1, . . . , xn) = (x1, x0, x2, x3, . . . , xn). Then deg(f) = −1.


(4) Let f : Sn → Sn : (x0, . . . , xn) 7→ (xτ(0), . . . , xτ(n)) for some permutation τ ∈ Σn+1. Thendeg(f) = sg(f) is the signature of the permutation τ .

Proof. Part (1) follows Example 3.(1) and Proposition 4 together with the observation that allreflections at hyperplanes are homotopic. The remaining parts are an immediate consequenceof (1) and the multiplicativity of the degree (to obtain (4), write an arbitrary permutation as asequence of transpositions). �

Lemma 6. Let f, g : X → Sn ⊆ Rn+1. If f(x) 6= −g(x) for all x ∈ X then f ' g.

Proof. Let H : X × [0, 1]→ Rn+1 be given by H(x, t) = (1− t)f(x) + tg(x). Then for a fixed x, thepartial map H(x,−) : [0, 1]→ Rn+1 is the line from f(x) to g(x). By assumption, this line does notpass through the origin, so we can normalize H to obtain the map

K : X × [0, 1]→ Sn : (x, t) 7→ H(x, t)/||H(x, t)||,

a well-defined homotopy from f to g. �

Corollary 7. Let f : Sn → Sn.

(1) If f has no fixed point, then deg(f) = (−1)n+1.(2) If f has no antipodal point (a point x with f(x) = −x), then deg(f) = 1.

Proof. For the first statement, if f(x) 6= x for all x, then f is homotopic to the antipodal map adefined by a(x) = −x, according to the lemma. But since the degree is homotopy-invariant we canconclude by Proposition 5.(2). The proof of the second statement is similar, since by the lemma,f(x) 6= −x for all x implies that f is homotopic to the identity. �

Corollary 8. If n is even, then any f : Sn → Sn has a fixed point or an antipodal point.

Proof. If f has neither a fixed point nor an antipodal point, then, by the previous corollary, thedegree of f has to be −1 and 1 which is impossible. �

Corollary 9. Let n ∈ N be even, then any vector field v on Sn has a zero.

Proof. Such a vector field assigns to any x ∈ Sn a vector v(x) based at x and lying in the hyperplanetangent to Sn ⊆ Rn+1 at x. If v(x) 6= 0 for all x, then the map v = v/||v|| : Sn → Sn (obtained bynormalizing the vector field and considering the vectors as attached to the origin of Rn+1) is suchthat each v(x) is a unit vector parallel to the hyperplane tangent to Sn at x. But this contradictsthe previous corollary, since v would have neither a fixed point nor an antipodal point. �

For n = 2 this is sometimes referred to as the hairy ball theorem: you cannot comb a hairyball without a parting. The corollary tells us that there are no no-where vanishing vector fields oneven-dimensional spheres. Thus one might wonder what happens for odd-dimensional spheres. Itis easy to construct a no-where vanishing vector field on S2n+1,

v(x0, x1, . . . x2n, x2n+1) = (−x1, x0,−x2, x3, . . . ,−x2n+1, x2n).

For a long time it was an open problem in algebraic topology to determine the maximal number ofeverywhere linearly independent vector field on spheres, which was finally solved by Adams usingfairly advanced techniques. Using singular homology we managed to obtain first partial results inthat direction.

LECTURE 11: THE DEGREE OF A MAP AND THE CELLULAR BOUNDARIES 5

2. The cellular boundary operator

If we want to be able to calculate the cellular homology of CW complexes which have cellsin subsequent dimensions, then it is helpful to have a more geometric description of the cellularboundary homomorphism. Such a description can be obtained by means of the homological degreesof self-maps of the spheres, as discussed in Section 1. Let us recall that the cellular chain groupsare free abelian groups, i.e., we have isomorphisms

⊕Jn

Z ∼= Ccelln (X) where Jn denotes the index

set for the n-cells of X. Under these isomorphisms, the cellular boundary maps hence correspondto homomorphisms ⊕

Jn

Z ∼= Ccelln (X)

∂celln→ Ccell

n−1(X) ∼=⊕Jn−1

Z.

This composition sends every n-cell σ of X to a sum

σ 7→ Στ∈Jn−1zσ,ττ

for suitable integer coefficients zσ,τ . To conclude the description of this assignment we thus have tospecify these coefficients. For that purpose, let us fix an n-cell σ and an (n− 1)-cell τ . The n-cell σcomes with an attaching map

Sn−1

��

χσ // X(n−1)

��

Dn // X(n).

Now, associated to this attaching map we can consider the following composition

fσ,τ : Sn−1 χσ→ X(n−1) → X(n−1)/X(n−2) ∼=∨Jn−1

Sn−1 → Sn−1

in which the last arrow maps all copies of the spheres constantly to the base point except the onebelonging to the index τ ∈ Jn−1 on which the map is the identity. Thus, for each such pair of cellswe obtain a pointed self-map fσ,τ of Sn−1 and its degree turns out to coincide with zσ,τ . Note thatsince Sn−1 is compact it follows that for any n-cell σ there are only finitely many (n − 1)-cells τsuch that fσ,τ is not the constant map. Thus, the sums in the next proposition are well-defined.Working out the details, by using the long exact sequence of the homology of the pair, one caneasily verify the following result:

Proposition 10. Under the above isomorphisms the cellular boundary homomorphism is given bythe map ⊕

Jn

Z→⊕Jn−1

Z : σ 7→ Στ∈Jn−1deg(fσ,τ )τ.

Hence in the context of a specific CW complex in which we happen to be able to calculate all thedegrees showing up in the proposition, the problem of calculating the homology of the CW complexis reduced to a purely algebraic problem.

Let us give a brief discussion the example of the real projective spaces RPn, n ≥ 0. We begin byrecalling that RPn is obtained from Sn by identifying antipodal points. Hence, there are quotientmaps p = pn : Sn → RPn. The real projective space RPn can be endowed with a CW structure suchthat there is a unique k-cell in each dimension 0 ≤ k ≤ n. One can check that the cellular boundaryhomomorphism ∂ : Ccell

k (RPn)→ Ccellk−1(RPn), 0 < k ≤ n is zero if k is odd and multiplication by 2

if k is even. From this one can derive the following calculation.


Example 11. The homology of an even-dimensional real projective space is given by

Hk(RP 2m) ∼=

Z , k = 0,Z/2Z , k odd, 0 < k < 2m0 , otherwise.

In particular, the top-dimensional homology group H2m(RP 2m) is zero. The homology of odd-dimensional real projective spaces looks differently and is given by

Hk(RP 2m+1) ∼=

Z , k = 0, 2m+ 1Z/2Z , k odd, 0 < k < 2m+ 10 , otherwise.

In this case, the top-dimensional homology group is again simply a copy of the integers. Anygenerator of this group is called fundamental class of RP 2m+1.

Note that these are our first examples of spaces in which the homology groups have non-trivialtorsion elements. This should not be considered as something exotic but instead it is a generalphenomenon. We conclude this lecture with a short outlook. There is an axiomatic approach tohomology which is due to Eilenberg and Steenrod. By definition a homology theory consists offunctors hn, n ≥ 0, from the category of pairs of topological spaces to abelian groups together withnatural transformations (called connecting homomorphisms)

δ : hn(X,A)→ hn−1(A, ∅), n ≥ 1.

This data has to satisfy the long exact sequence axiom, the homotopy axiom, the excision axiom,and the dimension axiom. We let you guess the precise form of the first three axioms, but we wantto be specific about the dimension axiom. It asks that hk(∗, ∅) is trivial in positive dimensions.Thus, the only possibly non-trivial homology group of the point sits in degree zero and that grouph0(∗, ∅) is referred to as the group of coefficients of the homology theory. So, parts of this coursecan be summarized by saying that singular homology theory defines a homology theory in the senseof Eilenberg-Steenrod with integral coefficients. In the sequel to this course we study closely relatedalgebraic invariants of spaces, namely homology groups with coefficients and cohomology groups.

lecture 1: definition of singular homology · lecture 1: definition of singular homology 3 more...

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