Math 540, Topology

Instructor: Francis Bonahon

Fall 2016

Course outline

This is an outline of what we have covered in the semester so far, with maindefinitions and statements. No proofs!

Version as of: December 1, 2016

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CHAPTER 1

A crash course in pointset topology

This material will not be covered in class, and is only intended as review.

1. Topological spaces

Definition. A topological space consists of a set X and of a family T of subsetsof X such that:

1. the empty set ∅ and the whole space X are both elements of T ;2. if Uii∈I is a (possibly infinite) family of elements of T , their union

⋃i∈I Ui

is also in T .3. if Uii=1,2,...,n is a finite family of elements of T , their intersection

⋂i=1,2,...,n Ui

is also in T .The family T is then called a topology on the set X. The elements of

T are called open subsets of X for this topology.

Recall that, given a family Xii∈I of sets Xi indexed by a set I, their union⋃i∈I is the set of all x for which there exists an i ∈ I with x ∈ Xi. Their intersection⋂i∈I is the set of all x such that x ∈ Xi for every i ∈ I. Note that I does not need

to be finite.

Fundamental example. Let (X, d) be a metric space, namely a set X endowedwith a function d: X ×X → R such that

1. d(x, y) > 0 for every x, y;2. d(x, y) = 0 if and only if x = y;3. d(y, x) = d(x, y) for every x, y;4. d(x, z) 6 d(x, y) + d(y, z) for every x, y, z.

Given x in such a metric space X and given ε > 0, let the open ball of radius εcentered at x be B(x, ε) = y ∈ X; d(x, y) < ε.

Define T by the property that U ⊂ X is an element of T if and only if, forevery x ∈ U , there exists a ball B(x, ε) which is contained in U . Then, T is atopology for X.

Definition. If x is an element of a topological space X, a neighborhood of x is asubset W ⊂ X for which there exists an open subset U of X with x ∈ U ⊂W .

Note that the terminology is not standard. This one is used in most researcharticles. Many textbook require a neighborhood to be open, namely just an opensubset containing x.

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4 1. A CRASH COURSE IN POINTSET TOPOLOGY

Definition. A topological space X is Hausdorff if, for every x, y ∈ X with x 6= y,there exists a neighborhood U of x and a neighborhood V of y such that U∩V = ∅.

For instance, the topology of a metric space is always Hausdorff.

Convention. All topological spaces encountered in the course will be implicitlyassumed to be Hausdorff, except for those that are constructed explicitly (and forwhich we will need to prove the Hausdorff property).

2. Continuous functions

Definition. Let f : X → Y be a map between two topological spaces. The mapf is continuous if, for every open subset U of Y , its pre-image f−1(U) is an opensubset of X.

Recall that the pre-image of U is f−1(U) = x ∈ X; f(x) ∈ U.For metric spaces, this definition of continuity turns out to be equivalent to a

more familiar one:

Proposition. (ε − δ definition of continuity) Let f : X → Y be a map betweenmetric spaces (X, dX) and (Y, dY ). Then f is continuous in the above sense if andonly if, for every x ∈ X and every ε > 0, there is a δ > 0 such that dY (f(x), f(x′)) <ε for every x′ ∈ X with dX(x, x′) < δ.

Definition. A homeomorphism between two topological spaces X and Y is abijection f : X → Y such that f and f−1 are both continuous.

If f : X → Y is a homeomorphism and U is a subset of X, it follows fromdefinitions that U is open in X if and only if f(U) is open in Y . Therefore, festablishes a ‘dictionary’, which enables us to translate any topological property ofX to a similar property of Y . A homeomorphism is therefore an ‘isomorphism oftopological spaces’.

3. The subspace topology

Let X be a topological space, with topology T . If X ′ is a subset of X, it isimmediate that

T ′ = U ′ ⊂ X ′;∃U ∈ T , U ′ = X ′ ∩ Uis a topology for X ′.

Definition. The topology T ′ is the subspace topology induced on X ′ by thetopology of X.

4. The quotient topology

Let p: X → Y be a surjective map from a topological space X to a set Y . Thissituation typically arises when (and exactly when) Y is the quotient of X under anequivalence relation ∼, namely when Y is the set of equivalence classes of ∼, andwhen p is the natural projection.

4. THE QUOTIENT TOPOLOGY 5

The topology of X induces a topology T on Y , defined by

T = U ⊂ Y ; p−1(U) open in X.

Definition. The topology T is the quotient topology induced on Y by the topologyof X.

The quotient topology is specially designed for the following property.

Proposition. Let f : X → Z be a map such that p(x) = p(x′) ⇒ f(x) = f(x′).Then there is a unique continuous map g: Y → Z such that f = g p.

Xf//

p

Z

Y

g>>

CHAPTER 2

The fundamental group

Let X be a topological space.

Throughout the course, we will assume that all topological spaces consideredhave the Hausdorff Property. This means that, for any two distinct points x, y ∈ X,there exists U and V open such that x ∈ U , y ∈ V and U ∩ V = ∅.

Similarly, every map will be assumed to be continuous, unless you really haveto prove that it is continuous.

Advice for the course: Draw plenty of pictures. They help in understandingwhat we are talking about.

1. Paths and homotopies

Definition. A path in X is a continuous map α: [0, 1] → X, where [0, 1] is theinterval in R.

Stupid example. The constant path at x ∈ X is the map cx : [0, 1] → X definedby cx(s) = x for every s ∈ [0, 1].

This example will become not so stupid after all, and play an important rolein the fundamental group of X.

A less trivial example. The path in X = R defined by α(s) = s− s2.

Definition. Let α and β be two paths in X such that the final point α(1) is equalto the initial point β(0). The composition of α and β is the path α ∗ β: [0, 1]→ Xdefined by

α ∗ β(s) =

α(2s) if 0 6 s 6 1

2

β(2s− 1) if 12 6 s 6 1

Namely, α ∗ β is defined by chaining α and β together, but speeding them upso that each of them is covered in half the time.

To prove that α ∗ β is continuous, it is convenient to use the

Pasting Lemma. Let X and Y be two topological spaces. Suppose that Y is theunion of two closed subsets C1 and C2. Let f1: C1 → X and f2: C2 → X be twocontinuous maps which coincide on C1 ∩ C2, namely such that f1(x) = f2(x) forevery x ∈ C1 ∩ C2. Then, there exists a unique continuous map f : Y → X suchthat f|C1

= f1 and f|C2= f2.

Question: Do we really need C1 and C2 to be closed?

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8 2. THE FUNDAMENTAL GROUP

Definition. A path homotopy between the two paths α0 and α1 is a continuousmap H: [0, 1]× [0, 1]→ X such that

H(s, 0) = α0(s), ∀s ∈ [0, 1]

H(s, 1) = α1(s), ∀s ∈ [0, 1]

H(t, 0) = α0(0) = α1(0), ∀t ∈ [0, 1]

H(t, 1) = α0(1) = α1(1), ∀t ∈ [0, 1]

We then say that α0 and α1 are path homotopic, and we write α0 ' α1.

This definition is better understood if you write H(s, t) = αt(s). We then havea family of paths αt: [0, 1]→ X, depending continuously (whatever this means) ona “time” parameter t ∈ [0, 1], all with the same starting point α0(0) = α1(0) andthe same end point α0(1) = α1(1), and moving from α0 when t = 0 to α1 whent = 1.

Remark. If two paths are path homotopic, they necessarily have the same endpoints.

Example. Any two paths α and β in Rn with the same initial point α(0) = β(0)and the same final point α(1) = β(1) are path homotopic by the path homotopyH(s, t) = (1− t)α(s) + tβ(s).

Sort of example. In R2 − (0, 0), it appears that a path going from (−1, 0) to(1, 0) and passing “above” the hole (0, 0) is not path homotopic to a path joiningthe same points but passing “below” the hole. We will prove later on that this isindeed the case.

Question: Why can’t we use the path homotopy H(s, t) = (1 − t)α(s) + tβ(s) inR2 − (0, 0)? Where do things go wrong?

Lemma. On the set of all paths in X, the relation ' (= “is path homotopic to”)is an equivalence relation.

We will usually denote by [α] the equivalence class of the path α.

2. The fundamental group

Pick a point x0 ∈ X, called a base point. We will restrict attention to pathsgoing from x0 to x0.

Definition. The fundamental group of X based at x0 is the set of equivalenceclasses

π1(X;x0) = α: [0, 1]→ X path;α(0) = α(1) = x0/ ' .

Proposition. There is a unique group law ∗ on π1(X;x0) defined by the propertythat [α] ∗ [β] = [α ∗ β].

Note that we need the property that α(1) = β(0) = x0 for the compositionα ∗ β to be defined.

Here are the key steps of the proof of the Proposition. First we have to showthat the law is well defined.

4. GROUP HOMOMORPHISMS INDUCED BY MAPS 9

Lemma. If α ' α′, β ' β′ and α(1) = β(0), then α ∗ β ' α′ ∗ β′.Then we have to show that the law is transitive.

Lemma. If α(1) = β(0) and β(1) = γ(0), then (α ∗ β) ∗ γ ' α ∗ (β ∗ γ).

Note that in general the paths (α ∗ β) ∗ γ and α ∗ (β ∗ γ) are not equal, theyare just path homotopic.

We then have to find an identity element. For this, given x ∈ X, let cx: [0, 1]→X denote the constant path defined by cx(s) = x for every s ∈ [0, 1].

Lemma. cα(0) ∗ α ' α ' α ∗ cα(1) for every path α.

This proves that [cx0] is an identity element for ∗ in π1(X;x0). We will also

write [cx0] = 1 to express this fact.

Finally, we need an inverse. The opposite of a path α is the path obtained bytraveling backwards, namely defined by α(s) = α(1− s).

Lemma. α ∗ α ' cα(0) and α ∗ α ' cα(1) for every path α.

Therefore, [α]−1 = [α] in π1(X;x0).

This proves that ∗ is a group law for π1(X;x0).

Stupid example. If X = x0, then π1(X;x0) is the trivial group, usually denotedby 1.

3. Dependence on base points

Proposition. If there is a path γ going from x0 to y0, the map [α] 7→ [γ ∗ α ∗ γ]defines a group isomorphism Tγ: π1(X; y0)→ π1(X;x0).

Note that [γ ∗ (α ∗ γ)] = [(γ ∗ α) ∗ γ], so that we do not have to worry aboutwhere we put the parentheses.

Corollary. If x0 and y0 are in the same path component of X, the fundamentalgroups π1(X;x0) and π1(X; y0) are isomorphic.

Remark. If X0 is the path connected component of X that contains x0, thenπ1(X;x0) = π1(X0;x0).

4. Group homomorphisms induced by maps

Definition. The group homomorphism induced by the map f : X → Y is the mapf∗: π1(X;x0)→ π1(Y ; y0) defined by f∗([α]) = [f α], where y0 = f(x0).

Lemma. f∗ is well-defined, and is really a group homomorphism.

Definition. A homotopy between the maps f0, f1: X → Y is a mapH: X×[0, 1]→Y such that H(x, 0) = f0(x) and H(x, 1) = f1(x) for every x ∈ X. The maps f0

and f1 are then said to be homotopic. The homotopy is base point preserving ifH(x0, t) = f0(x0) = f1(x0) for every t ∈ [0, 1].

10 2. THE FUNDAMENTAL GROUP

Again, this definition is best understood by considering the family of mapsft: X → Y defined by ft(x) = H(x, t).

Proposition. If f and g are homotopic by a base point preserving homotopy then,for y0 = f(x0) = g(x0), the two homomorphisms f∗ and g∗: π(X;x0) → π1(Y ; y0)are equal.

How about homotopies which do not fix base points?

Proposition. If f and g : X → Y are homotopic by a homotopy H, let γ bethe path from y0 = f(x0) to z0 = g(x0) defined by γ(t) = H(x0, t). Then f∗ :π1(X;x0) → π1(Y ; y0) and g∗ : π1(X;x0) → π1(Y ; z0) are related by the propertythat g∗ = Tγ f∗, where Tγ : π1(Y ; y0) → π1(Y ; z0) is the change of base pointisomorphism defined by γ.

Theorem. Let the product X × Y of the two topological spaces X and Y beendowed with the product topology, and let p: X × Y → X and q: X × Y → Y bethe two projection maps defined by p(x, y) = x and q(x, y) = y. Then the productmap

p∗ × q∗: π1

(X × Y ; (x0, y0)

)→ π1(X;x0)× π1(Y ; y0)

is a group isomorphism.

5. Homotopy equivalences and deformation retracts

Definition. The map f : X → Y is a homotopy equivalence if there exists a mapg: Y → X such that g f is homotopic to the identity IdX and f g is homotopicto IdY . In this case, g is a homotopy inverse for f .

Proposition. If f : X → Y is a homotopy equivalence and y0 = f(x0), thenf∗: π(X;x0)→ π(Y ; y0) is an isomorphism.

A good source of homotopy equivalences comes from the following definition.

Definition. The space X deformation retracts to A ⊂ X, or A is a deformationretract of X, if there exists a homotopy H: X × [0, 1]→ X such that

H(x, 0) = x for every x ∈ XH(x, 1) ∈ A for every x ∈ XH(a, t) = a for every a ∈ A and every t ∈ [0, 1]

The map r : X → A defined by r(x) = H(x, 1) is the associated retraction bydeformation.

Lemma. If A is a deformation retract of X, the inclusion map i : A → X is ahomotopy equivalence, with homotopy inverse the retraction r : X → A. As aconsequence, i∗: π1(A;x0)→ π1(X;x0) is an isomorphism.

Example. The space X is contractible if it deformation retracts to a point. Then,π1(X;x0) ∼= π1(x0;x0) = 1.

5. HOMOTOPY EQUIVALENCES AND DEFORMATION RETRACTS 11

Example of the example. A subset X ⊂ Rn is star-shaped if there exists x0 ∈ Xsuch that, for every x ∈ X, the line segment [x, x0] is completely contained in X.Then X is contractible.

Example. The n–dimensional sphere Sn = x ∈ Rn+1; ‖x‖ = 1 is a deformationretract of Rn − 0, so that i∗: π1(Sn;x0) is isomorphic to π1(Rn+1 − 0;x0) forevery x0 ∈ Sn. Of course, this is pretty useless at this point since we don’t knoweither one of these fundamental groups (except when n = 0).

A weaker form of deformation retraction is the following.

Definition. A retraction of X to a subset A ⊂ X is a map r: X → A such thatr(a) = a for every a ∈ A. In this case, the space X retracts to A, or A is a retractof X.

Proposition. IfA is a retract ofX and x0 ∈ A, the homomorphism i∗: π1(A;x0)→π1(X;x0) induced by the inclusion map i: A→ X is injective.

CHAPTER 3

The fundamental group of the circle

The main tool to compute the fundamental group of the circle S1 is the mapp: R→ S1 defined by p(t) = (cos t, sin t).

A lift of a path α in S1 is a path α in R such that p α = α.

The two key tools are the following.

Proposition. (Path Lifting Property) Let α be a path in S1, beginning at x0 =α(0). Then, for every x0 ∈ p−1(x0), there exists a unique lift α of α beginning atα(0) = x0.

Namely, given a path α in the circle, we know that each of its points can bewritten as α(t) =

(cos t(s), sin t(s)

)for some t(s) ∈ R. However, we also know that

there are many possible choices for t(s), which is unique only modulo 2π. The PathLifting Property asserts that we can choose t(s) (called α(s) in the statement) todepend continuously on s, and that t(s) is then uniquely determined once we havemade a choice for t(0).

Proposition. (Homotopy Lifting Property) Let α and β be two path homotopic

path in S1, beginning at α(0) = β(0) = x0. Let α and β be their two lifts beginning

at α(0) = β(0) = x0 ∈ p−1(x0). Then α and β are path homotopic.

Similar properties will also play a critical role in our analysis of covering spaces(and will be proved there). The Homotopy Lifting Property has the followingimportant consequence.

Corollary. Under the hypotheses of the Homotopy Lifting Property, α and β

have the same final point α(1) = β(1).

Theorem. Pick a base point x0 in the circle S1. Then there is a group isomorphismρ: π1(S1;x0)→ Z defined by the property that ρ([α]) = 1

2π

(α(1)− α(0)

)for every

[α] ∈ π1(S1;x0) and every lift α of α.

Corollary. π1(R2 − 0;x0) is isomorphic to Z.

Corollary. There is no homeomorphism between the line R and the plane R2.

Consider the closed disk B2 = x ∈ R2, ‖x‖ 6 1.

Corollary. There is no retraction r: B2 → S1, namely no such map such thatr(x) = x for every x ∈ S1.

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14 3. THE FUNDAMENTAL GROUP OF THE CIRCLE

Corollary. (2–dimensional Brouwer Fixed Point Theorem) For every map f :B2 → B2, there exists at least one point x ∈ B2 such that f(x) = x.

CHAPTER 4

Computing fundamental groups

The key tool here is the van Kampen Theorem which, under suitable hypothe-ses, computes the fundamental group of a union X1∪X2 in terms of the fundamentalgroups of X1, of X2 and of the intersection X1∩X2. The statement of this theoremrequires some algebraic preliminaries.

1. Algebraic preliminaries: Free products, presentations, andamalgamated products

1.1. Free products and free groups. .

Definition. The free product A ∗ B of the groups A and B consists of all finitesequences x1x2 . . . xn where:

1. xi ∈ A− 1 or xi ∈ B − 1 for every i;2. if xi ∈ A then xi+1 ∈ B and, conversely, if xi ∈ B then xi+1 ∈ A.

We allow the empty sequence where n = 0.

Example. Let A and B by both isomorphic to Z. If is convenient to pick agenerator a ∈ A and a generator b ∈ B and to denote these groups multiplicatively,so that A = an;n ∈ Z and B = bn;n ∈ Z. Then b4 and ab−3a−5b2a−1 aretypical elements of A ∗B.

We endow A ∗ B with a multiplication law ∗ defined by concatenation andsimplification, namely

(x1x2 . . . xm)(y1y2 . . . yn) =

x1x2 . . . xmy1y2 . . . yn if xn and y1 are not

in the same set A or B

x1x2 . . . xm−k−1(xm−kyk+1)yk+2 . . . yn otherwise,

where k is the first i such that xm−i 6= y−1i+1.

Proposition. ∗ is a group law on A ∗B.

The only difficult (or at least annoying) part of the proof is that of the asso-ciativity, because of the cancellations. The identity element consists of the emptysequence, which will consequently be denoted by 1. The inverse of x1x2 . . . xn isx−1n x−1

n−1 . . . x−11 .

Definition. The free group on n generators is the free product Fn = Z∗Z∗· · ·∗Z.

In practice, if we choose a generator xi for the i–th cyclic group Z, each elementof Fn is of the form xn1

i1xn2i2. . . xnk

ikwhere ni ∈ Z− 0 and xij+1

6= xij .

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16 4. COMPUTING FUNDAMENTAL GROUPS

Remark. The group F0 is trivial. The group F1 is isomorphic to Z. The groupFn is non-commutative if n > 2.

More generally:

Definition. The free group F (S) generated by a set S consists of all finite se-quences xn1

1 xn22 . . . xnk

k of elements xi ∈ S and of superscripts ni ∈ Z − 0,where any two consecutive xi and xi+1 ∈ S are distinct. The same concatena-tion/simplification law turns F (S) into a group, with identity element the emptysequence.

Note that we have a natural inclusion map i : S → F (S), which to x ∈ Sassociates i(x) = x1 ∈ F (S).

Theorem. (Universal Property of Free Groups)

1. Every map f : S → G from the set S to a group G has a unique extensionto a group homomorphism ϕ: F (S)→ G, in the sense that ϕ i = f .

2. Up to isomorphism, the free group F (S) is the only group satisfying (1).Namely, suppose that we are given another group F ′ and a map i′: S → Fsuch that every map f ′: S → G′ from S to a group G′ uniquely extends toa group homomorphism ϕ′: F ′ → G′ with ϕ′ i′ = f ′. Then there exists agroup homomorphism ψ: F (S)→ F ′ such that i′ = ψ′ i.

Similarly, note that the free product A ∗B comes with natural inclusion homo-morphisms iA: A→ A ∗B and iB: A ∗B.

Theorem. (Universal Property of Free Products)

1. Given two group homomorphisms jA: A→ G and jB: B → G, there existsa unique group homorphism k : A ∗ B → G such that jA = k iA andjB = k iB .

2. Up to isomorphism, the free A∗B is the only group satisfying (1). Namely,suppose that we are given another group H ′ and homomorphisms i′A: A→H and i′B: B → H such that, for every wo group homomorphisms jA: A→G and jA : A → G, there exists a unique group homorphism h′ : H → Gsuch that jA = h i′A and jB = h i′B . Then there exists an isomorphismϕ: H → A ∗B such that iA = ϕ i′A and iB = ϕ i′B .

1.2. Group presentations. .

Definition. If x1, x2, . . . , xn are the standard generators of Fn and if we are givenr1, r2, . . . , rm ∈ Fn, the group defined by the generators x1, x2, . . . , xn and by therelators r1, r2, . . . , rm is the quotient of Fn/K of Fn by the subgroup K normallygenerated by the ri. Recall that the fact that K is normally generated by the rimeans that K is the smallest normal subgroup containing all the ri; equivalently,every element of K can be written as a product of conjugates γr±1

i γ−1 with γ ∈ Fn.

In this case, we write Fn/K = 〈x1, x2, . . . , xn; r1 = r2 = · · · = rm = 1〉.

Definition. A (finite) presentation for a group G is an isomorphism

ϕ: G→ Fn/K = 〈x1, x2, . . . , xn; r1 = r2 = · · · = rm = 1〉.

1. ALGEBRAIC PRELIMINARIES 17

In practice, we have a homomorphism ψ: Fn → G, defined as the composittionof the quotient map Fn → Fn/K and of ϕ−1 : Fn/K → G. The fact that ϕ−1

is surjective implies that every g ∈ G can be written as g = gn1i1gn2i2. . . gnk

ikwith

ni ∈ Z and gi = ψ(xi), namely that the gi generate G. The fact that ϕ−1 isinjective implies that, every time gn1

i1gn2i2. . . gnk

ik= 1 in G, the corresponding element

xn1i1xn2i2. . . xnk

ik∈ Fn is actually in K. In other words, any relation gn1

i1gn2i2. . . gnk

ik= 1

between the generators gi can be deduced by the usual rules of algebra from theproperty that ψ(r1) = ψ(r2) = · · · = ψ(rm) = 1

Examples.

1. 〈a; an = 1〉 ∼= Z/n.2. 〈a, b; 〉 = Z ∗ Z.3. The group 〈a, b; aba−1b−1 = 1〉 is isomorphic to the direct product Z× Z.4. Let ∆2n be the dihedral group, namely the subgroup of the orthogonal

group O(2) generated by the rotation A of angle 2πn and by the reflection

B across a line L. Then ∆2n admits the presentation 〈A,B;An = B2 =(AB)2 = 1〉.

Theorem. (Markov) There exists no algorithm which, given m relators r1, r2, . . . ,rm in the free group Fn and given an element x ∈ Fn, decides whether the elementof G = 〈x1, x2, . . . , xn; r1 = r2 = · · · = rm = 1〉 defined by x is trivial or not.

One can similarly define group presentations with infinitely many generatorsand/or infinitely many relations by replacing the free group Fn by the group F (S)freely generated by a possibly infinite set S, and taking as relators the elements ofa possibly infinite subset R of F (S).

1.3. Amalgamated products. .

Definition. If we are given three groups A, B and C and two group homomor-phisms iA: C → A and iB: C → B, the free product of A and B amalgamated alongC using the homomorphisms iA and iB is the quotient A ∗C B = (A ∗B)/K of thefree product A ∗ B by the subgroup K normally generated by all the elements ofthe form iA(c)iB(c)−1 with c ∈ C. Note that the notation A ∗C B is somewhatincomplete, inasmuch as A ∗C B depends also on the homomorphisms iA and iB ,not just on the groups A, B and C.

In practice, if

A = 〈x1, x2, . . . , xn; r1 = r2 = · · · = rm = 1〉,

if

B = 〈y1, y2, . . . , yp; s1 = s2 = · · · = sq = 1〉,

and if C is generated by c1, c2, . . . , ct, then A ∗C B has the presentation

A ∗C B = 〈x1, x2, . . ., xn, y1, y2, . . . , yp;

r1 = r2 = · · · = rm = s1 = s2 = · · · = sq = 1,

iA(c1)iB(c1)−1 = iA(c2)iB(c2)−1 = · · · = iA(ct)iB(ct)−1 = 1〉.

18 4. COMPUTING FUNDAMENTAL GROUPS

Note that the amalgamated product comes with two natural group homomor-phism jA: A → A ∗C B and jB → A ∗C B. For instance, jA is the composition ofthe inclusion map A→ A ∗B with the quotient map A ∗B → A ∗C B = A ∗B/K.

Theorem. (Universal Property of Amalgamated Products)

1. For any two group homomorphisms kA : A → G and kB : B → G suchthat kA(iA(c)) = kB(iB(c)) for each c ∈ C, there exists a unique grouphomomorphism k: A ∗C B → G such that kA = k jA and kB = k jB .

2. Up to isomorphism, A ∗C B is uniquely determined by Property (1).

Exercise. Write a detailed statement of the uniqueness property in (2) above.

2. The van Kampen Theorem(s)

If X1 and X2 are two subspaces of the topological space X and if we pick abase point x0 in X1 ∩X2, the appropriate inclusion maps induce homomorphisms

i1∗: π1(X1 ∩X2;x0)→ π1(X1;x0) i2∗: π1(X1 ∩X2;x0)→ π1(X2;x0)

k1∗: π1(X1;x0)→ π1(X1 ∪X2;x0) k2∗: π1(X2;x0)→ π1(X1 ∪X2;x0)

such that k1∗ i1∗ = k2∗ i2∗. The Universal Property of Amalgamated Productsthen provides a group homomorphism

k: π1(X1;x0) ∗π1(X1∩X2;x0) π1(X2;x0)→ π1(X1 ∪X2;x0).

Theorem. (van Kampen Theorem I) Suppose that X1 and X2 are open in X, andthat X1, X2 and X1 ∩X2 are all path connected. Then the above homomorphism

k: π1(X1;x0) ∗π1(X1∩X2;x0) π1(X2;x0)→ π1(X1 ∪X2;x0).

is an isomorphism.

The proof uses the following result from real analysis and/or pointset topology.

Proposition. (Lebesgue Number Lemma) Let U = Ui; i ∈ I be an open coveringof the compact metric space (X, d). (Namely, each Ui is open in X, and X =⋃i∈I Ui.) Then, there exists a number ε > 0 such that, for every x ∈ X, the ball

Bd(X, ε) is completely contained in some Ui.

The number ε is a Lebesgue number for the covering U .

Because we tend to prefer compact spaces, and because compact spaces tend toprovide closed subsets, it is desirable to have a version of the van Kampen Theoreminvolving compact subsets. However, an additional condition is then necessary.

Definition. A subset A is a neighborhood deformation retract in X if there existsan open subset U of X such that U deformation retracts to A. (In particular, Ucontains A.)

In general, most “reasonable” subsets are neighborhood deformation retracts.However, the Cantor set is not a neighborhood deformation retract in R.

Theorem. (van Kampen Theorem II) Suppose that X1 and X2 are closed in X,and that X1, X2 and X1 ∩ X2 are all path connected. Suppose in addition that

3. FINITE CELL COMPLEXES 19

X1 ∩ X2 is a neighborhood deformation retract in both X1 and X2. Then thehomomorphism

k: π1(X1;x0) ∗π1(X1∩X2;x0) π1(X2;x0)→ π1(X1 ∪X2;x0).

is an isomorphism.

Example. The figure eight has fundamental group 〈a, b; 〉 = Z ∗ Z.

Example. The sphere Sn = (x0, x1, . . . , xn) ∈ Rn+1;∑ni=0 x

2i = 1 has funda-

mental group

π1(Sn;x0) =

0 if n 6= 1

Z if n = 1

Example. The torus S1×S1 has fundamental group 〈a, b; aba−1b−1 = 1〉 ∼= Z×Z.

Example. The Klein bottle has fundamental group 〈a, b; abab−1 = 1〉.

Example. Let K0 be the circle S1 ⊂ R2 ⊂ R3. Then the complement R3−K0 hasfundamental group isomorphic to Z.

Example. Let K1 be the closed curve in R3 parametrized by

t 7→((R+ r cos 3t) cos 2t, (R+ r cos 3t) sin 2t, r sin 3t

)with R > r > 0. The fundamental group of R3−K1 has presentation 〈a, b; a2 = b3〉.

Corollary. There is no homeomorphism R3 → R3 sending the overhand knot K1

(also called trefoil knot) to the unknot K0.

3. Finite cell complexes

Recall the definition of the n–dimensional ball Bn = x ∈ Rn; ‖x‖ 6 1,delimited by the (n− 1)–dimensional sphere Sn−1 = x ∈ Rn; ‖x‖ = 1 in Rn.

Definition. An n–dimensional cell complex is a space X which can be written asX = X(0) ∪X(1) ∪ · · · ∪X(n) in such a way that:

1. X(0) consists of finitely many points;2. each X(i+1) is obtained by gluing to X(i) finitely many copies Bi+1

1 , Bi+12 ,

. . . , Bi+1ki

of the i+ 1–dimensional ball Bi+1 by continuous maps ϕj: Sij →

X(i) from the corresponding spheres Sij ⊂ Bi+1j to X(i); namely, given such

maps ϕj , X(i+1) is the quotient space

X(i+1) = X(i) tBi+11 tBi+1

2 t · · · tBi+1ki

/ ∼

quotient of the disjoint union of X(i) and the Bi+1j by the equivalence

relation ∼ that identifies each x ∈ Sij to ϕj(x) ∈ X(i).

The images of the balls Bi+1j are the (i+ 1)–dimensional cells of X, and X(i) is its

i–dimensional skeleton.

20 4. COMPUTING FUNDAMENTAL GROUPS

Definition. A finite graph is a 1–dimensional cell complex. Traditionally, the0–dimensional cells of a graph are also called its vertices, and its 1–dimensionalcells are its edges.

Theorem. If X = X(1) is a connected graph with v vertices and e edges, itsfundamental group π1(X;x0) is isomorphic to the free group Fn where n = e−v+1.

Theorem. Let X be an n–dimensional cell complex, whose 2–skeleton X(2) isobtained by attaching 2–dimensional cells B2

1 , B22 , . . . , B2

k to the 1–skeleton X(1)

by gluing maps ϕi : S1i → X(1). Then, the fundamental group of X is isomor-

phic to the quotient of the fundamental group of the graph X(1) by the subgroupnormally generated by the ϕi∗(ai), where ai is a generator of the fundamentalgroup π1(Si;xi) ∼= Z. In particular, the 1-skeleton X(1) provides generators, the2–dimensional cells give relators, and the higher dimensional cells do not contributeto the fundamental group of X.

CHAPTER 5

Covering spaces

1. Definitions

Definition. A covering map is a map p: X → X such that, for every x ∈ X, thereexists an open neighborhood U of x for which:

1. the preimage p−1(U) can be written as union of disjoint open subsets

Uii∈I ;2. for every i ∈ I, the restriction p|Ui

: Ui → Ui is a homeomorphism.

A covering is a triple (X, X, p) consisting of two topological spaces X and X and

of a covering map p: X → X. The space X is the total space or covering space ofthe covering, while X is its base.

A subset U ⊂ X that satisfies the properties of the definition above is said tobe evenly covered.

Examples.

1. The map p : R → S1 defined by p(t) = (cos t, sin t), which we used tocompute the fundamental group of the circle, is a covering map.

2. The restriction p : [0, 6π) → S1 of the above map is not a covering map.Note that p−1(x) consists of 3 points for every x ∈ S1.

3. The map p: C− 0 → C− 0 defined by p(z) = z7 is a covering map.4. The map p: C→ C defined by p(z) = z7 is not a covering map.5. Let X be a topological space, and let F be a set endowed with the discrete

topology. The projection p: X × F → X to the second factor is a coveringmap, called the trivial covering with base X and fiber F . Note that everycovering is locally of this type, but not necessarily globally.

Definition. Two coverings p: X → X and p′: X ′ → X over the same base X are

isomorphic if there exists a homeomorphism ϕ: X → X ′ such that p′ ϕ = p.

Our goal is to classify all coverings of a given topological space X, up to iso-morphism.

2. Lifting properties

The theory of covering spaces with base X turns out to be strongly connectedto the fundamental group π1(X;x0) because of the following two fundamental prop-erties.

21

22 5. COVERING SPACES

Theorem.(Path Lifting Property) Let p : X → X be a covering map, and letα : [0, 1] → X be a path beginning at x0 = α(0). Then, for every x0 ∈ p−1(x0),

there exists a unique path α: [0, 1]→ X such that

1. α lifts α in the sense that p α = α;2. α(0) = x0.

Theorem. (Homotopy Lifting Property) Let p: X → X be a covering map, and letα, β: [0, 1] → X be two paths beginning at the same point x0 = α(0) = β(0). For

x0 ∈ p−1(x0), let α, β : [0, 1] → X be the two lifts beginning at x0 = α(0) = β(0)provided by the Path Lifting Property. If, in addition, α and β are path homotopic,

then α and β are path homotopic; in particular, α and β end at the same point

α(1) = β(1).

Corollary. The homomorphism p∗: π1(X; x0)→ π1(X;x0) induced by a covering

map p: X → X is injective.

The (proofs of the) Path Lifting Property and Homotopy Lifting Property showus how to lift maps α: [0, 1] → X and H : [0, 1] × [0, 1] → X to a covering space

X. The following generalizes these lifting properties, and explains what is specialabout [0, 1] and [0, 1]× [0, 1].

Theorem. (Lifting Criterion) Let p: X → X be a covering map, and consider amap f : Y → X where Y is path connected and locally path connected. Choose base

points x0 ∈ X, x0 ∈ X and y0 ∈ Y such that x0 = p(x0) = f(y0). The followingare equivalent.

1. There exists a lift f : Y → X of f (namely a map f such that p f = f)

such that f(y0) = y0.

2. In π1(X;x0), the subgroup f∗(π1(Y ; y0)

)is contained in p∗

(π1(X; x0)

).

In particular, a lift always exists if Y is simply connected, in the following sense.

Definition. A space X is simply connected if it is path connected and locally pathconnected, and if π1(X;x0) = 1 for some (or all) base point x0 ∈ X.

Although apparently much more general than the Path and Homotopy LiftingProperties, this Lifting Criterion actually is a relatively simple consequence of theseproperties.

3. Connected coverings

We now restrict attention to coverings p: X → X where the total space X isconnected. It may useful to remember that a connected space that is locally pathconnected is also path connected.

Theorem. Assume that X is connected and locally path connected. Let p: X → X

and p′ : X ′ → X be two coverings where X and X ′ are path connected, and pick

two base points x0 ∈ X and x′0 ∈ X ′ such that p(x0) = p′(x′0) = x0. There exists a

3. CONNECTED COVERINGS 23

covering isomorphism ϕ: X0 → X ′ sending x0 to x′0 if and only if p∗(π1(X; x0)

)=

p′∗(π1(X ′; x′0)

)in π1(X0, x0).

In other words, when X is connected and locally path connected, the map

which to a path connected covering p: X → X and base point x0 ∈ X associates

the subgroup p∗(π1(X; x0)

)is injective, up to covering isomorphism sending base

point to base point. We will show later in this section that this map is surjective,under a (mild) additional hypothesis on X.

First, let us consider what happens if we do not require covering isomorphismsto send base point to base point.

Theorem. Assume that X is connected and locally path connected. Let p: X → X

and p′: X ′ → X be two coverings where X and X ′ are path connected, with base

points x0 ∈ X and x′0 ∈ X ′ such that p(x0) = p′(x′0) = x0. These two coverings are

isomorphic if and only if the subgroup p′∗(π1(X; x0)

)is conjugate to p∗

(π1(X ′; x′0)

)in π1(X0, x0).

Definition. A space X is semi-locally simply connected if it, for every x ∈ X andevery neighborhood U of x, there exists a smaller neighborhood V ⊂ U of x whichis path connected and such that the homomorphism π1(V ;x) → π1(X;x) inducedby the inclusion map has trivial image (namely every path from x to x in V is pathhomotopic to the constant path cx in X, although not necessarily in V ).

Examples.

1. Every topological manifold is semi-locally simply connected. (Recall thata topological manifold of dimension n is a space where every point admitsa neighborhood that is homeomorphic to an open subset of Rn.)

2. Consider the Hawaiian rings X =⋃n∈N Cn, where Cn is the circle of radius

1n centered at ( 1

n , 0) in the plane. This space X is not semi-locally simplyconnected.

Theorem. Let X be path connected and semi-locally simply connected. For every

subgroup G ⊂ π1(X;x0), there exists a path connected covering p: X → X suchthat p(x0).

Corollary. Up to isomorphism, a space X that is path-connected and semi-

locally simply connected space admits a unique covering p : X → X where X issimply connected.

This simply connected covering is the universal covering space of X.

By realizing a free group as the fundamental group of a 1–dimensional cellcomplex, one obtains a purely algebraic result.

Corollary. (Nielsen-Schreier Theorem) Every subgroup of a free group is iso-morphic to a free group.

24 5. COVERING SPACES

4. Coverings with a given fiber

The fiber of a covering p : X → X with base point x0 ∈ X is the pre-imageF = p−1(x0).

Consider a point x ∈ F and an element [α] ∈ π1(X;x0). Use the Path Lifting

Property to lift the path α: [0, 1]→ X to a path α: [0, 1]→ X with p α = α andα(0) = x. Then α(1) is another element of the fiber F , usually distinct from thestarting point α(0) = x.

Let ρ([α]): F → F be the map which to x ∈ F associates the element α(1) ∈ Fdefined as above. The Homotopy Lifting Property shows that ρ([α]) is independentof the choice of the path α used to represent [α] ∈ π1(X;x0).

Proposition. This defines a group antihomomorphism ρ : π1(X;x0) → Bij(F ),valued in the group Bij(F ) of all bijections F → F . This is called the monodromyantihomomorphism of the covering.

The fact that ρ is an antihomomorphism just means that ρ([α] ∗ [β]) = ρ([β]) ∗ρ([α]) for every [α], [β] ∈ π1(X;x0).

Theorem. Let X be a path connected and semi-locally simply connected space,let F be a set, and consider an antihomomorphism ρ: π1(X;x0) → Bij(F ). Then,

there exists a covering p: X → X with fiber p−1(x0) = F and whose monodromyantihomomorphism is equal to ρ. In addition, this covering is unique up to isomor-

phism in the following sense: For any other covering p′: X ′ → X with monodromy

ρ, there is a covering isomorphism ϕ: X → X ′ (with p′ ϕ = p) which coincideswith the identity on F = p−1(x0) = p′−1(x0).

Corollary. If X is simply connected, every covering p: X → X is isomorphic tothe trivial bundle X × F → X.

CHAPTER 6

Singular homology

Homology is the science of the statement “B is the boundary of A”, motivatedby Stokes’s theorem.

1. Simplices

Definition. The standard simplex of dimension n is

∆n = (t0, t1, . . . , tn) ∈ Rn+1;

n∑i=0

ti = 1 and ti > 0,∀i.

Namely ∆n is the convex hull of the n+ 1 points that are the tips of the coordinatevectors of Rn+1.

We can embed the (n − 1)–dimensional simplex ∆n−1 in the n–dimensionalsimplex ∆n, but there are n− 1 natural ways to do so.

Definition. For i = 0, . . . , n, the i–face map is the map Fi: ∆n−1 → ∆n definedby

Fi(t0, t1, . . . , tn−1) = (t0, . . . , ti−1, 0, ti, . . . , tn−1),

where the 0 is inserted in the (i+ 1)–th position.

Note that the boundary ∂∆n is the union of the images Fi(∆n−1) of the n+ 1face maps.

Definition. A (singular) n–simplex in a topological space X is a continuous mapσ: ∆n → X.

The set of all singular n–simplices is denoted by Sn(X).

2. Cochains

This is here just as a motivation for the definition of singular chains in the nextsection.

Let R be a commutative ring with a unit denoted by 1.

Definition. A singular n–cochain valued in R is a map c: Sn(X)→ R.

The set Cn(X) of singular cochains valued in R is clearly an R–module.

Historic example. If X is a manifold and if we restrict attention to differentiablesingular simplices, a differential form ω ∈ Ωn(X) of degree n defines a cochain

25

26 6. SINGULAR HOMOLOGY

cω: Sn(X)→ R by

cω(σ) =

∫∆n

σ∗(ω)

for every differentiable simplex σ: ∆n → X.

Definition. The coboundary of a cochain c ∈ Cn(X;R) is the cochain dc ∈Cn+1(X;R) defined by the formula

dc(σ) =

n+1∑i=0

(−1)ic(σ Fi) ∈ R

for every (n + 1)–simplex σ. Note that each face map Fi: ∆n → ∆n+1 defines ann–simplex σ Fi: ∆n → X.

The formula is specially designed so that dcω = cdω in the historical exampleabove, where dω is the exterior derivative of the differential form ω. The sign (−1)i

is then justified by the possible discrepancy, on the face Fi(∆n) of the boundary∂∆n+1, between the boundary orientation of ∂∆n+1 and the orientation comingfrom the orientation of ∆n.

These observations were designed as an introduction to what follows. Cochainswill reappear later when we discuss cohomology.

3. Chains

From now on, we fix a commutative ring R with a unit element 1. This ring Rwill be Z most of the time, R or Q sometimes, Z2 sometimes too, and a general Znmuch less frequently.

Definition. A singular n–chain in X and with coefficients in R is an element ofthe free R–module Cn(X) generated by the set Sn(X). Recall that the free R–module Cn(X) consists of all the maps c: Sn(X) → R such that c(σ) = 0 for allbut finitely many elements of Sn(X).

In practice, there is a canonical inclusion map χ: Sn(X) → Cn(X), which toσ ∈ Sn(X) associates the characteristic function χσ defined by χσ(σ′) = 0 if σ′ 6= σand χσ(σ) = 1. If we identify σ ∈ Sn(X) to χσ ∈ Cn(X), then any a ∈ Cn(X) canbe written as

c =∑

σ∈Sn(X)

aσσ =

k∑i=1

aiσi

where aσ = c(σ) ∈ R, where the set of σ ∈ Sn(X) with c(σ) 6= 0 is contained in thefinite set σ1, σ2, . . . , σk, and where ai = c(σi) = aσi

∈ R. In particular, the firstsum makes sense since only finitely many of its terms are non-zero.

Definition. The n–dimensional boundary map is the unique linear map ∂n :Cn(X)→ Cn(X) such that, for every simplex σ ∈ Sn(X) ⊂ Cn(X),

∂nσ =

n∑i=0

(−1)iσ Fi.

Note the analogy with the coboundary map of the previous section.

5. HOMOMORPHISMS INDUCED BY CONTINUOUS MAPS 27

In practice, we usually drop the subscript n, and write ∂n = ∂.

Lemma. ∂ ∂ = 0, or more precisely ∂n−1 ∂n(c) = 0 for every c ∈ Cn(X).

4. Homology modules

Definition. The set of closed n–chains, or the set of n–cycles, is the kernel

Zn(X) = c ∈ Cn(X); ∂nc = 0

of the boundary map ∂n. The set of exact n–chains, or the set of n–boundaries isthe image

Bn(X) = c ∈ Cn(X);∃c′ ∈ Cn+1(X), c = ∂n+1c′

of ∂n+1. Note that Zn(X) and Bn(X) are both submodules of Cn(X), and thatBn(X) ⊂ Zn(X) since ∂ ∂ = 0.

Definition. The n–th homology module of X with coefficients in R is

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

We just write Hn(X) when the coefficient ring R is clear from the context. WhenR = Z, the Hn(X) are the homology groups of X. (Recall that a Z–module is justan abelian group),

Silly example. If X = ∅, then Hn(X) = 0 for every n.

Example. Let X consist of a single point x0. Then

Hn(x0) =

R if n = 0

0 if n 6= 0

Proposition. Let Xii∈I be the set of path connected components of X. Then

Hn(X) ∼=⊕i∈I

Hn(Xi).

This enables us to restrict attention to path connected spaces.

Theorem. If X is path connected and non-empty, then H0(X) is isomorphic to R.

5. Homomorphisms induced by continuous maps

Let f : X → Y be a (continuous) map. It determines a (set-theoretic) mapSn(X) : Sn(X) → Sn(Y ) defined by σ 7→ f σ. This map then defines an R–module homomorphism (= a linear map) Cn(f): Cn(X)→ Cn(Y ), which to a chain∑i aiσi ∈ Cn(X), with ai ∈ R and σi ∈ Sn(X), associates

∑i aif σi ∈ Cn(Y ).

The following is automatic.

Lemma. ∂n−1 Cn(f) = Cn−1(f) ∂m.

Corollary. The map Cn(f): Cn(X)→ Cn(Y ) sends Zn(X) to Zn(Y ), and Bn(X)to Bn(Y ). As a consequence, it induces a linear map Hn(f): Hn(X)→ Hn(Y ).

28 6. SINGULAR HOMOLOGY

Definition. The above map Hn(f) : Hn(X) → Hn(Y ) is the homomorphisminduced in n–th homology by f .

6. Homotopy invariance

Recall that two maps f0, f1: X → Y are homotopic if there exists a continuousmap H: X× [0, 1]→ Y , called a homotopy such that H(x, 0) = f0(x) and H(x, 1) =f1(x) for every x ∈ X.

Theorem. (Homotopy Invariance) If f0, f1: X → Y are homotopic, then for everyn the induced homomorphisms Hn(f0), Hn(f1): Hn(X)→ Fn(Y ) are equal.

The key step in the proof is the following.

Lemma. (Construction of a chain homotopy) Under the above hypotheses, thereexists a family of linear maps Kn: Cn(X)→ Cn+1(Y ) such that

∂n+1 Kn +Kn−1 ∂n = Cn(f1)− Cn(f0).

Proof. Associate to the singular n–simplex σ: ∆n → X the chain

Kn(σ) =

n∑i=0

(−1)iσi

where the (n+ 1)–simplices σi: ∆n+1 → Y are defined by

σi(x0, x1, . . . , xn+1) = H(σ(x0, . . . , xi−1, xi+xi+1, xi+2, . . . , xn+1), xi+1+· · ·+xn+1

).

After this geometric step, the rest of the proof is a simple algebraic manipula-tion.

Corollary. If f : X → Y is a homotopy equivalence, then the induced homomor-phism Hn(f): Hn(X)→ Hn(Y ) is an isomorphism.

Corollary. If X ⊂ Rn is star-shaped,

Hn(X;R) ∼=

0 if n 6= 0

R if n = 0.

CHAPTER 7

Basic homological algebra

1. Chain complexes

Definition. A chain complex is a bi-infinite sequence C = Cn, ∂nn∈Z of R–modules Cnand linear maps ∂n : Cn → Cn−1 such that ∂n−1 ∂n = 0 for everyn.

In particular, in Cn, the kernel Zn(C) = Ker ∂n of ∂n contains the imageBn(C) = Im ∂n+1 of ∂n+1, and we can consider the quotient R–module Hn(C) =Zn(C)/Bn(C).

Definition. The quotient Hn(C) = Zn(C)/Bn(C) is the n–th homology moduleof the chain complex C = Cn, ∂nn∈Z.

The chain complex C = Cn, ∂nn∈Z is an exact sequence if Hn(C) = 0, namelyif Ker ∂n = Im ∂n+1 for every n.

Fundamental example. Cn = Cn(X) for some topological space X, with theconvention that Cn(C) = 0 if n < 0 and with the usual boundary homomorphisms∂n : Cn(X) → C−n(X). The homology module Hn(C) of the chain complex C isthen the singular homology module Hn(X).

2. Another example: reduced homology

Consider the homomorphism ∂0 : C0(X) → R defined by the property that

∂0(∑ki=1 aixi) =

∑ki=1 ai. For n ∈ Z, define

Cn(X) =

Cn(X) if n > 0

R if n = −1

0 if n 6 −2

and let ∂n: Cn(X)→ Cn−1(X) by the property that

∂n =

∂n if n > 0

the above ∂0 if n = 0

0 if n 6 −1.

One can show that C(X) = Cn(X), ∂nn∈N is a chain complex.

Definition. The homology modules Hn(C(X)) of the above chain complex are

the reduced homology modules Hn(X) of X.

29

30 7. BASIC HOMOLOGICAL ALGEBRA

Theorem.

1. Hn(X) = Hn(X) if n 6= 0;

2. H0(X) = 0 if X is path connected;

3. H0(X) ∼= Rk−1 if X has k path connected components.

3. Chain maps

Definition. A chain map f : C → C ′ from the chain complex C = Cn, ∂nn∈Z tothe chain complex C = Cn, ∂nn∈Z is a family of linear maps fn: Cn → C ′n suchthat ∂′n fn = fn−1 ∂n for every n. Namely, the infinite ladder of arrows providedby the data is a commutative diagram.

It is immediate from definitions that fn sends Zn(C) to Zn(C ′) and Bn(C) toBn(C ′). We can therefore consider:

Definition. The homomorphism induced in n–dimensional homology induced bythe chain map f : C → C ′ is the linear map Hn(f): Hn(C) → Hn(C ′) induced byfn.

Definition. A chain homotopy from the chain map f : C → C ′ to the chain mapg: C → C ′ is a family of linear maps Kn: Cn → C ′n+1 such that

∂′n+1 Kn +Kn−1 ∂n = gn − fnfor every n. The chain maps f : C → C ′ and g: C → C ′ are chain homotopic whenthere exists such a chain homotopy.

Proposition. If the chain maps f : C → C ′ and g: C → C ′ are chain homotopic,the induced homomorphisms Hn(f), Hn(g): Hn(C)→ Hn(C ′) are equal.

Corollary. A map f : X → Y induces a homomorphism Hn(f): Hn(X)→ Hn(Y )between the reduced homology modules of X and Y . If f and g : X → Y are

homotopic, then Hn(f) = Hn(g).

4. Exact sequences of chain complexes

Recall that a short exact sequence of R–modules is a sequence of modules andmaps

00 // A

f// B

g// C

0 // 0

where Ker f = Im 0 (so that f is injective), Ker g = Im f and Ker 0 = Im g (so thatg is surjective).

Definition. A short exact sequence of chain complexes is a seqence of chaincomplexes and chain maps

00 // A

f// B

g// C

0 // 0

such that each 00 // An

fn // Bngn // Cn

0 // 0 is a short exact sequence ofR–modules.

4. EXACT SEQUENCES OF CHAIN COMPLEXES 31

In particular, we have a commutative grid1

0

∂

∂

∂

0

00 //

0

An+1

fn+1//

∂

Bn+1

gn+1//

∂

Cn+10 //

∂

0

0

00 //

0

Anfn //

∂

Bngn //

∂

Cn0 //

∂

0

0

00 //

0

An−1

fn−1//

∂

Bn−1

gn−1//

∂

Cn−10 //

∂

0

0

where each vertical line is a chain complex, and where each horizontal line is a shortexact sequence of R–modules.

Theorem. (Snake Lemma) Let 00 // A

f// B

g// C

0 // 0 be a shortexact sequence of chain complexes. Then there exists a long exact sequence

δn+2

rrHn+1(A)

Hn+1(f)// Hn+1(B)

Hn+1(f)// Hn+1(C)

δn+1

ssHn(A)

Hn(f)// Hn(B)

Hn(f)// Hn(C)

δn

ssHn−1(A)

Hn−1(f)// Hn−1(B)

Hn−1(f)// Hn−1(C)

δn−1

rr

between the corresponding homology modules.

Complement. The homomorphism δn : Hn(C) → Hn−1(A) is defined as follows.Let cn ∈ Cn represent [cn] ∈ Hn(C). Then δn([cn]) = [an−1] where fn−1(an−1) =∂bn for an arbitrary bn ∈ Bn such that gn(bn) = cn.

The homomorphism δn is called the connecting homomorphism of the longexact sequence.

1I hope you are impressed by my commutative diagrams. It’s a first time I am using this setof TEX macros, and I am really proud of myself.

32 7. BASIC HOMOLOGICAL ALGEBRA

The proof of the Snake Lemma is by “diagram chasing”. It uses all nodes andmaps of the grid

Bn+1

gn+1//

∂

Cn+10 //

∂

0

Anfn //

∂

Bngn //

∂

Cn0 //

∂

0

00 // An−1

fn−1//

∂

Bn−1

gn−1//

∂

Cn−1

An−2

fn−2// Bn−2

5. The Five Lemma

The same kind of diagram chasing proof gives:

Theorem. (Five Lemma) Suppose that we have a commutative diagram of modulesand linear maps

Af//

α ∼=

Bg//

β ∼=

Ch //

γ

Dk //

δ ∼=

E

ε ∼=

A′f ′// B′

g′// C ′

h′ // D′k′ // E′

where the horizontal rows are exact sequences. If α, β, δ and ε are isomorphisms,then the middle map γ is also an isomorphism.

(Not all the hypotheses are needed here, but the statement is easier to rememberin this way.)

CHAPTER 8

Relative homology

1. Relative homology

If A ⊂ X, any singular simplex valued in A can also be considered as a simplexvalued in X. We can therefore consider Sn(A) as a subset of Sn(X), and Cn(A) asa sub-module of Cn(X).

Definition. The module of relative chains in X modulo A is the quotient moduleCn(X,A) = Cn(X)/Cn(A).

The boundary map ∂n : Cn(X) → Cn−1(X) clearly sends Cn(A) to Cn−1(A),and consequently induces a quotient map ∂n: Cn(X,A)→ Cn−1(X,A). It is imme-diate that ∂n ∂n−1 = 0 since ∂n ∂n−1 = 0, so that C(X,A) = Cn(X,A), ∂nn∈Nis a chain complex.

Definition. The n–th relative homology module of X modulo A is the homologymodule Hn(X,A) of the above chain complex C(X,A).

In particular, Hn(X,A) is a quotient of quotients, which is not very convenient.The following is more convenient to understand what we are doing.

Definition. The module of relative n–cycles of X modulo A is

Zn(X,A) = c ∈ Cn(X); ∂c ∈ Cn(A).

The module of relative n–boundaries of X modulo A is

Bn(X,A) = c ∈ Cn(X);∃c′ ∈ Cn+1(X), c− ∂c′ ∈ Cn(A).

Theorem.

Hn(X,A) ∼= Zn(X,A)/Bn(X,A).

Remark. If A = ∅, Hn(X,∅) = Hn(X).

Proposition. If Xii∈I is the family of path connected components of X, and ifAi = A ∩Xi, then

Hn(X,A) =⊕i∈I

Hn(Xi, Ai).

2. Homomorphism induced by maps

Suppose that A ⊂ X and B ⊂ Y .

33

34 8. RELATIVE HOMOLOGY

Definition. A map f : (X,A) → (Y,B) from the pair (X,A) to the pair Y,B) isa (continuous) map f : X → Y such that f(A) ⊂ B.

It is immediate that the homomorphism Cn(f) : Cn(X) → Cn(Y ) induceshomomorphisms Cn(f): Cn(X,A)→ Cn(Y,B) and Hn(f): Hn(X,A)→ Hn(Y,B).

Definition. Hn(f): Hn(X,A)→ Hn(Y,B) is the homomorphism induced in n–threlative homology by the map f : (X,A)→ (Y,B).

3. Relative homology in dimension 0

Theorem. If X is path connected

H0(X,A) ∼=

0 if A 6= ∅R if A = ∅

4. The long exact sequence in relative homology

By definition, there is a short exact sequence of chain complexes

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

Applying the Snake Lemma then automatically gives the following long exact se-quence

Theorem. (Long Exact Sequence in Relative Homology) If A ⊂ X, there is a longexact sequence

δn+1

rrHn(A) // Hn(X) // Hn(X,A)

δn

rrHn−1(A) // Hn−1(X) // Hn−1(X,A)

δn−1

rr

Complement. In the above long exact sequence, the connecting homomorphismδn associates to the homology class [c] ∈ Hn(X,A), represented by c ∈ Zn(X,A),the homology class [∂c] ∈ Hn−1(A).

Note that ∂c is the boundary of something in X (so that [∂c] = 0 ∈ Hn−1(X)),but is not necessarily the boundary of something in A so that [∂c] ∈ Hn−1(A) isnot necessarily 0.

This kind of phenomenon indicates that extreme care should be used whenwriting equalities between homology classes. It is a good idea to systematicallywrite the homology modules where such equalities take place.

6. EXAMPLES 35

5. Reduced relative homology

One can define reduced relative homology by the convention that

Cn(X,A) =

Cn(X,A) if n 6= −1

0 if n = −1 and A 6= ∅R if n = −1 and A = ∅

The homology modules of the chain complex so defined are the reduced relative

homology modules Hn(X,A) of X modulo A. Note that Hn(X,A) is just thestandard homology module Hn(X,A) when A 6= ∅, and the reduced homology

module Hn(X) when A = ∅.

The definitions are designed so that we still have a short exact sequence of chaincomplexes, and a Long Exact Sequence in Reduced Relative Homology obtainedfrom the Long Exact Sequence in Relative Homology above by putting tildes ev-erywhere. This long exact sequence tends to have more zero modules when n = 0,which makes it more convenient for computations.

6. Examples

Recall that Bn = x ∈ Rn; ‖x‖ 6 1 is the (closed) unit ball in Rn, boundedby the (n− 1)–dimensional sphere Sn−1 = x ∈ Rn; ‖x‖ = 1.

Proposition.

Hp(Bn, Sn−1) ∼= Hp−1(Sn−1) if p > 1

H1(Bn, Sn−1) ∼=

0 if n > 1

R if n = 1.

Remark. We can combine the two cases (and their proofs) by using reduced

homology. Then, Hp(Bn, Sn−1) ∼= Hp−1(Sn−1) for every p > 0 and every n > 1.

In the sphere Sn, we can also consider the lower hemisphere Bn−, consisting ofthose x ∈ Sn ⊂ Rn+1 whose last coordinate is 6 0 (and homeomorphic to the ballBn by projection to Rn).

Proposition.

Hp(Sn, Bn−) ∼= Hp(S

n) if p > 0

H0(Sn, Bn−) ∼=

0 if n > 0

R if n = 0.

These two propositions are still pretty useless right now. However, when com-bined with the Excision Theorem of the next chapter, they will enable us to computeall homology modules of spheres.

CHAPTER 9

The Excision Theorem

1. Excisions

Suppose that B ⊂ A ⊂ X. We now have an inclusion map i: (X−B,A−B)→(X,A).

Definition. The subset B can be excised from the pair (X,A), or the inclusionmap i: (X −B,A−B)→ (X,A) is an excision, if the induced homomorphism

Hn(i): Hn(X −B,A−B)→ Hn(X,A)

is an isomorphism for every n.

Theorem. (Excision Theorem) Suppose that B is “deep inside of A”, in the sensethat the closure of B is contained in the interior of A. Then B can be excised from(X,A).

The key lemma in the proof, which is of independent interest, is the following.Let U = Uii∈I be an open covering of X, meaning that X =

⋃i∈I Ui and each

Ui is open in X. Let CUn (X) ⊂ Cn(X) be the submodule generated by the Cn(Ui).Namely, CUn (X) consists of linear combinations of singular n–simplices in X whoseimage is completely contained in one of the Ui. The CUn (X) are clearly well-behavedunder the boundary maps, and therefore form a chain complex with homologymodules HUn (X). The inclusion maps CUn (X)→ Cn(X) form a chain map, inducinghomomorphisms HUn (X)→ Hn(X).

Theorem. The above homomorphisms HUn (X)→ Hn(X) are isomorphisms.

2. Subdivisions

Let Sn be the symmetric group of order n, considered as the group of bijectionsof the set 0, 1, . . . , n − 1. For every s ∈ Sn+1, consider the map δs : ∆n → ∆n

defined by

δs(x0, x1, . . . , xi, . . . , xn) =(xs(0)

1 ,xs(0)+xs(1)

2 , . . . ,∑i

j=0 xs(j)

i+1 , . . . ,∑n

j=0 xs(j)

n+1

).

In particular, the δs(∆n) are n–simplices in Rn+1, whose vertices are barycen-ters of subsets of the set of vertices of the standard simplex ∆n. By induction, oneeasily sees that, as s ranges over all permutations of Sn, the δs(∆n) cover all of∆n, and have disjoint interiors in ∆n.

37

38 9. THE EXCISION THEOREM

Define a linear map Σn : Cn(X) → Cn(X) by the property that, for everysingular simplex σ: ∆n → X,

Σn(σ) =∑

s∈Sn+1

sign(s)σ δs

where sign(s) = ±1 is the signature of s.

Lemma. The Σn form a chain map Σ: C(X)→ C(X).

Lemma. The chain map Σ: C(X)→ C(X) is chain homotopic to the identity.

For every k > 0, let Σk = Σ Σ · · · Σ be the chain map C(X) → C(X) beobtained by composing Σ with itself k times.

Lemma. For every k > 0, the chain map Σk: C(X)→ C(X) is chain homotopic tothe identity.

For every singular simplex σ: ∆n → X, the chain Σk(σ) is a linear combinationof simplices σ δs1 δs2 · · · δsk where s1, s2, . . . , sk ∈ Sn.

Lemma. Endow Rn+1 with the distance d(x, y) = maxi |xi − yi|. For every s1, s2,. . . , sk ∈ Sn, the image of δs1 δs2 · · · δsk : ∆n → ∆n has diameter 6 (n−1

n )k.

The proof of the Excision Theorem follows from the following result, appliedto U = int(A) and V = X − cl(B).

Proposition. Let U and V be two open subsets of X such that X = U ∪ V . LetCUVn (X) ⊂ Cn(X) be the submodule generated by all singular simplices σ: ∆n →X whose image is, either completely contained in U , or completely contained inV . Let HUV

n (X) be the homology module of the chain complex formed by theCUVn (X). Then, the homomorphism HUV

n (X) → Hn(X) induced by the inclusionmap CUVn (X)→ Cn(X) is an isomorphism.

3. Application: homology modules of spheres

As an application of the Excision Theorem, consider the upper and lower hemi-spheres Bn± of the sphere Sn.

Lemma. Hp(Sn, Bn−) ∼= Hp(B

n+, S

n−1).

Combining this with the isomorphismsHp(Bn, Sn−1) ∼= Hp−1(S−1) andHp(S

n, Bn−) ∼=Hp(S

n) provided by the Long Exact Sequence in Relative Homology gives the fol-lowing computation.

Theorem.

Hp(Sn) ∼=

0 if p 6= n, 0

R if p = n, 0 and n > 0

R⊕R if p = n = 0

Corollary. There is no retraction r : Bn → Sn−1.

3. APPLICATION: HOMOLOGY MODULES OF SPHERES 39

Corollary. (Brouwer Fixed Point Theorem) Every continuous map f : Bn → Bn

has a fixed point, namely a point x ∈ X with f(x) = x.

A n–dimensional topological manifold is a space M such that every x ∈ Madmits a neighborhood that is homeomorphic to an open subset of Rn. For instance,an open subset of Rn is an n–dimensional topological manifold.

lem. If M is an n–dimensional topological manifold and if x ∈M ,

Hp(M,M − x) ∼=

0 if p 6= n

0 if p = n

CHAPTER 10

The Mayer-Vietoris exact sequence

1. The Mayer-Vietoris long exact sequence

Theorem. (Mayer-Vietoris Exact Sequence) For X1, X2 ⊂ X, consider the inclu-sion maps i1 : X1 ∩ X2 → X1, i2 : X1 ∩ X2 → X2, j1 : X1 → X1 ∪ X2, j2 : X2 →X1 ∪X2, and suppose that the inclusion map (X1, X1 ∩X2)→ (X1 ∪X2, X2) is anexcision. Then there exists a long exact sequence

δn+2

qqHn+1(X1 ∩X2)

fn+1

// Hn+1(X1)⊕Hn+1(X2)gn+1

// Hn+1(X1 ∪X2)

δn+1

qqHn(X1 ∩X2)

fn

// Hn(X1)⊕Hn(X2)gn // Hn(X1 ∪X2)

δn

qqHn−1(X1 ∩X2)

fn−1// Hn−1(X1)⊕Hn−1(X2)

gn−1// Hn−1(X1 ∪X2)

δn−1

qq

where fn = Hn(i1)⊕Hn(i2), gn = Hn(j1)−Hn(j2) and δn is a connecting homo-morphism provided by the Snake Lemma.

The key step in the proof is the following result.

Lemma. Let CX1X2n (X1 ∪ X2) = Cn(X1) + Cn(X2) ⊂ Cn(X) consist of linear

combinations of n–simplicies that are, either completely contained in X1, or com-pletely contained in X2. Let HX1X2

n (X1 ∪ X2) be the homology module of thecorresponding chain complex. If the inclusion map (X1, X1 ∩X2)→ (X1 ∪X2, X2)is an excision, the homomorphism HX1X2

n (X1 ∪ X2) → Hn(X1 ∪ X2) induced bythe inclusion CX1X2

n (X1 ∪X2) ⊂ Cn(X) is an isomorphism.

Complement. The connecting homomorphism δn : Hn(X1∪X2)→ Hn−1(X1∩X2)is defined as follows. By the Lemma, each homology class [c] ∈ Hn(X1 ∪X2) canbe written as [c] = [c1 + c2] ∈ Hn(X1 ∪ X2) with c1 ∈ Cn(X1) and c2 ∈ Cn(X2).Then δn

([c1 + c2]

)= [∂c1] = [−∂c2] ∈ Hn−1(X1 ∩X2).

41

42 10. THE MAYER-VIETORIS EXACT SEQUENCE

2. Some computations

Proposition. Let Tn = S1 × S1 × · · · × S1 be the n–dimensional torus. Then

Hp(Tn) ∼= R(n

p)

where(np

)is the binomial coefficient.

Proposition. Let τ : Sn → Sn be the reflection across 0×Rn defined by τ(x0, x1, . . . , xn) =(−x0, x1, . . . , xn). Then Hn(τ) acts by multiplication by −1 on Hn(Sn) ∼= R.

Proposition. Let K be the Klein bottle. In the coefficient ring R, let ψ2 : R→ Rbe the multiplication by 2 defined by ψ2(a) = 2a. Then,

Hn(K) ∼=

kerψ2 if n = 2

R⊕(R/imψ2

)if n = 1

R if n = 0

0 if n 6= 0, 1, 2.

For instance, H1(K;Z) ∼= Z⊕Z2 and H2(K;Z) = 0, H1(K;R) ∼= R and H2(K;R) =0, and H1(K;Z2) ∼= Z2 ⊕ Z2 and H2(K;Z2) = Z2.

CHAPTER 11

Homology of manifolds

1. Manifolds

Definition. A (topological) manifold of dimension n is a topological space X suchthat every x ∈ X admits a neighborhood Ux which is homeomorphic to an opensubset Vx of Rn.

In particular, there exists a family A = (Ui, ϕi)i∈I where:

1. each Ui is open in X and X =⋃i∈I

Ui

2. each ϕi: Ui → Vi is a homeomorphism from Ui to an open subset Vi ⊂ Rn.

Such a family A is a topological atlas for X.

Proposition. If X is an n–dimensional manifold, Hn(X,X − x) ∼= R for everyx ∈ X.

Proof. If y ∈ Vx ⊂ Rn corresponds to x,

Hn(X,X − x) ∼= Hn(Ux, Ux − x) ∼= Hn(Vx, Vx − y)∼= Hn(Rn,Rn − y) ∼= Hn−1(Sn−1) ∼= R

by the Excision Theorem (twice) and the Long Exact Sequence in Relative Homol-ogy.

2. Orientation-preserving homeomorphisms

Definition. A homeomorphism ϕ: U → V between two open subsets U , V ⊂ Rnis R–orientation preserving at x ∈ U if the composition

Hn−1(Sn−1) ∼= Hn(Rn,Rn − x) ∼= Hn(U,U − y)Hn(ϕ)−→ Hn(V, V − ϕ(x)) ∼= Hn(Rn,Rn − ϕ(x)) ∼= Hn−1(Sn−1)

is the identity map of Hn−1(Sn−1). The homeomorphism ϕ is R–orientation pre-serving if it is R–orientation preserving at each x ∈ U .

Examples.

1. If R = Z2, every homeomorphism ϕ: U → V is Z2–orientation preserving(since the identity is the only isomorphism of Hn−1(Sn−1) = Z2).

43

44 11. HOMOLOGY OF MANIFOLDS

2. Let ϕ: U → V be the restriction of a linear map Rn → Rn. If there exists ana ∈ R with 2a 6= 0 (for instance R = Z, R or Z3), then ϕ is R–orientationpreserving at x ∈ U if and only if det(ϕ) > 0.

3. Let ϕ : U → V be differentiable at x ∈ U , and assume in addition that

the jacobian matrix Txϕ =(∂ϕj(x)

∂xi

)is invertible. If, again, there exists an

a ∈ R with 2a 6= 0, then ϕ is R–orientation preserving at x ∈ U if and only

if det(∂ϕj(x)

∂xi

)> 0.

Proposition. Let ϕ : U → V be a homeomorphism between connected opensubsets U , V ⊂ Rn. Then ϕ is R–orientation preserving at some x ∈ U if and onlyif it is R–orientation preserving at every x ∈ U .

3. Oriented manifolds

Definition. An atlas A = (Ui, ϕi)i∈I for a manifold X is R–oriented if, forevery i, j ∈ I, the coordinate change homeomoprhism

ϕj ϕ−1i : ϕi(Ui ∩ Uj)→ ϕj(Ui ∩ Uj)

is R–orientation preserving.

Definition. The manifold X is R–orientable if it admits an R–oriented atlas. AnR–orientation for X is an equivalence class of atlases of R–oriented atlases, for theequivalence relation ∼ defined by the property that A ∼ A′ if and only if A∪A′ isR–oriented.

In particular, every manifold X is Z2–orientable, and admits a unique Z2–orientation.

Lemma. A connected manifold X admits exactly zero or two Z–orientations.

If you know what this words mean, you will notice that a differentiable manifoldis Z–orientable if and only if it is orientable in the differentiable sense.

4. Homology of manifolds

Thm. If X is a manifold of dimension n, then Hp(X) = 0 for every p > n.

We now consider the n–dimensional homology. Suppose that X is oriented,with oriented atlas (Ui, ϕi)i∈I . For every x ∈ X, we now have a composition

Hn