Rijksuniversiteit Groningen

Bachelor thesis in mathematics

Fibre Bundles: Trivial or Not?

Timon Sytse van der Berg

Prof. Dr. Henk W. Broer (first supervisor)Prof. Dr. Jaap Top (second supervisor)

July 12, 2016

Contents

1 Introduction to Bundles 11.1 Fibre Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Vector Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Principal Bundles . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Bundle Morphisms and Triviality 52.1 Fibre Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Vector Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Principal Bundles . . . . . . . . . . . . . . . . . . . . . . . . . 7

3 Examples of Bundles and Their Triviality 93.1 Tangent Bundles of S1 and S2 . . . . . . . . . . . . . . . . . . 93.2 The Mobius Band . . . . . . . . . . . . . . . . . . . . . . . . . 113.3 The Klein Bottle . . . . . . . . . . . . . . . . . . . . . . . . . 133.4 The Hopf Fibration . . . . . . . . . . . . . . . . . . . . . . . . 14

A 17A.1 Group Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . 17A.2 The Fundamental Group . . . . . . . . . . . . . . . . . . . . . 17

1

Abstract

When are fibre bundles globally isomorphic to a product space? We developsome theory to answer this question for vector bundles and principal bundlesand consider examples such as the Mobius band, the Klein Bottle, and theHopf fibration.

Chapter 1

Introduction to Bundles

In this chapter we develop some general theory and give criteria for thetriviality of vector bundles and principal bundles.

1.1 Fibre Bundles

Definition 1.1.1 (Fibre Bundle). A fibre bundle is a four-tuple (E,B, π, F )consisting of manifolds1 (E,B, F ) and a smooth surjection π : E → B suchthat the following conditions are satisfied.

1. Every x ∈ E has a neighborhood Ux ⊂ B such that there is a dif-feomorphism φ : π−1(Ux) → Ux × F . The neighborhood Ux and thediffeomorphism φ constitute a local trivialisation;

2. If we let proj1 : B × F → B be the map proj1(a, b) = a then thediffeomorphism φ of condition 1 satisfies proj1◦φ = π. This correspondsto the commutativity of the diagram seen below.

π−1(Ux) Ux ⊂ B

Ux × F

π

φproj1

The spaces E,B and F are called the total space, base space and fibre,respectively. The set π−1(b) is called the fibre at b ∈ B.

Definition 1.1.1 says that the total space E of a fibre bundle (E,B, π, F ) islocally isomorphic to B × F . If we view topology as the study of properties

1Manifold will mean smooth (real and C∞) manifold from now on.

1

that are invariant under homeomorphisms and ignore the C∞ structure, thenwe can say that π−1(Ux) is topologically “the same” as the product spaceUx × F .

Remark 1.1.1 (Notation). The fibre bundle (E,B, π, F ) is often representedschematically as

F → Eπ−→ B.

Example 1.1.1 (Product Bundle). If we let E = B×F , then (E,B, proj1, F )is a fibre bundle. Specifically, the identity id serves as the diffeomorphism ofDefinition 1.1.1. In this case, a single diffeomorphism works for the whole ofE. This is called a global trivialisation.

Definition 1.1.2 (Trivial). We call a fibre bundle that admits a global triv-ialisation a trivial fibre bundle.

Definition 1.1.3 (Section). A section σ : B → E over a fibre bundle(E,B, π, F ) is a smooth right inverse of π : E → B. Specifically,

π(σ(x)) = x

for all x ∈ B.

Example 1.1.2 (Sections of the product bundle). A section of the productbundle is a smooth function σ : B → B × F such that proj1 ◦ σ = id.Specifically, if we write

σ : a 7→ (b, c),

proj1 : (b, c) 7→ b,

we see that σ can be any smooth function of the form a 7→ (a, c). Hencesections of the product bundle are graphs of smooth functions.

1.2 Vector Bundles

Definition 1.2.1 (Vector Bundle). A vector bundle over the field F is a fibrebundle (E,B, π, F ) satisfying the following conditions.

1. The fibre F is a k-dimensional vector space over F. We call k the rankof the vector bundle.2

2. The map v 7→ φ(x, v) for v ∈ F is a linear isomorphism between F andπ−1(x).

2Some authors do not require k to be constant. The rank of the vector bundle is thennot always well-defined, unless it is assumed that the base space is connected.

2

Remark 1.2.1. A vector bundle over R is called a real vector bundle. Likewise,if F = C we call the vector bundle complex.

The tangent and cotangent bundles of differentiable manifolds form an im-portant example of vector bundles.

Definition 1.2.2 (Tangent Bundle). Let M be a differentiable manifold andsuppose we write TxM for the tangent space of M at the point x ∈M . Thenthe tangent bundle TM of M is defined

TM =⊔x∈M

TxM =⋃x∈M

{(x, y) | y ∈ TxM}.

Remark 1.2.2. A section of the tangent bundle TM is a vector field on M .

Definition 1.2.3 (Cotangent Bundle). Let M be a differentiable manifoldand suppose we write T ∗xM for the cotangent space of M at the point x ∈M .Then the cotangent bundle T ∗M of M is defined

T ∗M =⊔x∈M

T ∗xM =⋃x∈M

{(x, y) | y ∈ T ∗xM}.

Remark 1.2.3. A section of the cotangent bundle TM is a differential one-form on M .

Theorem 1.2.1 (TM as vector bundle). Suppose M is a k-dimensionalmanifold. Then (TM,M, proj1,Rk) is a vector bundle.

Proof. Let {x1, . . . , xn} be local coordinates for the open neighborhood Uxcontaining x. We recall that

TxM = spanR

{∂

∂x1, . . . ,

∂

∂xk

}.

Hence for (a1, . . . , ak) ∈ Rk the bijection

(a1, a2, . . . , ak)↔ a1∂

∂x1+ . . .+

∂

∂xk,

is an isomorphism between the vector spaces TxM and Rk.We now let φ : TM →M × Rn be the map

φ

(x, a1

∂

∂x1+ . . .+ ak

∂

∂xk

)= (x, a1, . . . , ak) .

We note that both φ and φ−1 are smooth. This means that φ is a diffeomor-phism between TM and M ×Rn. Moreover, we have proj1 ◦ φ = id. Finally,v 7→ φ(x, v) is a linear isomorphism since for v =

∑vi

∂∂xi

and w =∑wi

∂∂xi

we have that φ(x, v) + φ(x,w) = (x, v1 +w1, . . . , vk +wk) = φ(x, v +w).

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Remark 1.2.4. The cotangent bundle is a vector bundle. We will show laterthat the tangent bundle is isomorphic to the cotangent bundle.

Definition 1.2.4 (Structure group). Suppose that (Ui, φi) and (Uj, φj) aretrivialisations of the vector bundle (E,B, π, F ) such that Ui ∩ Uj 6= ∅. Thenthe composite functions

φ−1i ◦ φj : (Ui ∩ Uj)× F → (Ui ∩ Uj)× F

are of the formφ−1i ◦ φj(x, y) = (x,Ψij(x)y).

Here Ψij : (Ui ∩ Uj) → GLk are called coordinate transformations. If themaps Ψij(x) all belong to a subgroup G ⊂ GLk, then we call G the structuregroup of the vector bundle.

More generally, if the transition functions of a fibre bundle are well-definedand members of a group G, then we call G the structure group.

1.3 Principal Bundles

A principal bundle is a bundle for which the fibre is the structure group.

Definition 1.3.1 (Principal Bundle). A Principal Bundle is a fibre bundle(E,B, π, F ), where the fibre F is a Lie group equipped with a smooth rightgroup action on E such that the following conditions are satisfied.

1. The group action of F on E is free and transitive on the fibres π−1(b)for b ∈ B;

2. The orbits of F in E are identified with B:

B = E/G.

Remark 1.3.1. Let (E,B, π, F ) be a principal bundle and ? its group action.Given some fixed y ∈ E, we can write any x that lies in the fibre of y uniquelyin the form x = y ? g for some g ∈ F .

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Chapter 2

Bundle Morphisms andTriviality

2.1 Fibre Bundles

A fibre bundle morphism is a map between two fibre bundles that ‘respects’fibers.

Definition 2.1.1 (Fibre Bundle morphism). Let (E,B, π, F ) and (E ′, B′, π′, F ′)be fibre bundles. Then a fibre bundle morphism is a pair (g, f) of smoothmaps

g : E → E ′, f : B → B′,

such that π′ ◦ g = f ◦ π. This condition means that the following diagram iscommutative.

E E ′

B B′

g

π π′

f

If a fibre bundle morphism has an inverse that is also a bundle morphism wespeak about a fibre bundle isomorphism.

Theorem 2.1.1. The condition π′g = fπ implies

gπ−1(b) ⊂ (π′)−1f(b).

That is, the fibre at the point b ∈ B is mapped to the fibre at f(b) ∈ B′.

Proof. We first note that

(π′)−1f(b) = {x′ ∈ E ′ | π′(x) = f(b)}

5

andg(π−1b) = {g(x) | x ∈ E such that π(x) = b}.

Now suppose that y = g(x) ∈ gπ−1(b). Then y ∈ g(E) ⊂ E ′. Moreover, wehave that

π′(y) = π′(g(x)) = f(π(x)) = f(b).

It follows that y ∈ π−1(f(b)).

Remark 2.1.1. (compare Definition 1.1.2.) A fibre bundle (E,B, π, F ) istrivial if and only if it is isomorphic to the product (B×F,B, proj1, F ). Thefibre bundle isomorphism (g, f) gives us a diffeomorphism g : E → B × F .For the converse we note that we can take id and the diffeomorphism φ ofthe global trivialisation to obtain a fibre bundle isomorphism (φ, id).

2.2 Vector Bundles

Definition 2.2.1 (Vector bundle morphism). Let V = (E,B, π, F ) and W =(E ′, B′, π′, F ′) be vector bundles. Then a vector bundle morphism betweenV and W is a pair (g, f) of smooth maps

g : E → E ′, f : B → B′,

such that π′ ◦ g = f ◦ π and the map

π−1(b) 7→ (π′)−1(f(b)),

is linear. See the diagram below.

E E ′

B B′

g

π π′

f

If a vector bundle morphism has an inverse which is also a vector bundlemorphism, we speak about a vector bundle isomorphism.

Remark 2.2.1. Just like every vector bundle is a fibre bundle, every vectorbundle morphism is a fibre bundle morphism. The extra requirements on avector bundle morphism are such that the linear structure is preserved.

Theorem 2.2.1. TM is isomorphic to T ∗M .

6

Proof. Let 〈·, ·〉x be a Riemannian metric on M . We construct a vectorbundle isomorphism (g, f). TM and T ∗M have the same base space, so welet f = id. For (x, v) ∈ TM, the map

g(x, v) = (x, 〈v, ·〉x)

is a diffeomorphism.

Theorem 2.2.2 (Triviality of Vector Bundles). A vector bundle (E,B, π,Rn)is trivial if and only if there exist n linearly independent sections s1, . . . , sn,sj : B → E such that {s1, . . . , sn} is a basis for the fibre π−1(p) for everyp ∈ B.

Proof. Suppose (E,B, π,Rn) has n sections {s1, . . . , sn} that are everywhereindependent. Then let g : B × Rn → E be the map

(p, x1, . . . , xn) 7→ x1s1 + . . .+ xnsn.

The diagram below now describes a vector bundle isomorphism.

B × Rn E

B B

g

proj1 π

id

Remark 2.2.2. Since the zero vector does not span anything, the sections{s1, . . . , sn} of Theorem 2.2.2 have to be non-zero everywhere.

2.3 Principal Bundles

Definition 2.3.1 (Principal Bundle Morphism). Let V = (E,B, π, F ) andW = (E ′, B′, π′, F ) be two principal bundles. Then a principal bundle mor-phism from V to W is a fibre bundle morphism (g, f) with g : E → E ′ andf : B → B′ such that g is F -equivariant. That is, given any h ∈ F we havethat

g(x) ? h = g(x ∗ h),

where ? is the action of F on E ′ and ∗ is the action of F on E.

Theorem 2.3.1 (Triviality of Principal Bundles). A principal bundle (E,B, π, F )is trivial if and only if there exists a smooth section s : B → E.

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Proof. Suppose that s : B → E is a smooth section for the principal bundle(E,B, π, F ). Denote the action of F on E by ?. We will show that

(b, g)↔ g ? s(b)

defines a diffeomorphism B × F → E.If g ? s(b) = g′ ? s(b′) then s(b′) is in the orbit of s(b). The orbits of F inE are identified with B, so s(b′) and s(b) then have the same base pointb = b′. This shows that the identification given above is injective. Because ?is transitive, it is also surjective.Given x ∈ E we can recover (b, g) explicitly as (b, g) = π(g−1 ? x). The fourmaps s, ?, π, and g 7→ g−1 are smooth, so we have constructed a diffeomor-phism.

Remark 2.3.1. A vector space contains a zero element. Therefore, a vectorbundle always has a global section, namely the zero section σ(x) = (x, 0). Itfollows through Theorem 2.3.1 that any vector bundle that is also a principalbundle must be trivial.

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Chapter 3

Examples of Bundles and TheirTriviality

In this chapter we consider the tangent bundles TS1 and TS2, showing thatTS1 is trivial while TS2 is not.We then construct the Mobius Band as an example of a vector bundle, theKlein Bottle as an example of a line bundle and the Hopf Fibration as anexample of a non-trivial principal circle bundle.

3.1 Tangent Bundles of S1 and S2

Example 3.1.1 (TS1 is Trivial). We will show that the map F : TS1 →S1 × R given by

F (x, v) = (x, v/ix)

is a diffeomorphism.We first parametrise S1 by t 7→ eit for t ∈ R. Using this parametrisation, thetangent space at x ∈ S1 is given by

TxS1 =[ieit]R .

Now let F be as above. For (x, v) = (eitx , αieitx) ∈ TS1 we have

F (x, v) = F (eitx , αieitx) = (eitx , αieitx/ieitx) = (x, α) ∈ S1 × R.

Hence, F is a map from TS1 into S1×R. Moreover, F is smooth since x 6= 0on S1. Finally, the inverse map F−1 : S1 × R→ TS1 is given by

F−1(x, α) = (x, αix)

and is also smooth.

9

Figure 3.1: The tangent bundle TS1. The vertical lines representthe tangent spaces attached disjointly to S1, represented by theblack circle.

Remark 3.1.1. We can view TS1 as a principal bundle with fibre (R,+).Specifically, ? defined for β ∈ R and (x, v) ∈ TS1 by

(x, v) ? β = (x, v + βix)

provides a natural group action of (R,+) on TS1. The action ? is free on thefibre since v+β1e

it = v+β2eit implies β1 = β2. Moreover, ? is also transitive;

See Figure 3.2.

Figure 3.2: The action of β adds a tangent vector of length β.Since we can reach any vector in the tangent space at eit, theaction is transitive.

It follows from Remark 2.3.1 that TS1 is trivial.

Remark 3.1.2. We can view TS1 as a vector bundle. The map

x 7→ ix,

is a global section. It follows from Theorem 2.2.2 that TS1 is trivial.

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Example 3.1.2 (TS2). The Hairy Ball Theorem (see for example Theorem10.15 of [14]) states that a nowhere zero section on TS2 does not exist. Itfollows from Remark 2.2.2 that the tangent bundle TS2 is non-trivial.

Figure 3.3: The tangent space of the two-sphere.

Remark 3.1.3. It was shown by Adams[1] that TSn is trivial only for n =1, 3, 7.

3.2 The Mobius Band

Construction. The cylinder S1 × R is trivial. We use the cylinder to con-struct a non-trivial vector bundle called the Mobius Band.

Let ∼ be the equivalence relation (p, x) ∼ (p + 2π,−x) on S1 × R. TheMobius band Mo is defined as the quotient space

Mo = (S1 × R)/ ∼ .

See Figure 3.4.

Figure 3.4: Construction of the Mobius band. Points on theleft side of the rectangle are identified, through reflection in thedashed line, with points on the right side of the rectangle.

Vector Bundle. The indentification ∼ has no effect on subsets of Moshorter than 2π. See Figure 3.5. Furthermore, the fibre R is a vectorspace and the fibres are isomorphic to R.

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1

2

Figure 3.5: Rectangle 1 is a subset of Mo. Rectangle 2 is asubset of S1 × R.

Not Trivial. By definition, sections σ : S1 →Mo are of the form

σ(x) = (x, f(x)),

where f : S1 → R. For σ to be continuous, f has to satisfy f(2π) = −f(0).See Figure 3.6. By the intermediate value theorem, there exists some ζ ∈[0, 2π] such that f(ζ) = 0. See Figure 3.6. It follows from Remark 2.2.2that the Mobius Band is not trivial.

Figure 3.6: Every section of the Mobius Band intersects the zerosection, which is represented by the dashed line.

Not Orientable.

Theorem 3.2.1. If two vector bundles are isomorphic and one of them isorientable, then so is the other.

Proof. See for example paragraph 38 of [6].

The Mobius band is not orientable. See Figure 3.7.

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Figure 3.7: The identification ∼ is orientation-reversing.

By contrast, the cylinder S1×R is orientable. It follows from Theorem 3.2.1that Mo is not homeomorphic to the cylinder, and hence not trivial.

3.3 The Klein Bottle

Construction. The Torus S1 × S1 is trivial. We will use the Torus to con-struct a non-trivial circle bundle called the Klein Bottle.

Let ∼2 be the equivalence relation

(0, y) ∼2 (2π, 2π − y),

(x, 0) ∼2 (x, 2π).(3.1)

We define the Klein bottle Kl as the product space

Kl = S1 × S1/ ∼2 .

See Figure 3.8.

Figure 3.8: By identifying the top of a rectangle with its bottom,a cylinder is obtained. If we identify points on the sides with theirreflections in the dashed line we obtain a Klein Bottle.

We claim that the Klein Bottle is a circle bundle. The reasoning is the sameas for the Mobius Band.

Not Orientable and Not Trivial. It was shown in Section 3.2 that theMobius band is not orientable. The Mobius band is an open subset of theKlein Bottle, see Figure 3.9. It follows that the Klein bottle is not orientable.

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Figure 3.9: The Mobius strip is an open subset of the KleinBottle.

Because the Torus S1 × S1 is orientable we can conclude from Theorem3.2.1 that the Klein Bottle is not homeomorphic to the Torus, and hencenot trivial.

3.4 The Hopf Fibration

The Hopf Fibration (S3, S2, πh,S1) is a way to view S3 as a principal circlebundle over S2.Construction. We identify R4 with C2. We can now describe S3 as

S3 ={

(z1, z2) ∈ C2∣∣ |z1|2 + |z2|2 = 1

}.

We also identify R3 with C× R to describe S2 as

S2 ={

(z, x) ∈ C× R∣∣ |z|2 + x2 = 1

}.

The Hopf projection πh : S3 → S2 is given by

πh(z1, z2) = (2z1z∗2 , |z1|2 − |z2|2),

where z∗2 is the complex conjugate of z2. We note that πh indeed maps intoS2 because for (z1, z2) ∈ S3 we have

|πh(z1, z2)| = (2z1z∗2)(2z1z

∗2)∗ + (|z1|2 − |z2|2)2

= 4|z2|2|z1|2 + |z1|4 + |z2|4 − 2|z1|2|z2|2

= (|z1|2 + |z2|2)2 = 1.

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Principal Circle Bundle. We claim that

(z1, z2) ? λ = (λz1, λz2)

provides an action of S1 3 λ on S3 3 (z1, z2). To see this, write

z1 = r1eiθ1 z2 = r2e

iθ2 ,

where r21 + r22 = 1. We have for λ ∈ S1 that

πh(λ(z1, z2)) = (2z1z∗2 , |z1|2 − |z2|2) = πh(z1, z2),

since the factor λ cancels in both components of πh. Conversely, if (w1, w2) =(r3e

iθ3 , r4eiθ4) ∈ S3 is such that

πh(z1, z2) = πh(w1, w2),

then r1r2 = r3r4 from the equality in the first coordinate and r21−r22 = r23−r24from the equality in the second coordinate. It follows that r1 = r3 andr2 = r4. Since z1z

∗2 = w1w

∗2 we also have that ei(θ1−θ2) = ei(θ3−θ4). It follows

that (w1, w2) = (λz1, λz2) for some λ ∈ S3.This makes the Hopf Fibration a principal bundle with fibre S1.

Triviality. Let Π1(X) denote the fundamental group of X. Theorem A.2.3says that if (E,B, π, F ) is trivial, then Π1(E) ∼= Π1(B) × Π1(F ). We willshow that Π1(S3) � Π1(S2)× Π1(S1).

Figure 3.10: Stereographic projection of S1 on R. In general, wecan project Sn onto Rn.

Lemma 3.4.1. Π1(S3) is the trivial group.

Proof. We use Seifert-van Kampen, Theorem A.2.4.

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Let x and y be any two antipodal points on S3. We define U and V by

U = S3 − {x} and V = S3 − {y}.

We can project U onto R3 through stereographic projection from x.

This provides a diffeomorphism between U and R3. Because Π1(R3) is trivialit follows that Π1(U) is trivial, see Theorem A.2.2. The same reasoningshows that Π1(V ) is trivial.

We now show that U ∩V = S3−{x, y} is path-connected. We stereograph-ically project S3 into R3 at x. The point x is already missing from thisprojection. Removing y also removes a single point from R3. We can easilyconstruct paths in R3 that avoid this missing point.

It follows from Theorem A.2.4 that Π1(S3) is the trivial group.

Lemma 3.4.2. Π1(S2)× Π1(S1) is not the trivial group.

Proof. A loop that winds around the circle once can not be continuouslydeformed to the constant loop. See Figure 3.11. This shows that Π1(S1) isnot trivial. It follows that Π1(S2)× Π1(S1) is not trivial.

Figure 3.11: It is impossible to continuously deform the loopdisplayed on the right to the constant loop displayed on the left.

Remark 3.4.1 (Π1(S1) = Z). The elements of Π1(S1) can be identified withthe number of times a loop wraps around the circle.

Since Π1(R3) is trivial and Π1(S2)×Π1(S1) is not trivial we have Π1(S3) �Π1(S2)× Π1(S1). We conclude that the Hopf fibration is not trivial.

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Chapter 4

Conclusion

In chapter 2 criteria for triviality of vector bundles and principal bundleswere given. In chapter 3 these were applied to several examples. It wasfound that TSn is trivial only for n = 1, 3, 7. The Mobius band was shownto be a non-trivial vector bundle, and the Klein bottle was shown to be anon-trivial circle bundle. Finally, the fundamental group was used to showthat the Hopf fibration is a non-trivial principal bundle.

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Appendix A

A.1 Group Actions

Definition A.1.1 (Right group action). If (G, ?) is a group and X is a set,then a right group action is a map σ : X × G → X satisfying the followingconditions.

1. If e is the identity of G, then σ(x, e) = x.

2. If g and h are elements of G, then

σ(σ(x, g), h) = σ(x, g ? h).

Remark A.1.1. It is common to write xg for σ(x, g). The conditions abovethen read 1) xe = x and 2) (xg)h = x(g ? h).

Definition A.1.2 (Free Action). Suppose (G, ?) is a group, and g and h areelements of G. A right group action of (G, ?) on X 3 x is free if xh = xgimplies that h = g.

Definition A.1.3 (Transitive Action). A group action ? of G on X is tran-sitive if for every pair x, y ∈ X there exists a g ∈ G such that x ? g = y.

A.2 The Fundamental Group

Definition A.2.1 (Loop). Let X be a topological space with x0 X. A loopat x0 is a continuous function f : [0, 1]→ X such that f(0) = x0 = f(1). SeeFigure A.1.

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Figure A.1: A loop at the point x0.

Definition A.2.2 (Homotopy). Let f : X → Y and g : X → Y be twocontinuous functions between to topological spaces X, Y . A homotopy h isa continuous function h : X × [0, 1]→ Y such that if x ∈ X, then

h(x, 0) = f(x) and h(x, 1) = g(x).

Two loops are called homotopy-equivalent if there exists a homotopy betweenthem.

Definition A.2.3 (The Fundamental Group). Let X be a topological spaceand x0 ∈ X a point. Let F be the set of all loops at x0. The fundamentalgroup of X at x0, denoted π1(X, x0) is the group

π1(X, x0) = F/h,

where h identifies two loops if they are homotopy-equivalent.

Remark A.2.1. The fundamental group Π1(X, x0) of a path-connected spaceX is independent of the base point x0.

Theorem A.2.1. Let X and Y be topological spaces with x0 ∈ X andy0 ∈ Y . Then

Π1(X × Y, (x0, y0)) = Π1(X, x0)× Π1(Y, y0),

where X × Y is the Cartesian product and Π1(X, x0)× Π2(Y, y0) is a directproduct of groups.

Theorem A.2.2 (Induced isomorphism). If X is homeomorphic to Y , thenΠ1(X) is isomorphic to Π1(Y ).

Theorem A.2.3. If (E,B, π, F ) is a trivial bundle, then Π1(E) ∼= Π1(B)×Π1(F ).

Proof. Combine Theorem A.2.1 and A.2.2.

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Theorem A.2.4 (Van Kampen). Let X = U1∪U2 be the union of two openand path-connected sets U1, U2 such that U1 ∩ U2 is path-connected. Leti12 : Π1(U1 ∩ U2) → Π1(U1) be the homomorphism induced by the inclusionU1 ∩ U2 ↪→ U1 and define i21 analogously. Then

Π1(X) ∼= Π1(U1) ∗ Π1(U1)/N,

where N is the normal subgroup generated by all elements of the formiαβ(ω)i−1αβ(ω) for ω ∈ Π1(U1 ∩ U2). Here ∗ denotes the free product.

Proof. See Theorem 1.20 of Hatcher [15].

Remark A.2.2. If both Π1(U1) = e1 and Π2(U2) = e2 are trivial, then Π1(U1)∗Π1(U2) contains only the reduced element e1e2. It then follows that Π1(X)is trivial without considering N .

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Bibliography

[1] J.F. Adams, On the Non-Existence of Elements of Hopf Invariant One.Annals of Mathematics, Second Series, Vol. 72, No. 1 (Jul., 1960), pp.20-104.

[2] Henk Broer, Meetkunde en Fysica. Epsilon Uitgaven, November 2013.

[3] Bert Mendelson, Introduction to Topology. Third edition, Dover Publica-tions, 1990.

[4] Manfredo Perdiggao Do Carmo, Differential Forms and Applications.First edition, Springer, 1994.

[5] Richard L. Bishop and Samual I. Goldberg, Dover Publications, 1980.Tensor Analysis on Manifolds.

[6] Norman Steenrod, The Topology of Fibre Bundles. Princeton UniversityPress, 1951

[7] David W Lyons, An elementary introduction to the Hopf fibration. Math-ematics Magazine 76(2), April 2003.

[8] Dale Husemoller, Fibre Bundles, Third Edition, Springer, 1991.

[9] Gert Vegter, Notes on Tangent spaces. Unpublished.

[10] Harold Simmons, An introduction to Category Theory. First edition,Cambridge University Press, November 2011

[11] Jimmie Lawson, Lecture notes on Differential Geometry, Spring 2006.Unpublished.

[12] Jimmie Lawson, The Tangent Bundle. Unpublished.

[13] Alexei Kovalev, Lecture notes on Differential Geometry. Unpublished.

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[14] Jeffrey M. Lee, Manifolds and Differential Geometry. American Mathe-matical Society, 2009.

[15] Allen Hatcher, Algebraic Topology. Cambridge University Press, Decem-ber 2001.

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