Manuel Kauers Hagenberg, March 3, 2003 RISC-Linz

Algebraic Topology

Summary to a Lecture of Günther Landsmann

Contents

1 Categories 2

2 Topology 3

2.1 The Category of Spaces . . . . . . . . . . . . . . . . . . . . . . . 3 2.1.1 Products and Coproducts . . . . . . . . . . . . . . . . . . 4 2.1.2 Internalization . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 The Category of Pairs . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3 The Category of Pointed Spaces . . . . . . . . . . . . . . . . . . . 6

3 Homotopy 7 3.1 Homotopic Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3.2 Homotopy Categories . . . . . . . . . . . . . . . . . . . . . . . . 8

3.2.1 The Category of Spaces . . . . . . . . . . . . . . . . . . . 8 3.2.2 Pairs and Pointed Spaces . . . . . . . . . . . . . . . . . . 8

3.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.3.1 Contractibility . . . . . . . . . . . . . . . . . . . . . . . . 9 3.3.2 Nontrivial Examples . . . . . . . . . . . . . . . . . . . . . 10 3.3.3 Consequences . . . . . . . . . . . . . . . . . . . . . . . . . 11

4 Functorial Constructions 12 4.1 Wedge Sum and Smash Product . . . . . . . . . . . . . . . . . . 12 4.2 Suspension and Loop space . . . . . . . . . . . . . . . . . . . . . 13

5 Homotopy Groups 15 5.1 Groups and H-Groups . . . . . . . . . . . . . . . . . . . . . . . . 15 5.2 𝔄-Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

6 The Theorem of Seifert-van Kampen 18 6.1 Free Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 6.2 Seifert-van Kampen’s Theorem . . . . . . . . . . . . . . . . . . . 19

1 Categories

For the presentation of Algebraic Topology which is planed to be given in this course, we want to use tools from category theory. This first section will provide the necessary background about categories.

Definition 1 Let ℭ consist of a class C of objects, of a set M(X, Y) = M_{ℭ}(X, Y) of morphisms for any X, Y ∈ C, and of a map ∘: M(X, Y) × M(Y, Z) → M(X, Z) for any three objects X, Y, Z. ℭ is called a category, if

1. for any X, Y, Z, U ∈ C and any f ∈ M(X, Y), g ∈ M(Y, Z), h ∈ M(Z, U), we have f ∘ (g ∘ h) = (f ∘ g) ∘ h, and

2. for any X ∈ C there is a morphism 1 = 1_{X} ∈ M(X, X) with the property 1_{X} ∘ f = f (f ∈ M(Y, X)) and g ∘ 1_{X} = g (g ∈ M(X, Z)).

Instead of X ∈ C we will also write X ∈ ℭ. Also, we will write ℭ = (C, M, ∘) to indicate that C be the class of objects, M(X, Y) be the set of morphisms from X to Y, and that ∘ be the composition function of morphisms.

In the present context, the class C will usually contain sets, and M(X, Y) will be a set of certain maps from X to Y. In this setting, when taking for ∘ the composition of maps, it is clear that C, M, and ∘ fulfill the two axioms and hence build a category. The easiest category we are interested in is 𝔖𝔢𝔱, the category of sets. Its objects are all sets (note that the objects need not belong a set, but only to a “class”), and M(X, Y) is the set of all maps from X to Y. In fact, many mathematical structurs can be viewed as categories: The category 𝔊𝔯𝔬𝔲𝔭 of groups (with homomorphisms as morphisms), the category 𝔙𝔢𝔠𝔖𝔭𝔞𝔠𝔢 of vector spaces (with linear maps as morphisms), and so on. The idea behind the notion of a category is that many similar questions that arise in different mathematical structures may be treated once and for all in a more abstract setting. For instance, the notion of isomorphism between objects can be defined on the level of categories.

Definition 2 Let ℭ be a category and X, Y be objects of ℭ. These objects are called isomorphic if there are morphisms f ∈ M(X, Y) and g ∈ M(Y, X) such that 1_{X} = g ∘ f and 1_{Y} = f ∘ g. Such f and g are called isomorphisms, and we write X ≅ Y if X and Y are isomorphic.

Clearly, X, Y ∈ 𝔖𝔢𝔱 are isomorphic iff they have the same cardinality, X, Y ∈ 𝔊𝔯𝔬𝔲𝔭 are isomorphic iff they are isomorphic in the usual sense, and so on. It is also easy to see that isomorphism is an equivalence relation on a category. In the next definition, we consider the relationship between two categories.

Definition 3 Let 𝔄, 𝔅 be categories.

1. A covariant functor F: 𝔄 → 𝔅 is a mapping that maps objects X ∈ 𝔄 to objects F (X) ∈ 𝔅 and morphisms m ∈ M_{𝔄}(X, Y) to morphisms F (m) ∈ M_{𝔅}(F (X), F (Y)) such that F (1_{X}) = 1_{F (X)} for all X ∈ 𝔄 and F (ψ ∘ φ) = F (ψ) ∘ F (φ) for all φ ∈ M_{𝔄}(X, Y), ψ ∈ M_{𝔄}(Y, Z).

2. A contravariant functor F: 𝔄 → 𝔅 is a mapping that maps objects X ∈ 𝔄 to objects F (X) ∈ 𝔅 and morphisms m ∈ M_{𝔄}(X, Y) to morphisms F (m) ∈ M_{𝔅}(F (Y), F (X)) such that F (1_{X}) = 1_{F (X)} for all X ∈ 𝔄 and F (ψ ∘ φ) = F (φ) ∘ F (ψ) for all φ ∈ M_{𝔄}(X, Y), ψ ∈ M_{𝔄}(Y, Z).

The difference between covariant and contravariant functor is that the contravariant functor also changes the direction of the morphisms, i.e., if φ is a morphism from X to Y, then F (φ) is a morphism from F (X) to F (Y) in case F is covariant, and F (φ) is a morphism from F (Y) to F (X) in case F is contravariant.

Example 1 The forgetful functor | · |: 𝔊𝔯𝔬𝔲𝔭 → 𝔖𝔢𝔱 maps a group G to its underlying set |G| and homomorphisms to the corresponding maps — it “forgets” the group structure. The forgetful functor can be analogously defined for other categories 𝔄 in place of 𝔊𝔯𝔬𝔲𝔭 as long as the objects of 𝔄 are sets with a certain structure.

The distinction between covariant and contravariant functors is not essential, but convenient. In fact, contravariant functors from a category 𝔄 to a category 𝔅 can also be viewed as covariant functors from 𝔄 to the dual category 𝔅^{∗} which is defined as follows.

Definition 4 Let ℭ be a category with objects in a class C, where M(X, Y) defines the morphisms from X to Y and ∘: M(X, Y) × M(Y, Z) → M(X, Z) the composition of morphisms. Further, let ℭ^{∗} be the category which consists of the same objects as ℭ does, but with M^{∗}(X, Y) := M(Y, X) as the set of morphisms and with f ∘^{∗} g := g ∘ f as composition function. Then ℭ^{∗} is called the dual category of ℭ.

We conclude this section by another example of a functor which will be used frequently in the sequel.

Example 2 Let ℭ = (C, M, ∘) be a category and A ∈ ℭ. The functors

hom_{ℭ,A}: ℭ → 𝔖𝔢𝔱,  hom_{ℭ,A}(X) := M_{ℭ}(A, X),
hom_{ℭ^{∗},A}: ℭ^{∗} → 𝔖𝔢𝔱, hom_{ℭ^{∗},A}(X) := M_{ℭ}(X, A)

are called the covariant hom functor and the contravariant hom functor of ℭ with respect to A, respetively. The “combination” of both,

hom := hom_{ℭ}: ℭ^{∗} × ℭ → 𝔖𝔢𝔱, hom(X, Y) := M_{ℭ}(X, Y)

is simply called the hom functor of ℭ.

2 Topology

2.1 The Category of Spaces

We turn to the objects of our interest, the topological spaces. By 𝔗𝔬𝔭 we denote the category of spaces. An object X ∈ 𝔗𝔬𝔭 is a topological space, that is, a set |X| together with a system T ⊆ P(|X|) of open sets. The morphisms φ ∈ M_{𝔗𝔬𝔭}(X, Y) are the continous functions form X to Y; we will write C(X, Y) instead of M_{𝔗𝔬𝔭}(X, Y).

2.1.1 Products and Coproducts

The category 𝔗𝔬𝔭 is closed under taking products and coproducts. If X_{i} (i ∈ I) are topological spaces then so are ∏_{i∈I} X_{i} and ∐_{i∈I} X_{i}. By π_{j} or π_{X_{j}} we will denote the projection from ∏_{i∈I} X_{i} to X_{j}. By i_{j} we denote the injection from X_{j} to ∐_{i∈I} X_{i}.

Theorem 1 Let X_{i} (i ∈ I) and Y be spaces.

1. Let f_{i} ∈ C(Y, X_{i}) for every i ∈ I. Then there exists a unique morphism h ∈ C(Y, ∏_{i∈I} X_{i}) such that f_{j} = h ∘ π_{j} for every j ∈ I.

2. Let f_{i} ∈ C(X_{i}, Y) for every i ∈ X_{i}. Then there exists a unique morphism h ∈ C(∐_{i∈I} X_{i}, Y) such that f_{j} = i_{j} ∘ h for every j ∈ I.

The situations are illustrated in the following diagrams.

X_{1}  X_{1}

[figure: f_{1}; π_{1}; ⋮; ⋮; f_{1}; i_{1}]

[figure: Y; h; ∏_{i∈I} X_{i}; ⋮; ⋮; ∐_{i∈I} X_{i}; h; Y]

[figure: f_{n}; π_{n}; ⋮; ⋮; i_{n}; f_{n}]

X_{n}  X_{n}

Theorem 1 might be taken as the defining property of product and coproduct in the following sense. The product topology ∏_{i∈I} X_{i} of X_{i} (i ∈ I) consists of the set theoretic cartesian product of the sets |X_{i}| (i ∈ I) equipped with the initial topology that makes the left diagram commute. Analogously, the coproduct ∐_{i∈I} X_{i} consists of the set

| ∐_{i∈I} X_{i}| = ⋃_{i∈I}(X_{i} × {i})

equipped with the final topology that makes the right diagram commute.

2.1.2 Internalization

Given a category ℭ whose objects are sets with a certain structure, we want to equip the set hom(X, Y) also with such a structure, i.e., we want it to become an object of ℭ. This is called internalization. At present, we deal with the category 𝔗𝔬𝔭 of topological spaces. For X, Y ∈ 𝔗𝔬𝔭, we have C(X, Y) as the set of morphisms. In order to get this set itself a member of 𝔗𝔬𝔭, we have to equip it with a topology. It will turn out that the following topology is suitable.

Definition 5 Let X, Y ∈ 𝔗𝔬𝔭. The compact open topology on C(X, Y) is defined to be the topology generated by

{f ∈ C(X, Y) : ∀ K ⊆ X compact, U ⊆ Y open : f(K) ⊆ U}

as a subbase. We will write X^{Y} for the space C(X, Y) with the compact open topology.

This is of course not the only way to define a topology on C(X, Y). But the compact open topology has the property that it “factors through 𝔗𝔬𝔭.” More formally, this means the following.

Theorem 2 Let the functor Hom be defined by

Hom: 𝔗𝔬𝔭^{∗} × 𝔗𝔬𝔭 → 𝔗𝔬𝔭, Hom(X, Y) := X^{Y} .

Then Hom ∘| · | = hom where hom is the functor of Example 2 and | · | is the forgetful functor defined analogously to Example 1.

𝔗𝔬𝔭^{∗} × 𝔗𝔬𝔭  \overline{hom} 𝔖𝔢𝔱

[figure: Hom; | · |]

𝔗𝔬𝔭

Theorem 3 (Exponential Law in 𝔗𝔬𝔭) Let X, Y, Z ∈ 𝔗𝔬𝔭, X Hausdorff, and Y locally compact T_{2}. Then

Z^{X×Y} ≅ (Z^{X})^{Y}

in 𝔗𝔬𝔭.

2.2 The Category of Pairs

We consider pairs (X, A) of topological spaces where A ⊆ X is a subspace of X. The category of pairs is denoted by 𝔗𝔬𝔭_{2}. Its objects are the pairs (X, A), and the morphisms are naturally defined by

M((X, A), (Y, B)) := hom((X, A), (Y, B)) := {f ∈ C(X, Y) : f(A) ⊆ B}.

It is clear that 𝔗𝔬𝔭 is a subcategory of 𝔗𝔬𝔭_{2} by writing spaces X ∈ 𝔗𝔬𝔭 in the form (X, ∅) ∈ 𝔗𝔬𝔭_{2}. We will find that 𝔗𝔬𝔭_{2} behaves similar to 𝔗𝔬𝔭. If we define

∏_{i∈I}(X_{i}, A_{i}) := ( ∏_{i∈I} X_{i}, ∏_{i∈I} A_{i}),  ∐_{i∈I}(X_{i}, A_{i}) := ( ∐_{i∈I} X_{i}, ∐_{i∈I} A_{i}),

we get in analogy to Theorem 1 the following result:

Theorem 4 Let (X_{i}, A_{i}) (i ∈ I) and (Y, B) be spaces.

1. Let f_{i} ∈ hom((Y, B), (X_{i}, A_{i})) for every i ∈ I. Then there exists a unique morphism h ∈ hom((Y, B), ∏_{i∈I}(X_{i}, A_{i})) such that f_{j} = h ∘ π_{j} for every j ∈ I.

2. Let f_{i} ∈ hom((X_{i}, A_{i}), (Y, B)) for every i ∈ X_{i}. Then there exists a unique morphism h ∈ hom(∐_{i∈I}(X_{i}, A_{i}), (Y, B)) such that f_{j} = i_{j} ∘ h for every j ∈ I.

Also the internalization of 𝔗𝔬𝔭_{2} is of interest. For this we have to interpret the set hom((X, A), (Y, B)) of morphisms from (X, A) to (Y, B) as a pair (Z, C) which itself is a pair of spaces. A construction similar to the one of Section 2.1.2 is possible:

Definition 6 Let (X, A), (Y, B) ∈ 𝔗𝔬𝔭_{2}, and let Z := {f ∈ Y^{X} : f(A) ⊆ B} be equipped with the inherited topology of Y^{X}, i.e., with the compact open topology. With C := B^{X}, regarded as a subspace of Z, define

(Y, B)^{(X,A)} := (Z, C).

Theorem 5 The functor

Hom: 𝔗𝔬𝔭_{2}^{∗} × 𝔗𝔬𝔭_{2} → 𝔗𝔬𝔭_{2}, Hom((X, A), (Y, B)) := (Y, B)^{(X,A)}

factors through 𝔗𝔬𝔭_{2}. This means that | · | ∘ Hom = hom.

Theorem 6 (Exponential Law in 𝔗𝔬𝔭_{2}) Let (X, A), (Y, B), (Z, C) ∈ 𝔗𝔬𝔭_{2}, X Hausdorff, Y locally compact T_{2}. Then

(Z, C)^{(X,A)×(Y,B)} ≅ ((Z, C)^{(Y,B)})^{(X,A)}

in 𝔗𝔬𝔭_{2}, where

(X, A) × (Y, B) := (X × Y, X × B ∪ A × Y).

2.3 The Category of Pointed Spaces

A pointed space is a pair (X, x_{0}) where X is a space and a base point x_{0} ∈ X. By 𝔗𝔬𝔭_{p} we denote the category of pointed spaces. Its objects are the pointed spaces (X, x_{0}), and the morphisms from (X, x_{0}) to (Y, y_{0}) are the continous maps f: X → Y mapping x_{0} to y_{0}. The category 𝔗𝔬𝔭_{p} is a subcategory of 𝔗𝔬𝔭_{2} which can be seen by rephrasing (X, x_{0}) as (X, {x_{0}}). Moreover, there is the so called contraction functor c: 𝔗𝔬𝔭_{2} → 𝔗𝔬𝔭_{p} which maps a pair (X, A) to the pointed space (X/A, A), i.e., c contracts A to a single point. As in 𝔗𝔬𝔭 and in 𝔗𝔬𝔭_{2}, we are intereted in products, coproducts, and internalization.

For products, the situation is as expected. If the product of pointed spaces X_{i} (i ∈ I) is defined by

∏_{i∈I}(X_{i}, x_{0}^{(i)}) := ( ∏_{i∈I} X_{i}, (x_{0}^{(i)})_{i∈I})

then the analogon of Theorems 1 (1) and 4 (1) also holds. However, for coproducts, the situation is more complicated. The problem is that the naive componentwise definition

∐_{i∈I}(X_{i}, x_{0}^{(i)}) :=^{?} ( ∐_{i∈I} X_{i}, {x_{0}^{(i)} : i ∈ I})

does not work since {x_{0}^{(i)} : i ∈ I} is not a singleton. The solution is to identify (glue) the base points to one single point. This leads to the concept of wedge sums.

Definition 7 Let (X, x_{0}), (Y, y_{0}) ∈ 𝔗𝔬𝔭_{p}. Then

(X, x_{0}) ∨ (Y, y_{0}) := ((X ∪^{·} Y)/{x_{0}, y_{0}}, [x_{0}])

is called the wedge sum of (X, x_{0}) and (Y, y_{0}). Here X ∪^{·} Y means the disjoint union (coproduct), [x_{0}] means the equivalence class of x_{0}, which is simply {x_{0}, y_{0}}.

More general, if (X_{i}, x_{0}^{(i)}) (i ∈ I) is a family of pointed spaces, we define

⋁_{i∈I}(X_{i}, x_{0}^{(i)}) := (∐_{i∈I} X^{i}/{x_{0}^{(i)} : i ∈ I}, {x_{0}^{(i)} : i ∈ I})

to be the wedge sum of the pointed spaces (X_{i}, x_{0}^{(i)}) (i ∈ I).

The internalization of 𝔗𝔬𝔭_{p} does not pose any problems if we regard pointed spaces (X, x_{0}) ∈ 𝔗𝔬𝔭_{p} as pairs (X, {x_{0}}) ∈ 𝔗𝔬𝔭_{2}. The results carry over from Section 2.2. In particular, we have also an exponential law in 𝔗𝔬𝔭_{p}:

Theorem 7 Let (X, x_{0}), (Y, y_{0}), (Z, z_{0}) ∈ 𝔗𝔬𝔭_{p}, Y locally compact T_{2}, X Hausdorff. Then

(Z, z_{0})^{(X,x_{0})×(Y,y_{0})} ≅ ((Z, z_{0})^{(Y,y_{0})})^{(X,x_{0})}

in 𝔗𝔬𝔭_{p} where

(X, x_{0}) × (Y, y_{0}) := (X × Y, X ∨ Y).

3 Homotopy

3.1 Homotopic Maps

Definition 8 Given two spaces X, Y ∈ 𝔗𝔬𝔭 and f, g ∈ C(X, Y). f and g are called homotopic, written as f ≃ g, if there exists a continous function H ∈ C(X × [0, 1], Y) such that H(·, 0) = f and H(·, 1) = g. Such an H is called a homotopy to f and g.

A homotopy H has to be understood intuitively as a continous deformation of f(X) into g(X). The second parameter can be interpreted as the time: In the beginning, H behaves like f, but at time 1, it behaves like g.

Example 3 Given X = Y = ℝ^{2} with the usual topology. The maps

f: ℝ^{2} → ℝ^{2}, f(x, y) = x · y, g: ℝ^{2} → ℝ^{2}, g(x, y) = x + y

are homotopic. It is shown by providing a homotopy.

H: ℝ^{2} × [0, 1] → ℝ^{2}, H(x, y, t) := (1 − t) · x · y + t · (x + y)

obviously does the job, and hence f ≃ g. In fact, a similar construction works for every choice of f and g in C(ℝ^{2}, ℝ^{2}), for all such f and g we have f ≃ g.

Theorem 8 Homotopy ≃ is a congruence relation on C(X, Y) (X, Y ∈ 𝔗𝔬𝔭), i.e., it is an equivalence relation and it is compatible with composition.

The rest of this sumary will be dedicated to the study of the equivalence classes of ≃, and to the definition of homotopy not only for 𝔗𝔬𝔭, but also for the categories 𝔗𝔬𝔭_{2} and 𝔗𝔬𝔭_{p}.

3.2 Homotopy Categories

3.2.1 The Category of Spaces

Definition 9 The category 𝔗𝔬𝔭′ := 𝔗𝔬𝔭/≃ is called the homotopy category. Its objects are the spaces X ∈ 𝔗𝔬𝔭, the morphism set M(X, Y) to spaces X, Y ∈ 𝔗𝔬𝔭 is the set C(X, Y)/≃, also written as [X, Y]. The composition is defined by the composition of representatives: [f] ∘ [g] := [f ∘ g].

The 0-component [x ↦ x_{0}], the homotopy class of a constant map, will be denoted by 0. There is a natural functor relating 𝔗𝔬𝔭 and 𝔗𝔬𝔭′ to each other which maps a space X ∈ 𝔗𝔬𝔭 to the same space X ∈ 𝔗𝔬𝔭′, and maps f ∈ C(X, Y) to its homotopy class [f] ∈ [X, Y]. Two spaces X, Y isomorphic in 𝔗𝔬𝔭′ are called homotopic equivalent, written as X ≃ Y. Two spaces X, Y being isomorphic in 𝔗𝔬𝔭 are also isomorpic in 𝔗𝔬𝔭′, i.e., they are homotopic equivalent. Isomorphisms in 𝔗𝔬𝔭′ are called homotopy equivalences.

3.2.2 Pairs and Pointed Spaces

Homotopy of morphisms in the category of pairs is defined analogously to homotopy of morphisms in 𝔗𝔬𝔭.

Definition 10 Let (X, A), (Y, B) ∈ 𝔗𝔬𝔭_{2} and f, g ∈ (X, A)^{(Y,B)}. f and g are called homotopic if there exists a continous map

H: (X, A) × I → (Y, B)

with H(·, 0) = f and H(·, 1) = g. If f is homotopic to g, we write again f ≃ g.

As in 𝔗𝔬𝔭, the idea is to continously deform the image f(X) into the image g(X). The additional requirement here is that H(A, t) ⊆ B for each t ∈ [0, 1], i.e., H is meant to be continous wrt. the topology of (Y, B)^{(X,A)×(I,I)}. It is not a surprise that also homotopy in 𝔗𝔬𝔭_{2} is a congruence relation. This allows us to define 𝔗𝔬𝔭′_{2} := 𝔗𝔬𝔭_{2}/≃ in analogy to Def. 9. Again, the category of pointed spaces can be treated by regarding a pointed space (X, x_{0}) as a pair (X, {x_{0}}). The definition of homotopy and the category 𝔗𝔬𝔭′_{p} = 𝔗𝔬𝔭_{p}/≃ are according to this point of view. If f, g: (X, x_{0}) → (Y, y_{0}) are homotopic by a homotopy

H: (X, x_{0}) × I → (Y, y_{0}),

then H(x_{0}, t) = y_{0} for all t ∈ [0, 1].

3.3 Applications

3.3.1 Contractibility

We start with recalling the definitions of some basic topological notions.

Definition 11 Let X ∈ 𝔗𝔬𝔭.

1. Two points x, x′ ∈ X are 0-connected (or path connected), if there exists an α ∈ X^{[0,1]} with α(0) = x and α(1) = x′.

2. Being 0-connected is an equivalence relation on X. The class [x] of x is called the 0-component of x in X. Further, we write

π_{0}(X) := {[x] : x ∈ X} = X/∼

for the set of 0-components.

3. X is 0-connected if there is only one 0-component (namely X itself).

The notion of contractible expresses the fact that a space can be continously deformed (contracted) to a single point. Formally, this is defined in the following definition.

Definition 12 Let X, Y ∈ 𝔗𝔬𝔭.

1. f ∈ C(X, Y) is 0-homotopic, if it is homotopic to a constant map.

2. X is contractible if the identity 1_{X} is homotopic to a constant map.

Example 4 The approach of Example 3 shows that ℝ^{n} is contractible. More general: Every star shaped subset of ℝ^{n} is contractible.

Similar considerations in a slightly more general setting lead to the following theorem, which sumarizes some first results.

Theorem 9 Let X, Y ∈ 𝔗𝔬𝔭.

1. If X is contractible, then X is 0-connected.

2. If Y is contractible, then [X, Y] = {0}.

3. If X is contractible, then [X, Y] = π_{0}(Y).

Example 5 If Y ≠ ∅ is 0-connected, then [ℝ^{n}, Y] = {0}. Conversely, for any X ∈ 𝔗𝔬𝔭′, we have [X, ℝ^{n}] = {0}.

More interesting examples are collected in the following section.

3.3.2 Nontrivial Examples

First we define some spaces. We already used the n dimensional real space ℝ^{n} with the classical topology. The spaces we are mainly interested in are all subspaces of ℝ^{n} with the inherited topology. We identify ℂ with ℝ^{2}.

Definition 13

1. I := [0, 1] ⊆ ℝ is called the unit interval.

2. 𝔻^{n} := {x ∈ ℝ^{n} : ||x|| ≤ 1} is the n dimensional disc.

3. 𝔻^{∘}^{n} := {x ∈ ℝ^{n} : ||x|| < 1} is the open n dimensional disc.

4. 𝕊^{n} := {x ∈ ℝ^{n+1} : ||x|| = 1} is the n dimensional real sphere.

The spaces defined in the definition above are interrelated in various ways. For

instance: 𝕊^{1} ≅ I/∂I, 𝔻^{∘}^{n} = (𝔻^{n})^{0}, 𝕊^{n} = ∂𝔻^{n+1}, etc.

The points (1, 0, …, 0) and (−1, 0, …, 0) are called northpole and southpole, respectively, of 𝔻^{n} or 𝕊^{n}. If 𝕊^{n} or 𝔻^{n} are regarded as pointed spaces, we will always take the northpole as the base point. The canonical base point of 𝔻^{∘}^{n} is the origin. We now study the structure of [𝕊^{1}, 𝕊^{1}]. Consider a continous function φ: I → ℝ with φ(0) = 0 and φ(1) = n ∈ ℤ. Then exp(2πiφ(0)) = exp(2πiφ(1)) = 1, which means that the diagram

[figure: I; φ; ℝ; exp(2πit); 𝕊^{1}]

[figure: φ̂]

I/∂I ≅ 𝕊^{1}

commutes. The function φ gives rise to a corresponding function

φ̂: 𝕊^{1} → 𝕊^{1},  φ̂(exp(2πit)) = exp(2πiφ(t)).

Theorem 10 Given f ∈ C(𝕊^{1}, 𝕊^{1}) there is a unique map φ ∈ C((I, 0), (ℝ, 0)) with the property that f(z) = f(1) · φ̂(z).

Since φ is uniquely determined for a given function f on the sphere, to each f there corresponds a number φ(1). For reasons of continuity, this number must be an integer, which we will call the degree of f.

Definition 14 Let f ∈ C(𝕊^{1}, 𝕊^{1}) and let φ ∈ C(I, ℝ) be the uniquely determined map from Theorem 10. Then deg f := φ(1) is called the degree of f.

Theorem 11

1. f, g ∈ C(𝕊^{1}, 𝕊^{1}) are homotopic if and only if deg(f) = deg(g).

2. For the monomial function f: z ↦ z^{n} (n ∈ ℤ) we have deg(f) = n.

3. The groups ([𝕊^{1}, 𝕊^{1}], ∘) and (ℤ, +) are isomorphic.

Sumarizing the results, we can say that every continous map f form 𝕊^{1} to itself starts at some point x on the sphere, and turns several times around it, eventually changing its direction in between, and finally comes back to the starting point x. f does exacty deg f rotations around, if clockwise rounds count −1, rounds in the other direction count 1, and every corresponding parts cancel if the direction has changed.

3.3.3 Consequences

Theorem 12 The sphere 𝕊^{1} is 0-connected but not contractible.

Theorem 13 There is no continous map f: 𝔻^{2} → 𝕊^{1} with f|_{𝕊^{1}} = id_{𝕊^{1}} .

Theorem 14 (Brower’s Theorem) Every f ∈ C(𝔻^{n}, 𝔻^{n}) has a fixed point.

Theorem 15 (Fundamental Theorem of Algebra) Every nontrivial polynomial has a root in ℂ.

Definition 15 To f ∈ C(𝕊^{1}, ℂ) and z ∈ ℂ \ f(𝕊^{1}) let

w_{z}: 𝕊^{1} → 𝕊^{1}, w_{z}(x) := \frac{f(x) − z}{|f(x) − z|},

and call w(f, z) := deg(w_{z}) the winding number of f around z.

Theorem 16 Let f ∈ C(𝕊^{1}, ℂ).

1. If two points z, z′ ∈ ℂ \ f(𝕊^{1}) are in the same connected component, then they have the same winding number.

2. ℂ \ f(𝕊^{1}) has exactly one unbounded 0-connected component K. The winding number for all z ∈ K is 0.

Theorem 17 (Theorem of Bursuk-Ulam) For every f ∈ C(𝕊^{2}, ℝ^{2}) there exists an x ∈ 𝕊^{2} such that f(x) = f(−x).

(Counter-)intuitively, the theorem of Bursuk-Ulam states that, for instance, there is always a place x on the earth ( = 𝕊^{2}) such that on the opposite side −x there is the same pressure and temperature (taking as f the map which maps each location x to the pair of pressure and temperature).

4 Functorial Constructions

4.1 Wedge Sum and Smash Product

We consider pointed spaces (X_{i}, x_{0}^{(i)}) (i ∈ I). The wedge sum of the spaces X_{i} has already been defined to

⋁_{i∈I}(X_{i}, x_{0}^{(i)}) := (∐_{i∈I} X^{i}/{x_{0}^{(i)} : i ∈ I}, {x_{0}^{(i)} : i ∈ I})

It is constructed to behave like the coproduct in the sense of Theorem 1. For the intuition, the wedge sum of spaces (X_{i}, x_{0}^{(i)}) is the disjoint union of the X_{i} where all the base points x_{0}^{(i)} are glued together. Applying the construction to only two spaces looks as follows.

(X_{1}, x_{1}) ∨ (X_{2}, x_{2}) = (X^{1} ∪^{·} X^{2}/{x_{1}, x_{2}}, [x_{1}])

Given four spaces X_{1}, X_{2}, Y_{1}, Y_{2} ∈ 𝔗𝔬𝔭_{p} and maps f_{1} ∈ Y_{1}^{X_{1}}, f_{2} ∈ Y_{2}^{X_{2}}, there exists exactly one map

f_{1} ∨ f_{2} : X_{1} ∨ X_{2} → Y_{1} ∨ Y_{2}

such that i∘f_{1} = (f_{1} ∨f_{2})∘i = i∘f_{2} where the i denote the canonical injections. This observation allows to regard the wedge sum as a functor

∨: 𝔗𝔬𝔭_{p} × 𝔗𝔬𝔭_{p} → 𝔗𝔬𝔭_{p}.

It maps a pair (X_{1}, X_{2}) of pointed spaces to its wedge sum X_{1} ∨ X_{2}, and a pair (f_{1}, f_{2}) ∈ X_{1}^{Y_{1}} × X_{2}^{Y_{2}} of morphisms to f_{1} ∨ f_{2}. Does this also work for the product? We have seen in Section 2.3 that the category 𝔗𝔬𝔭_{p} is closed under taking products. The base point of the space ∏_{i∈I}(X_{i}, x_{0}^{(i)}) is simply the point (x_{0}^{(i)})_{i∈I}. But does not yet lead to a product of morphisms, and hence not to a functor. A “refined” product definition is needed. Consider pointed spaces (X_{i}, x_{0}^{(i)}) (i ∈ I) and for each j ∈ I morphisms

β_{j}: (X_{j}, x_{0}^{(j)}) → ∏_{i∈I} X_{i}, β_{j}(u) := (u_{i})_{i∈I}

where u_{i} = x_{0}^{i} (i ≠ j) and u_{i} = u (i = j). The maps β_{j} can be regarded as the maps which map a space X_{j} to the corresponding “coordinate axis” in ∏_{i∈I} X_{i}. The union of the coordinate axis β_{j}(X_{j}) is just the wedge sum of the X_{j}. Clearly β_{j}(X_{j}) ≅ X_{j}, and the intersection of β_{i}(X_{i}) and β_{j}(X_{j}) (i ≠ j) contains only the origin (x_{0}^{(i)})_{i∈I} which is by construction the base point of the wedge sum. The smash product is obtained from the product space by identifying its coordinate axes.

Definition 16 Let (X_{i}, x_{0}^{(i)}) (i ∈ I) be a family of pointed spaces. Then

⋀_{i∈i}(X_{i}, x_{0}^{(i)}) := ∏_{i∈I}(X^{i}, x_{0}^{(i)})/⋁_{i∈I}(X_{i}, x_{0}^{(i)})

is called the smash product of the (X_{i}, x_{0}^{(i)}).

Now take again four pointed spaces X_{1}, X_{2}, Y_{1}, Y_{2} ∈ 𝔗𝔬𝔭_{p} with base points x_{1}, x_{2}, y_{1}, y_{2}, respectively, and two morphisms f_{1} ∈ Y_{1}^{X_{1}}, f_{2} ∈ Y_{2}^{X_{2}}. The product map

f_{1} × f_{2} : X_{1} × X_{2} → Y_{1} × Y_{2}

is simply defined component wise by (f_{1} × f_{2})(a, b) := (f_{1}(a), f_{2}(b)). In particular, since f_{1} and f_{2} are morphisms of 𝔗𝔬𝔭_{p}, their product f_{1} × f_{2} maps axis points to axis points, e.g., (f_{1} × f_{2})(x_{1}, t) = (y_{1}, f_{2}(t)). This allows to define the smash product of f_{1} and f_{2} to be the morphism

f_{1} ∧ f_{2} : X_{1} ∧ X_{2} → Y_{1} ∧ Y_{2}

which maps [a] × [b] to [f_{1}(a)] × [f_{2}(b)]. This makes the smash product a functor

∧: 𝔗𝔬𝔭_{p} × 𝔗𝔬𝔭_{p} → 𝔗𝔬𝔭_{p}

which — similar to the situation with the wedge sum — maps a pair (X_{1}, X_{2}) of pointed spaces to the pointed space X_{1} ∧X_{2}, and a pair (f_{1}, f_{2}) of morphisms to f_{1} ∧ f_{2}.

Theorem 18 Let X, Y ∈ 𝔗𝔬𝔭_{p}, X Hausdorff, Y Hausdorff and locally compact. Then

[X ∧ Y, Z] ≅ [X, Z^{Y}].

4.2 Suspension and Loop space

Definition 17 If X = (X, x_{0}) ∈ 𝔗𝔬𝔭_{p} is a pointed space, then

SX := S(X, x_{0}) := 𝕊^{1} ∧ X = (𝕊^{1}, 1) ∧ (X, x_{0})

is called the suspension of X.

Example 6 When trying to imagine concrete examples for the suspension, one soon runs out of dimensions. (Try to draw S𝔻^{2}, for instance.) Intuition is hence restricted to very trivial spaces only. We consider X = (I, 0). 𝕊^{1} × I is the hollow cylinder without covers. 𝕊^{1} ∨ I is obtained by glueing I and 𝕊^{1} together at their basepoints. In order to get to 𝕊^{1} ∧ I, we have to take the cylinder 𝕊^{1} × I and to glue together the contained sphere 𝕊^{1} × 0 and the edge 1 × I. The result is a hollow hemisphere with its base point at the equator. For another example, take X = (x_{0}, x_{0}) to be the pointed space only consisting of one point. Then SX ≅ X, because 𝕊^{1} × X ≅ 𝕊^{1} and 𝕊^{1} ∨ X ≅ 𝕊^{1} as well.

The suspension admits also a functor S: 𝔗𝔬𝔭_{p} → 𝔗𝔬𝔭_{p} which acts on spaces as defined above, and which maps a morphism f: X → Y to a morphism Sf: SX → SY which acts on x by

Sf([z, x]) = (1 ∧ f)([z, x]) = [(1 × f)(z, x)] = [z, f(x)].

The suspension functor can hence be viewed as the composition of the embeding functor 𝔗𝔬𝔭_{p} → 𝔗𝔬𝔭_{p} × 𝔗𝔬𝔭_{p} mapping X to (𝕊^{1}, X) and the smash product functor 𝔗𝔬𝔭_{p} × 𝔗𝔬𝔭_{p} → 𝔗𝔬𝔭_{p}, (X, Y) ↦ X ∧ Y.

Definition 18 To X ∈ 𝔗𝔬𝔭_{p}, the space ΩX := X^{𝕊^{1}} is called the loop space of X.

Example 7 If X consists of only one point, then ΩX ≅ X. The loop space Ω𝕊^{1} was studied in Section 3.3.2.

The loop spaces of X consists of all closed paths from x_{0} to itself. Ω can be regarded as a functor from 𝔗𝔬𝔭_{p} → 𝔗𝔬𝔭_{p}, called the loop functor. Indeed, it is

Ω: 𝔗𝔬𝔭_{p} → 𝔗𝔬𝔭_{p}, ΩX = Hom(𝕊^{1}, X)

where Hom: 𝔗𝔬𝔭_{p}^{∗} × 𝔗𝔬𝔭_{p} is the covariant hom functor in 𝔗𝔬𝔭_{p}, cf. Section 2.3. The following theorem interrelates the functors S and Ω. It is a direct consequence of Theorem 18.

Theorem 19 Let X, Y ∈ 𝔗𝔬𝔭_{p}, X Hausdorff. Then

[SX, Y] ≅ [X, ΩY].

5 Homotopy Groups

5.1 Groups and H-Groups

Similar to 𝔗𝔬𝔭_{p}, we can define the category 𝔖𝔢𝔱_{p} of pointed sets, whose objects are pairs (S, s) where S is a set and s ∈ S, and whose morphisms are the maps mapping base points to base points. In this sense, let (G, e) ∈ 𝔖𝔢𝔱_{p} be a pointed set, and let

μ: G × G → G, ν: G → G

be morphisms in 𝔖𝔢𝔱_{p}. We call (G, e, μ, ν) a group, if the following diagrams are commutative. In these diagrams, we mean e to be the constant map mapping everything to the base point e, and id is the identity.

[figure: G; e × id; G × G; id ×e; G]

[figure: id; id; (A)]

G \begin{matrix}μ \\  \end{matrix}
G × G × G \overline{\begin{matrix}μ × id \\  \end{matrix}}G × G

(B)

[figure: G × G; id ×μ; μ; G; μ]

[figure: G; ν × id; G × G; id ×ν; G]

[figure: e; e; (C)]

G \begin{matrix}μ \\  \end{matrix}

These diagrams express the behaviour of the identity (A), the associativity (B), and the behaviour of inverses (C). If, furthermore, the diagram

G × G  \overline{T} G × G

[figure: μ; μ; (D)]

G

commutes, where T is meant to be the map which interchanges dimensions, i.e., (D) expresses commutativity, then G is naturally called abelian group. There is nothing special to the category 𝔖𝔢𝔱_{p} in this definition, and the purpose of this chapter is to study groups defined by the diagrams (A), (B), and (C), but living in different categories.

Definition 19 A tuple (X, x_{0}, μ, ν) where (X, x_{0}) ∈ 𝔗𝔬𝔭_{p} and

μ: X × X → X, ν: X → X

are morphisms preserving base points is called an H-group, if the diagrams (A), (B), and (C) with X in place of G and x_{0} in place of e commute in 𝔗𝔬𝔭′_{p}. If the diagrams commute in 𝔗𝔬𝔭_{p}, then (X, x_{0}, μ, ν) is called a topological group.

Example 8 For any Y ∈ 𝔗𝔬𝔭_{p}, ΩY is an H-group, if multiplication μ is defined as concatenation of loops, and inversion ν is reversion of a loop’s direction.

Theorem 20 Let (K, k_{0}) be an H-group with multiplication μ: K × K → K and inverse ν: K → K, and let (X, x_{0}) ∈ 𝔗𝔬𝔭′_{p}. If to any f, g ∈ [(X, x_{0}), (K, k_{0})], the product [f] · [g] is defined by

[f] · [g] := [μ ∘ (f × g) ∘ Δ],

with

Δ: X → X × X, Δ(x) := (x, x)

and if [f]^{−1} := [ν ∘ f], then ([(X, x_{0}), (K, k_{0})], ·, ∘^{−1}) is a group. Its identity is [x ↦ k_{0}]. The association (X, x_{0}) → [(X, x_{0}), (K, k_{0})] defines a contravariant functor from 𝔗𝔬𝔭^{∗} to 𝔊𝔯𝔬𝔲𝔭.

5.2 𝔄-Groups

Rather than doing the definition of groups for every pointed category separately, it is preferable to define the notion of group directly in an arbitrary setting leaving the category unspecified. But for general categories, the definition does not work. We need pointed categories.

Definition 20 A category 𝔄 is pointed if there is a choice function which selects from each morphism set M(X, Y) a morphism m such that the composition of m with any other morphism m′ ∈ M(Y, Z) or m′′ ∈ M(Z, X) gives the selected morphism of M(X, Z) or M(Z, Y), respectively.

If a category has a choice function, then this choice function is uniquely determined. If a pointed category has a zero object 0, i.e., an object where |M(0, X)| = |M(X, 0)| = 1 for any X, then the selected morphism m ∈ M(X, Y) is the morphism obtained by composition of the element of M(X, 0) with the element of M(0, Y). This morphism will be denoted by 0_{XY} or simply by 0. For instance, the categories 𝔖𝔢𝔱_{p} and 𝔗𝔬𝔭_{p} are pointed, their zero objects are the sets/spaces ({x_{0}}, x_{0}), and the selected morphism 0_{(X,x_{0}),(Y,y_{0})} maps every x ∈ X to y_{0} ∈ Y. The categories 𝔖𝔢𝔱 and 𝔗𝔬𝔭 are not pointed. We will now define 𝔄-groups to be groups in the pointed category 𝔄.

Definition 21 Let 𝔄 be a pointed category whose object class is closed under taking binary products. A triple (G, μ, ν), where G ∈ 𝔄, μ: G × G → G, ν: G → G are morphisms, is called an 𝔄-group if the diagrams (A), (B), and (C) commute, where e is meant to be the selected morphism from G to itself. If, in addition, (D) is commutative, then G is called abelian. If G and H are 𝔄-groups, and φ: G → H is a morphism that makes the diagram

[figure: G × G; φ × φ; H × H; G; μ_{G}; φ; H; μ_{H}]

commute, then φ is called 𝔄-homomorphism between G and H.

Example 9 An H-group is a 𝔗𝔬𝔭′_{p}-group, because 𝔗𝔬𝔭′_{p} is a pointed category with binary products.

Theorem 21 If 𝔄 is a pointed category and K ∈ 𝔄 is an 𝔄-group, then the contravariant hom functor factors through 𝔊𝔯𝔬𝔲𝔭, i.e., the diagram

𝔄^{∗} × 𝔄 \overline{{hom \atop }} 𝔖𝔢𝔱

X ↦ (K, X) [figure: hom(·, K)]

[figure: 𝔄^{∗}; ≅; 𝔊𝔯𝔬𝔲𝔭; | · |]

commutes.

Definition 22 If 𝔄 is a pointed category with binary coproducts, then 𝔄^{∗}- groups are called 𝔄-cogroups.

If ((K, k_{0}), μ, ν) is a 𝔄-cogroup, then they fulfill the diagrams obtained from (A), (B), and (C) by reversing the arrow directions, replacing K by G, e by k_{0}, and × by the binary coproduct ⊔. Note the reverse of the direction: A multiplication μ takes values in K and has images in K ⊔ K.

Example 10 TO BE: cogroup example (Is there any nontrivial cogroup?)

The notion of cogroups allows to state an analogon to Example 8:

Theorem 22 For any pointed space X ∈ 𝔗𝔬𝔭_{p}, the suspension SX is a 𝔗𝔬𝔭′_{p}- cogroup. TO BE: what is μ and ν here? (Do the diagrams commute here?)

The interesting result is that Theorem 19 also holds for groups:

Theorem 23 Let X, Y ∈ 𝔗𝔬𝔭_{p}, X Hausdorff. Then [SX, Y], [X, ΩY] are groups, and

[SX, Y] ≅ [X, ΩY]

in 𝔊𝔯𝔬𝔲𝔭.

Theorem 24 Let X, Y ∈ 𝔗𝔬𝔭_{p}. Then

1. [SX, ΩY] is abelian.

2. Ω^{n}Y is a commutative H-group for n ≥ 2.

3. S^{n}X is a commutative H-cogroup for n ≥ 2.

6 The Theorem of Seifert-van Kampen

6.1 Free Products

Let X ∈ 𝔖𝔢𝔱 and X′ be a disjoint copy of X. Consider the sequences (words) of the elements of X ∪ X′. Such a word is called reduced if it does not contain xx′ or x′x for any x ∈ X as a subword. Let’s declare that x′′ = x for convenience. Let F_{X} be the set of all these reduced words over the alphabet X ∪ X′. The operation ∘ on F_{X} is defined as concatenation plus reduction. This means that if (a, b, c) and (d, e, f) are reduced words in F_{X}, then (a, b, c) ∘ (d, e, f) is (a, b, c, d, e, f) or (a, b) ∘ (e, f) if c = d′. (F_{X}, ∘) is a group with identity () and with (x_{1}, …, x_{n})^{−1} = (x′_{n}, …, x′_{1}).

Example 11 A group G can be specified by a set X of generators and a set K ⊆ F_{X} of relations in the way G := F^{X}/K. For instance for X = {a, b} and K = {aba′b′} we have ℤ × ℤ ≅ F^{X}/K.

Let G_{1}, G_{2} be groups. Without loss of generality, we may assume that G_{1} and G_{2} only share the unit element, which we will denote by 1. The free product G_{1}∗G_{2} of G_{1} with G_{2} the set of all finite sequences (x_{1}, …, x_{n}) over G_{1} ∪ G_{2} where consecutive elements x_{i}, x_{i+1} do not belong to the same generating group G_{1} or G_{2}. None of the x_{i} be the unit element 1. For a, b ∈ G_{1} ∗ G_{2}, a ∘ b is defined as concatenation where …, a_{n}, b_{1}, … is replaced by …, a_{n} · b_{1}, … if a_{n} and b_{1} belong to the same underlying group.

Definition 23 Let G_{1}, G_{2} be groups with G_{1} ∩ G_{2} = {1}.

1. If X = G_{1} ∪ G_{2} and

K := {(a, b, (ab)^{−1}) : a, b ∈ G_{1}} ∪ {(a, b, (ab)^{−1}) : a, b ∈ G_{2}},

then G_{1} ∗ G_{2} := F^{X}/K is called the free product of G_{1} and G_{2}.

2. If H is another group and f_{1} : G_{1} → H, f_{2} : G_{1} → H are morphisms, then f_{1}∗f_{2} : G_{1}∗G_{2} → H is defined to be the morphism which maps a sequence (x_{1}, …, x_{n}) ∈ G_{1} ∗ G_{2} to ∏_{i=1}^{n} f(x_{i}) ∈ H

f(x) := \{ \begin{matrix}f_{1}(x) & if x ∈ G_{1} \\ f_{2}(x) & if x ∈ G_{2} \end{matrix} .

Example 12 ℤ_{2} ∗ ℤ_{2} contains the elements

(), (a), (b), (ab), (ba), (aba), (bab), (abab), (baba), …

with which computation can be done as expected. For instance

aba · abab = abaabab = abbab = aab = b.

ℤ_{2} ∗ ℤ_{2} is called the infinite Dieder group.

6.2 Seifert-van Kampen’s Theorem

Definition 24 By π_{1}(X, x_{0}) := ΩX we denote the homotopy group of (X, x_{0}) ∈ 𝔗𝔬𝔭_{p}.

Definition 25 A space X ∈ 𝔗𝔬𝔭 is simply connected if it is path connected and every closed path is zero homotopic.

Let (X, x_{0}) be a pointed space, U, V ⊆ X open, and U ∩ V ⊇ {x_{0}} path connected. Let i: U ∩ V ↪ U, i′: U ↪ X, j: U ∩ V ↪ V, j′: V ↪ X denote the injection maps from U ∩ V to U and V and from there to X. These maps correspond to homomorphisms between the corresponding homotopy groups. We will denote them by i_{∗}, i′_{∗}, j_{∗}, j′_{∗}, respectively.

[figure: U; i; X; π_{1}(U, x_{0}); i_{∗}; π_{1}(X, x_{0}); U ∩ V; i′; j′; V; j; π_{1}(U ∩ V, x_{0}); i′_{∗}; j′_{∗}; π_{1}(V, x_{0}); j_{∗}]

Theorem 25 (Seifert-van Kampen) Let (X, x_{0}), U, V, i, i′, i_{∗}, i′_{∗}, j, j′, j_{∗}, j′_{∗} be as above. Then

i_{∗} ∗ j_{∗}: π_{1}(U, x_{0}) ∗ π_{1}(V, x_{0}) → π_{1}(X, x_{0})

is surjective,

π(X, x_{0}) ≅ π^{1}(U, x^{0}) ∗ π^{1}(V, x^{0})/ker(i_{∗} ∗ j_{∗}),

and

ker(i_{∗} ∗ j_{∗}) = N ({i′_{∗}(ω)j′_{∗}(ω) : ω ∈ π_{1}(U ∩ V, x_{0}}),

where N (S) denotes the normal subgroup generated by S.