ALGEBRAIC TOPOLOGY - uni-regensburg.de · ALGEBRAIC TOPOLOGY 3 16.2. The orientation cover of a non-orientable manifold 265 17. Complex manifolds 268 18. Hyperbolic geometry 274 18.1.

ALGEBRAIC TOPOLOGY

STEFAN FRIEDL

Contents

References 4Initial remarks 71. Topological spaces 81.1. The definition of a topological space 81.2. Constructions of more topological spaces 161.3. Further examples of topological spaces 201.4. The two notions of connected topological spaces 231.5. Local properties 261.6. Graphs and topological realizations of graphs 281.7. The basis of a topology 311.8. Manifolds 331.9. The classification of 1-dimensional manifolds 361.10. Orientations of manifolds 402. Differential topology 432.1. The Tubular Neighborhood Theorem 432.2. The connected sum operation 472.3. Knots and their complements 483. How can we show that two topological spaces are not homeomorphic? 534. The fundamental group 574.1. Homotopy classes of paths 574.2. The fundamental group of a pointed topological space 635. Categories and functors 705.1. Definition and examples of categories 705.2. Functors 725.3. The fundamental group as functor 746. Fundamental groups and coverings 796.1. The cardinality of sets 796.2. Covering spaces 816.3. The lifting of paths 946.4. The lifting of homotopies 966.5. Group actions and fundamental groups 1026.6. The fundamental group of the product of two topological spaces 108

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2 STEFAN FRIEDL

6.7. Applications: the Fundamental Theorem of Algebra and the Borsuk-UlamTheorem 111

7. Homotopy equivalent topological spaces 1147.1. Homotopic maps 1147.2. The fundamental groups of homotopy equivalent topological spaces 1167.3. The wedge of two topological spaces 1218. Basics of group theory 1288.1. Free abelian groups and the finitely generated abelian groups 1288.2. The free product of groups 1348.3. An alternative definition of the free product of groups 1419. The Seifert-van Kampen theorem I 1449.1. The Seifert–van Kampen theorem I 1449.2. Proof of the Seifert-van Kampen Theorem 9.2 1519.3. More examples: surfaces and the connected sum of manifolds 15610. Presentations of groups and amalgamated products 16210.1. Basic definitions in group theory 16210.2. Presentation of groups 16310.3. The abelianization of a group 16810.4. The amalgamated product of groups 17211. The general Seifert-van Kampen Theorem 17711.1. The formulation of the general Seifert-van Kampen Theorem 17711.2. The fundamental groups of surfaces 17911.3. Non-orientable surfaces 18511.4. The classification of closed 2-dimensional (topological) manifolds 18811.5. The classification of 2-dimensional (topological) manifolds with boundary 18911.6. Retractions onto boundary components of 2-dimensional manifolds 19412. Examples: knots and mapping tori 19812.1. An excursion into knot theory (∗) 19812.2. Mapping tori 20313. Limits 21113.1. Preordered and directed sets 21113.2. The direct limit of a direct system 21213.3. Gluing formula for fundamental groups and HNN-extensions (∗) 22413.4. The inverse limit of an inverse system 22913.5. The profinite completion of a group (∗) 23714. Decision problems 23915. The universal cover of topological spaces 24215.1. Local properties of topological spaces 24215.2. Lifting maps to coverings 24315.3. Existence of covering spaces 24816. Covering spaces and manifolds 26116.1. Covering spaces of manifolds 261

ALGEBRAIC TOPOLOGY 3

16.2. The orientation cover of a non-orientable manifold 26517. Complex manifolds 26818. Hyperbolic geometry 27418.1. Hyperbolic space 27418.2. Angles in Riemannian manifolds 28018.3. The distance metric of a Riemannian manifold 28118.4. The hyperbolic distance function 28518.5. Complete metric spaces 28719. The universal cover of surfaces 28919.1. Hyperbolic surfaces 28919.2. More hyperbolic structures on the surfaces of genus g ≥ 2 (∗) 29319.3. More examples of hyperbolic surfaces 29519.4. The universal cover of surfaces 30019.5. Proof of Theorem 19.9 I 30119.6. Proof of Theorem 19.9 II 30519.7. Picard’s Theorem 30820. The deck transformation group (∗) 31221. Related constructions in algebraic geometry and Galois theory (∗) 32321.1. The fundamental group of an algebraic variety (∗) 32321.2. Galois theory (∗) 325

4 STEFAN FRIEDL

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409–448.[Bo] R. Bott, The stable homotopy of the classical groups, Ann. of Math. 70 (1959), 313–337.[Br] G. Bredon, Geometry and Topology, Sprinter Verlag (1993)[BZH] G. Burde, H. Zieschang and M. Heusener, Knots, 3rd fully revised and extended edition. De Gruyter

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279–315.[FW] G. Francis and J. Weeks, Conway’s ZIP Proof, Amer. Math. Monthly 106 (1999), 393–399.[Fr] M. Freedman, The topology of four-dimensional manifolds, J. Diff. Geom. 17 (1982), 357–453.[FJ] M. Fried and M. Jarden, Field arithmetic. Revised by Moshe Jarden. 3rd revised ed. Ergebnisse der

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(1989), no. 2, 371–415.[Gv] M. Gromov, Hyperbolic groups, in Essays in group theory, volume 8 of Math. Sci. Res. Inst. Publ.,

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6 STEFAN FRIEDL

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(1983)


Initial remarks

These are the lecture notes for the course Algebraic Topology I that I taught at theUniversity of Regensburg in the winter term 2016/2017. This course builds on the coursesAnalysis I-IV that I had taught in the previous terms. For the most part I only assumestandard results in general topology from the earlier courses. One unusual feature is that Iuse the homotopy invariance of path integrals of holomorphic functions, that I had provedin Analysis III, to quickly show that the fundamental group of S1 is non-trivial.

These course notes are meant to be “open source lecture notes”, i.e. they can be usedand modified by anybody. The tex-files and the files for the figures, which were producedwith winfig, can be found at

http://www.uni-regensburg.de/Fakultaeten/nat_Fak_I/friedl/

http://www.uni-regensburg.de/Fakultaeten/nat_Fak_I/friedl/

8 STEFAN FRIEDL

1. Topological spaces

1.1. The definition of a topological space. We recall the definition of a topologicalspace from Analysis IV.

Definition. A topological space is a pair (X, T ), where X is a set and T is a topology on X,i.e. T is a set of subsets of X with the following properties:

(1) the empty set and the entire set X are contained in T ,(2) the intersection of finitely many sets in T is again a set in T ,(3) the union of arbitrarily many sets in T is again a set in T .

The sets in T are called open.Example.

(1) Let (X, d) be a metric space. A subset U of X is called open if for every x ∈ Uthere exists an ϵ > 0 such that Bϵ(x) := {y ∈ X | d(x, y) < ϵ} is contained in U .We had already seen in Analysis II that

T := all open subsets of (X, d)is a topology on X. In the following we consider Rn as a metric space with theeuclidean metric and we always view Rn with the resulting topology, unless we sayexplicitly otherwise.

(2) Let X be a set, then T = {∅, X} is a topology on X. This topology is sometimescalled the trivial topology on X.

(3) Let X be a set and let T be the power set of X, i.e. the set of all subsets of X.Then T is also a topology on X. Put differently, T is the topology such that allsubsets are open. This topology is usually referred to as the discrete topology on X.

(4) Let X = R and let T be defined as follows:U ∈ T :⇐⇒ either U = ∅ or U is the complement of finitely many points in R.

For example the sets ∅,R\{π},R\{−1,√2} and also R (since it is the complement

of zero points) lie in T . It follows easily from elementary set theory that T is atopology on X = R.

(5) We consider the setX := Rn ∪ {∞},

i.e. X consists of Rn and an extra point ∞. We say U ⊂ X is open1, if both of thefollowing two conditions are satisfied:(a) for each point x ∈ U ∩ Rn there exists an ϵ > 0 such that Bϵ(x) ⊂ U ,(b) if ∞ ∈ U , then there exists a C > 0 such that {x ∈ Rn | ∥x∥ > C} ⊂ U .It is straightforward to see that this defines indeed a topology on X. For n = 1 wehad introduced this topological space in Analysis IV and we had referred to it asthe “line with a point at infinity”. We now refer to Rn ∪ {∞} as “Rn with a pointat infinity”.

1If we want to specify a topology, it suffices to specify which subsets are called “open”.


(6) We consider the set

X := R ∪ {∗},i.e. X consists of R and an extra point ∗. We say U ⊂ X is open, if the followingtwo conditions are satisfied:(a) for each point x ∈ U ∩ R there exists an ϵ > 0 such that (x− ϵ, x+ ϵ) ⊂ U ,(b) if ∗ ∈ U , then there exists an ϵ > 0 such that (−ϵ, 0) ∪ (0, ϵ) ⊂ U .We had seen in Analysis IV that this is indeed a topology on X. We refer to thistopological space as the “line with two zeros”.

(7) If (X, T ) is a topological space and if Y ⊂ X is a subset, then

S := {Y ∩ U |U ∈ T }

is a topology on Y . We refer to S as the subspace topology on Y . Unless we saysomething else we consider each subset Y of Rn always as a topological space withrespect to the subspace topology.

Now we recall several definitions from Analysis IV.

Definition. Let X be a topological space.2

(1) Let A ⊂ X be a subset. We say U ⊂ X is a neighborhood of A if there exists anopen set V such that A ⊂ V ⊂ U . We say U is an open neighborhood of A, if U isfurthermore open.

(2) We say X is Hausdorff, if given any two points x ̸= y there exist disjoint openneighborhoods U of x and V of y.

Example.(1) If X = R and A = [0, 2), then U = (−1, 3] and V = (−2,∞) are neighborhoods of

A in X.(2) We had already seen in Analysis II Proposition 1.8 that metric spaces are Hausdorff.

Furthermore we had seen in Analysis IV that the line with a point at infinity isalso Hausdorff and the same argument shows that Rn with a point at infinity isHausdorff. On the other hand we had seen in Analysis IV that the line with twozeros is not Hausdorff.

(3) A straightforward exercise shows that a topological space X is Hausdorff if and onlyif the diagonal D = {(x, x) |x ∈ X} is a closed subset of X ×X.

Definition. Let X be a topological space and let A be a subset of X.

(1) The interior◦A is defined as the union of all open sets of X that are contained in A.

(2) We say A is closed, if X \ A is open.(3) The closure A of A is defined as the intersection of all closed sets in X that contain A.

(4) The boundary of A in X is defined as ∂A := A \◦A.

2As usual we suppress the topology from the notation, i.e. we write “let X be a topological space”instead of the more precise “let (X, T ) be a topological space”.

10 STEFAN FRIEDL

Example. We consider X = R and A is the half-open interval [−1, 2). Then the interior of Ais the open interval (−1, 2) and the closure of A is the closed interval [−1, 2]. Furthermore∂A = {−1, 2}.

It follows immediately from the axioms of a topology that the interior of a set is an openset. Furthermore it is straightforward to see that the union of finitely many closed sets isagain closed and that the intersection of arbitrarily many closed sets is again closed. Itfollows easily that the closure of a subset is closed.

Definition. Let X be a topological space. An open covering of X is a family {Ui}i∈I ofopen subsets of X with

X =∪i∈I

Ui.

We say a topological space X is compact if for each open covering {Ui}i∈I of X there existfinitely many indices i1, . . . , ik ∈ I such that

X = Ui1 ∪ · · · ∪ Uik.Examples. The Heine–Borel Theorem says that a subset A of Rn is compact if and only ifit is bounded and closed.

We recall the following well-known lemma.

Lemma 1.1. Let X be a topological space and let A ⊂ X be a compact subset. If X isHausdorff, then A is a closed subset of X.

For completeness’ sake we provide the proof.

Proof. Let X be a Hausdorff space and let A ⊂ X be a compact subset. We want to showthat X \ A is open. It suffices to prove the following claim.

Claim. Let x ∈ X \ A. Then there exists an open neighborhood V of x that is containedin X \ A.

We apply the Hausdorff-property to x and every y ∈ A. For every y ∈ A we obtaindisjoint open neighborhoods e Uy of y and Vy of x. Evidently we have

A =∪y∈A{y} ⊂

∪y∈A

(Uy ∩ A) ⊂ A.

Thus we see that {Uy ∩ A}y∈A is an open covering of A. Since A is compact there existy1, . . . , yk such that

A =k∪i=1

(Uyi ∩ A).

Now we consider

V :=k∩i=1

Vyi .


Since V is the intersection of finitely many open sets, it is open itself. Furthermore V doesnot intersect any of the Uyi , i = 1, . . . , k. Hence it V is disjoint from von A ⊂ Uy1∪· · ·∪Uyk .This concludes the proof of the claim. �Definition. We say a map f : X → Y between two topological spacesX and Y is continuous,if for each open set U in Y the preimage f−1(U) is open in X.

It is straightforward to see that the composition of two continuous maps is again con-tinuous. For maps between metric spaces we obtain the same notion of continuity as inAnalysis II.The following lemma states perhaps the most important feature of compact sets. We

had proved the lemma in Analysis II for metric spaces, the proof for topological spaces isverbatim the same.

Lemma 1.2.

(1) Let f : X → Y be a continuous map. If X is compact, then f(X) is also compact.(2) Let f : X → R be a continuous map. If X is compact, then f assumes its maximum

and its minimum.

Definition. We say a map f : X → Y between two topological spaces X and Y is a homeo-morphism if the following three properties are satisfied:

(1) f is continuous,(2) f is bijective,(3) f−1 : Y → X is also continuous.

If there exists a homeomorphism between X and Y we say that X and Y are homeomorphicand sometimes we write X ∼= Y .

The following proposition, that we had proved in Analysis IV, gives an often usefulcriterion for showing that a map is a homeomorphism.

Proposition 1.3. Let f : X → Y be a bijective continuous map between topological spaces.If X is compact and if Y is Hausdorff, then f is a homeomorphism.

Example. We consider the map

Φ: Sn → Rn ∪ {∞}

(x1, . . . , xn+1) 7→

{ (x1

1− xn+1, . . . ,

xn1− xn+1

), if xn+1 < 1,

∞, if xn+1 = 1.where we equip Rn∪{∞} with the topology that we had introduced on page 8. Outside ofthe “North pole” (0, . . . , 0, 1) this map is just the stereographic projection that is illustratedin Figure 1. This map is easily seen to be continuous3 and a bijection. Furthermore Sn

is compact by Heine-Borel and Rn ∪ {∞} is Hausdorff, as we had just pointed out above.Hence it follows from Proposition 1.3 that Φ is a homeomorphism.

3Is that really so easy?

12 STEFAN FRIEDL

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stereographic projection of P

North pole N = (0, 0, 1)

Pray emanating from N through P

Figure 1. Stereographic projection from S2 \ {(0, 0, 1)} onto R2.

Remark. If two topological spaces are homeomorphic, then they have the same topologicalproperties, i.e. they share all properties that are defined purely in terms of the topology.For example, if X and Y are homeomorphic, then X is Hausdorff if and only if Y isHausdorff, X is compact if and only if Y is compact and so on.

Convention. Henceforth any map between two topological spaces is assumed to be contin-uous, unless we say explicitly otherwise.

Definition.

(1) We say that a subset A ⊂ Rn is convex, if for any two distinct points P and Q in Athe segment PQ := {tP + (1− t)Q | t ∈ [0, 1]} also lies in A.

(2) Given a subset S of Rk the convex hull of S is defined as the intersection of allconvex subsets of Rk that contain S. Since the intersection of convex sets is againconvex we see that the convex hull of S is a convex subset of Rk.

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convex hull of Snot convex

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Q

subset S of R2convex subset of R2

Figure 2.

The following lemma gives a useful criterion for showing that subsets of Rn are homeo-morphic to an open or to a closed ball.

Lemma 1.4.

(1) Let A be a bounded open convex subset of Rn, then A is homeomorphic to the openn-dimensional ball Bn := {x ∈ Rn | ∥x∥ < 1}.

(2) Let A be a bounded closed convex subset of Rn such that the interior of A is non-empty. Then it follows that A is homeomorphic to the closed n-dimensional ballBn= {x ∈ Rn | ∥x∥ ≤ 1}. More precisely there exists a homeomorphism f : A→ Bn

with Φ(∂A) = Sn−1.


Examples.

(1) It follows from Lemma 1.4 that the open cube (0, 1)n is homeomorphic to Bn. Moregenerally, it follows from Lemma 1.4 that for any r, s ∈ N0 the product of ballsBr ×Bs ⊂ Rr × Rs = Rr+s is homeomorphic to Br+s.

(2) It follows from Lemma 1.4 that any triangle, i.e. any subset of R2 of the formA = {P + sv + tw | s, t ∈ [0, 1] and s + t ≤ 1} where P ∈ R2 and v, w are twolinearly independent vectors, is homeomorphic to B

2.

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(0, 1)2 B2 B2

is homeomorphic to is homeomorphic to

Figure 3.

Proof.

(2) Let A be a bounded closed convex subset of Rn such that the interior of A is non-empty. After a translation we can assume that 0 lies in the interior of A. Givenx ∈ A \ {0} we define

ρ(x) := sup{∥rx∥

∣∣ r ∈ R>0 and rx ∈ A}and

f(x) := x · ρ(x)∥x∥ .

Since A is closed we have f(x) ∈ A.Claim. The map ρ : A→ R is positive, bounded and continuous.Since A is bounded it follows that ρ is bounded. It follows from the definition of

ρ that ρ(x) is always positive. Therefore it remains to show that ρ is continuous.Now let x ∈ A and let ϵ > 0. It suffices to show the following two statements:(a) there exists an open neighborhood U of x such that ρ(y) > ρ(x) − ϵ for all

y ∈ U ,(b) there exists an open neighborhood V of x such that ρ(y) < ρ(x) + ϵ for all

y ∈ V .We first show the existence of U . After possibly replacing ϵ by min{1

2ρ(x), ϵ} we

can suppose that ϵ ∈ (0, ρ(x)).Since 0 lies in the interior of A there exists an η > 0 such that Bη(0) ⊂ A. Since A

is convex, the convex hull C of Bη(0)∪{f(x)} is still contained4 in A. Furthermore,since Bη(0) is open it is straightforward to see that C

′ := C \ {f(x)} is an open4Why is that the case?

14 STEFAN FRIEDL

subset of Rn. We denote by Sn−1ρ(x)−ϵ the sphere of radius ρ(x) − ϵ around 0. Thepoint x · ρ(x)−ϵ∥x∥ lies on S

n−1ρ(x)−ϵ and it lies in C

′. Since C ′ is open there exists an open

neighborhood U ′ on Sn−1ρ(x)−ϵ that is contained in C′. We set

U := {rz | z ∈ U ′ and r ∈ (0, 1)}.This is an open neighborhood of x and for any y ∈ U we have (ρ(x) − ϵ) · y ∈ A,i.e. for any y ∈ A we have ρ(y) ≥ ρ(x) − ϵ. Thus we have shown that the desiredneighborhood U exists. The existence of V is proved in a very similar way. We referto [Be, Chapter 11.3] for full details. This concludes the proof of the claim.

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x

f(x)

Bη(0)

0A A

0

Bη(0)

f(x)

C := convex hull of Bη(0) and f(x)

circle with radius ρ(x) around 0

circle S1ρ(x)−ϵ with radius ρ(x)− ϵ around 0

Figure 4. Illustration for the proof of Lemma 1.4.

We now consider the map

Φ: A → Bn

x 7→

{x · 1ρ(x) , if x ̸= 0,0, if x = 0.

It is straightforward to verify that this map is bijective5 and it follows from theabove claim6 that Φ is continuous. It now follows from Proposition 1.3 that Φ is ahomeomorphism. We leave it as an exercise to verify that Φ(∂A) = Sn−1.

(1) The first claim follows from (2) by applying (2) to the closure A7 and by restrictingthe resulting homeomorphism Φ: A→ Bn to the interior. �

We recall two definitions about metric spaces.

5It follows easily from the convexity of A, the hypothesis that◦A ̸= ∅ and the definition of Φ that for

each v ∈ Sn−1 the map Φ restricts to a bijection on the “ray defined by v”, i.e. it restricts to a bijectionA ∩ {rv | r ∈ R>0} → B

n ∩ {rv | r ∈ R>0}.

6From the continuity of δ it follows that Φ is continuous on A \ {0}. From the fact that δ is bounded itfollows easily that Φ is continuous in 0.

7Why is the closure of a convex set again a convex set?


(1) Given a subset A of a metric space X the diameter is defined as

diameter(A) := sup{d(a, b) | a, b ∈ A} ∈ R≥0 ∪ {∞}.(2) For any metric space X, any x ∈ X and any non-empty subset A of X we refer to

d(x,A) := inf{d(x, a) | a ∈ A} ∈ R≥0as the distance from x to A.

For example the diameter of the cube [0, 1]n ⊂ Rn is easily seen to be the n-th root of 2.The following Lemma of Lebesgue was already stated and proved in Analysis II. As a

warm-up exercise we prove it again.

Lemma 1.5. (Lebesgue’s Lemma) Let K be a compact metric space and let {Ui}i∈Ibe an open covering of K. Then there exists a δ > 0 such that for every subset A withdiameter(A) < δ there exists an i ∈ I with A ⊂ Ui.

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Axx

Um

Bδ(x) ⊂ X \ UmUi

Br(x) ⊂ UiK

Figure 5. Illustration of the proof of Lebesgue’s Lemma.

Proof. Since K is compact we can cover K with finitely many of the Ui’s. Put differently,without loss of generality we can assume that I = {1, . . . , n} is a finite set.If there exists an i ∈ {1, . . . , n} with K = Ui, then any δ > 0 has the desired property.

Now suppose that this is not the case, i.e. we suppose that for any i we have Ui ( K.8 Wefirst prove the following claim.

Claim. The functionf : K → R

x 7→ f(x) := 1nn∑i=1

d(x,K \ Ui)

is continuous and positive.

It follows easily from the definitions and the triangle inequality that f is continuous.Now let x ∈ K. We want to show that f(x) > 0. Since K = U1 ∪ · · · ∪ Un there existsan i with x ∈ Ui. Since Ui is open there exists an r > 0 with Br(x) ⊂ Ui. It follows thatd(x,K \ Ui) ≥ r, hence f(x) ≥ rn . This concludes the proof of the claim.Since f is continuous and since K is compact it follows from Lemma 1.2 that the function

f has a global minimum δ on K. It follows from the claim that this minimum δ is greaterthan 0. Now we want to show that this δ has the desired property. More precisely, we wantto prove the following claim.

8Where do we use in the subsequent argument that Ui ( K?

16 STEFAN FRIEDL

Claim. Let A be a subset of K with diameter(A) < δ. Then there exists an i ∈ {1, . . . , n}with A ⊂ Ui.Let x ∈ A. We choose an m ∈ {1, . . . , n} so that d(x,K \ Um) is maximal. We want to

show that A ⊂ Um. Since the diameter of A is less than δ we have A ⊂ Bδ(x). Thus itsuffices to show that Bδ(x) ⊂ Um. Put differently, it we need to show that d(x,X \Um) ≥ δ.Indeed we have

d(x,K \ Um) ≥1

n

n∑i=1

d(x,K \ Ui) = f(x) ≥ δ.↑ ↑

since d(x,K \ Um) ≥ d(x,K \ Ui) by the choice of δ

This concludes the proof of the claim. �In many cases the following corollary is even more useful.

Corollary 1.6. Let f : [0, 1]n → X be a map from the cube [0, 1]n to a topological space Xand let {Vi}i∈I be an open covering of X. Then there exists an N > 0 such that for anya1, . . . , an ∈ {0, . . . , N − 1} there exists an i ∈ I such that

f([a1N ,

a1+1N

]× · · · ×

[anN ,

an+1N

])⊂ Vi.

The corollary thus says that if f : [0, 1]n → X is a map, and if we are given an opencovering of X, then we can always find a small grid on the cube, such that each cube ofthe grid gets mapped into one of the open sets covering X.

Proof. Let f : [0, 1]n → X be a map from the cube [0, 1]n to a topological space and let{Vi}i∈I be an open covering of X. We apply Lemma 1.5 to the open covering Ui := f−1(Vi),i ∈ I of K and we obtain a δ > 0 such that for each subset A with diameter(A) < δ thereexists an i ∈ I with A ⊂ Ui = f−1(Vi), which means that f(A) ⊂ Vi. The corollary followsfrom the observation that for sufficiently large N any cube of side length 1

Nhas diameter

less than δ. �1.2. Constructions of more topological spaces. In this section we will recall severalways to build new examples of topological spaces out of existing topological spaces.

Definition. Let X1, . . . , Xk be topological spaces. We say U ⊂ X1 × · · · ×Xk is open if foreach (x1, . . . , xk) ∈ U there exist open neighborhoods Ui of xi in Xi with U1× . . . Uk ⊂ U .In Analysis IV we had seen that the above definition does indeed define a topology on

X1× · · ·×Xk. We refer to this topology as the product topology on X1× · · ·×Xk. We hadseen in Analysis IV that the product X1 × · · · ×Xk is Hausdorff if and only if each Xi isHausdorff and that the product X1×· · ·×Xk is compact if and only if each Xi is compact.Example. The topological space Rk × Rl is easily seen to be homeomorphic to Rk+l. Moreprecisely, the obvious map

Rk × Rl → Rk+l((x1, . . . , xk), (y1, . . . , yl)) 7→ (x1, . . . , xk, y1, . . . , yl)


is a homeomorphism.

In the following we refer to S1×S1 as the torus and we refer to (S1)n as the n-dimensionaltorus. We had seen in Analysis IV that the torus is indeed homeomorphic to the “standardtorus” in R3.

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S1 × S1 is homeomorphic tosubset of R3

Figure 6.

Definition. Let ∼ be an equivalence relation on a topological space X. We denote byp : X → X/ ∼ the canonical projection map from X onto the set of equivalence classesX/ ∼. We say U ⊂ X/ ∼ is open if p−1(U) ⊂ X is open.

It is straightforward to verify that this does indeed define a topology on X/ ∼ and thatthe projection map p : X → X/ ∼ is continuous. We refer to this topology on X/ ∼ as thequotient topology and we refer to X/ ∼ as a quotient space of X. If X is compact, then itfollows from Lemma 1.2 that X/ ∼ is also compact.

Example. Let X be a topological space and let A ⊂ X be a subset. For P,Q ∈ X we define

P ∼ Q :⇐⇒ P = Q or P,Q both lie in A.

This is easily seen to be an equivalence relation on X. We write X/A := X/ ∼. InAnalysis IV we had for example seen that B2/S1 is homeomorphic to S2. In exercisesheet 1 we will see that Bn/Sn−1 is homeomorphic to Sn.

We also recall the following definition from Analysis IV.

Definition. Let X be a set and let G be a group with trivial element e.

(1) An action of G on X is a map

G×X → X(g, x) 7→ g · x

with the following properties

e · x = x, for all x ∈ X,g · (h · x) = (gh) · x, for all x ∈ X and g, h ∈ G.

(2) The action is called free, if g · x = x for some x ∈ X implies that g = e.(3) We say G acts transitively, if for any x and y in X there exists a g ∈ G with g ·x = y.

18 STEFAN FRIEDL

(4) If X is a topological space, then we say that the action is continuous, if for everyg ∈ G the map

X → Xx 7→ g · x

is continuous.9

It is straightforward to verify that if a group G acts on a set X, then

x ∼ y :⇐⇒ there exists a g ∈ G such that g · x = y

is an equivalence relation on X. We write

X/G := X/ ∼ .

If X is a topological space, then we view X/G as a topological space equipped with thequotient topology. On page 17 we had pointed out that the projection map X → X/G iscontinuous. In Lemma 18.9 of Analysis IV we had seen the projection map is also open.

Examples.

(1) The group G = Zn acts on X = Rn by addition. This action is evidently free andcontinuous. In Analysis IV we had seen that

Rn/Zn → (S1)n[(t1, . . . , tn)] 7→

(e2πit1 , . . . , e2πitn

)is a homeomorphism.

(2) Let X = R× (−1, 1) and G = Z. The map

Z× (R× [−1, 1]) → R× [−1, 1](n, (x, y)) 7→ (x+ n, (−1)ny)

defines an action that is free and continuous. The quotient is homeomorphic to theMöbius band.10

(3) Let X = Sn and G = {±1}. The map

{±1} × Sn → Sn(ϵ, P ) 7→ ϵ · P

gledefines an action that is free and continuous. We refer to the quotient space Sn/{±1}as the n-dimensional real projective space RPn.11 In exercise sheet 1 we will show thatRP1 is homeomorphic to S1.

9If the action is continuous, then the map x 7→ g · x is in fact a homeomorphism with inverse map givenby x 7→ g−1 · x.

10Depending on the point of view, this can of course also be taken as the definition of the Möbius band.11For n = 2 we sometimes refer to RP2 as the real projective plane.


(4) Let X = Rn+1\{0} and let G = R\{0}, where we view G as a group via multiplication.The map

(R \ {0})× (Rn+1 \ {0}) → Rn+1 \ {0}(r, P ) 7→ r · P

defines an action that is free and continuous. It is straightforward to see that the map

RPn = Sn/{±1} → (Rn+1 \ {0})/(R \ {0})[P ] 7→ [P ]

defines a homeomorphism. Our new point of view on real projective spaces has theadvantage that we can create a different class of examples of topological spaces byreplacing R by C. More precisely, let X = Cn+1 \ {0} and let G = C \ {0}. The map

(C \ {0})× (Cn+1 \ {0}) → Cn+1 \ {0}(z, P ) 7→ z · P

defines an action that is free and continuous. The quotient space (Cn+1\{0})/(C\{0})is called the n-dimensional complex projective space CPn. In exercise sheet 1 we willshow that CP1 is homeomorphic to S2.12

(5) Let X = R and G = {±1}. The map{±1} × R → R

(ϵ, x) 7→ ϵ · xdefines an action that is continuous. But this action is not free, since (−1) · 0 = 0, but−1 is not the trivial element in the group G = {±1}. In Analysis IV we had shownthat R/{±1} is homeomorphic to the half-open interval [0,∞).

Finally we recall Lemma 17.4 from Analysis IV which is used so frequently, that later onwe will no longer cite it explicitly.

Lemma 1.7. Let ∼ be an equivalence relation on a topological space X and let f : X → Ybe a continuous map with the property that f(x) = f(y) whenever x ∼ y. Then there existsa unique continuous map g : X/ ∼→ Y , such that f = g ◦ p, i.e. such that the followingdiagram of maps commutes:

Xp //

f ''NNNNN

NNNNN X/ ∼

g��Y.

12The definition of projective space makes sense for any field, even for fields of non-zero characteristic.If F is a field of characteristic p we can still define FPn = (Fn+1 \ {0})/(F \ {0}). A priori this is just a set,but together with the Zariski topology it actually becomes an unusual, but interesting topological space.For any field the map

FPn = (Fn+1 \ {0})/(F \ {0}) → {all one-dimensional subspaces of Fn+1}[v] 7→ F · v

is a bijection. Thus we can view FPn as the set of all lines through the origin.

20 STEFAN FRIEDL

1.3. Further examples of topological spaces. In this section we will recall several otherexamples of topological spaces that we had already introduced in Analysis IV. Before wedo so, we recall the following definition: Suppose we are given a relation ∼ on a set X.13We say x, y ∈ X are equivalent if there exists a sequence x = x1, . . . , xk = y of elementsin X such that for all i = 1, . . . , k − 1 the following holds: either xi ∼ xi+1 or xi+1 ∼ xi.We then write again x ∼ y if x and y are equivalent. It is straightforward to see that thisis now indeed an equivalence relation. We say it is generated by the initial relation.

Example. The relations x ∼ (x + 1) with x ∈ R on R generate the familiar equivalencerelation x ∼ y ⇔ x− y ∈ Z.

We will now see that many familiar spaces can be described as quotient spaces.

(1) We consider X = [0, 1]× [0, 1] ⊂ R2 and the equivalence relation which is generated by

(x, 0) ∼ (x, 1) for all x ∈ [0, 1].

The quotient topological space X/ ∼ is obtained from the square X = [0, 1] × [0, 1]by identifying each point on the upper edge with the corresponding point on the loweredge. Put differently, “we glue the upper edge to the lower edge”. Using Proposition 1.3one can easily show that the map

X/ ∼ → S1 × [0, 1](s, t) 7→

(e2πt, s

)is a homeomorphism. We refer to X/ ∼ and also to S1 × [0, 1] as a cylinder or anannulus.

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ALGEBRAIC TOPOLOGY - uni-regensburg.de · ALGEBRAIC TOPOLOGY 3 16.2. The orientation cover of a non-orientable manifold 265 17. Complex manifolds 268 18. Hyperbolic geometry 274 18.1.

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