Topological invariants of principal G-bundles with singularities

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Topological Invariants of Principal

G-bundles with Singularities

Fabi´an Antonio Arias Amaya

Thesis submitted to the Departamento de Matem´aticas for the degree of PhD in Mathematics

Advisor:

Mikhail Malakhaltsev

Universidad de Los Andes

Facultad de Ciencias

Departamento de Matem´

aticas

Bogot´

a

2016

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Topological Invariants of Principal

G-bundles with Singularities

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Dedicatory

This thesis is dedicated to my wife and my children for their long wait during my absence , and my

parents and brothers for their great support at all time.

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Acknowledgement

I thank ...

God for having joined my family all the time,

my advisor for his great support during the development of this work,

Universidad de los Andes, and specially the Departamento de Matem´aticas for the financial support and its teaching,

Universidad Tecn´ologica de Bol´ıvar for the financial support.

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Abstract

Arias Amaya, F. A.Topological Invariants of Principal G-bundles with Singularities. 2016. 88 pages. Thesis of doctorate. Departamento de Matem´aticas, Universidad de los Andes, Bogot´a Colombia, 2016.

In this work we present an introduction to the theory of principal bundles with singularities, i.e. principal bundles which reduce to a closed subgroup of the structure group outside of a closed subset of the base space.

In the first part we give the definition of these new structures, and we define morphisms between them. Then, we prove that these structures and their morphisms form a category which contains the bundles induced by transversal maps, and we use the obstruction theory to construct characteristic class of principal bundles with singularities.

In the second part we present a version of Gauss-Bonnet-Hopf-Poincar´e Formula for locally trivial fiber bundles over 2-dimensional manifolds, and we prove that this result generalizes the classical Gauss-Bonnet theorem. Then, we define branched sections of locally trivial fiber bundles, index of a singular point of a branched section, and give examples of its calculation, in particular for branched sections defined by binary differential equations. We also define a resolution of singularities of a branched section, and prove an analog of Gauss-Bonnet-Hopf-Poincar´e formula for the branched sec-tions admitting a resolution in case the manifold M has dimension two.

Keywords: Principal bundles with singularities, obstruction theory, branched section, index of singular point, binary differential equations, curvature.

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Notations

Rn n-dimensional Euclidian space

Dn n-disk in the Euclidian spaceRn

Sn−1 n−1-sphere in The Euclidian space Rn

|x| the norm of xin _Rn

ϕ|X Restriction of ϕtoX

p·g right action of g onp

L(M) the bundle of linear frames on M

O(M) the bundle of positive oriented orthonormal frames

SO(M) the bundle of orthonormal frames on M

pr1 :X×Y →X the canonical projection over X

pr2 :X×Y →Y the canonical projection over Y

I the interval [0,1]

In n-dimensional cube [0,1]n

∂_In _{boundary of the} _n_{-dimensional cube}

We assume all the manifolds to be smooth, and all the maps between manifolds to be smooth, as well.

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Introduction.

The classical theory ofG-structures deals with regular geometrical structures, this means there hold some nondegeneracy conditions, as for example for Riemannian or symplectic structures. However when considering various geometrical problems there naturally arise “singular points”, where the regularity conditions fail to be true. For example, a vector field can have zeros, a symplectic form ω

restricted to a submanifold has points wheredet(ω) = 0, the lift of a Riemannian metric to the bundle of orthonormal frames degenerates at some points, the curvature lines of a surface are not defined in umbilic points, etc., therefore it is natural to study the singularities of geometrical structures and their differential and topological invariants.

In terms of principal bundles, a singularity of a geometrical structure can be described as follows. Assume that ¯P → M is a ¯G-structure and Σ ⊂ M is a subset. We say that this bundle is a G -structure with singularities Σ if ¯P reduces to a G-principal subbundle, where G ⊂ G¯, over M \Σ. For example, if V is a vector field on a two-dimensional oriented Riemannian manifold M, and Σ is the set of zeros of V, then the bundle SO(M) of the positively oriented orthonormal frames of M

reduces to the trivial subgroup {e} ⊂SO(2) overM\Σ. Another natural example is that, if M is a surface in the three-dimensional Euclidean space, the bundle SO(M) reduces to a discrete subgroup over M \Σ, where Σ is the set of umbilic points of M.

In this thesis we find topological invariants of singularities of principal bundles and prove some analogs of the Gauss-Bonnet and the Hopf-Poincar´e theorems.

The thesis consists of three chapters. In the first chapter we give a brief description of the main definitions and results used in the work. This chapter begins with the definition of the homotopy groups, then we introduce an action of the fundamental group on the higher homotopy groups which allows us to give a definition of simple. The next section deals with CW-complexes, cohomology of

CW-complexes and the extension problem for sections of locally trivial fiber bundles. After that, we give a brief review of principal bundles. At the end we provide a review of the obstruction theory.

In the second chapter we give the definitions of principal G-bundles with singularities and

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phisms between them, and show that these objects and morphisms form a category. This chapter also contains a definition of characteristic classes of principal G-bundles with singularities. After that, we construct characteristic classes of these structures which allow us to associate an index to each singular point. The definitions and constructions are illustrated with examples. The main result of this chapter is Theorem 2.2.7.

In the third chapter we find a generalization of the Gauss-Bonnet-Hopf-Poincar´e formula for sections with singularities of locally trivial bundles, and we prove that this result reduces to the classical one, if the fiber bundle is the tangent bundle of an oriented surface. Then we apply this result to principal bundles with singularities. The chapter finishes with a generalization of the previous result to branched sections of locally trivial fiber bundles, in particular to the branched section determined by a binary differential form. The main results of this chapter are Theorem 3.2.7 and Theorem 3.8.2.

The results of this work we reported at the following scientific events: 1. Congreso Colombiano de Matemáticas, Manizales, Colombia, 2015. 2. XI Encuentro Internacional de Matemáticas, Barranquilla, 2015. 3. Congreso Latinoamericano de Matemáticas, Barranquilla, 2016.

The results of the thesis were partially published in [10], [11]. Also the ideas close to the main lines of the work were used to define topological index of a singular point of 3-web in [12]

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Chapter 1 Preliminaries.

1.1 Homotopy Groups.

This section contains the results on homotopy theory which will use throughout the work.

1.1.1 Definition and examples

In this section we will consider topological spaces and continuous maps. For details we refer the reader to [21], [1].

Definition 1.1.1. Let X and Y be topological spaces, A⊂ X, and let f :X →Y and g : X →Y

be two continuous maps. We say that f and g are homotopic relative to A if f|A = g|A, and there

exists a continuous map H:X×I →Y such thatH(x,0) =f,H(x,1) = g, andf|A=H(., t) =g|A

for all t ∈I.

The relative homotopy is an equivalence relation in the set [(X, A),(Y, B)] of continuous maps

f :X→Y such that f(A)⊂B, and its equivalence classes are called relative homotopy classes. By [f] we denote the relative homotopy class off, and by πn(X, x0), the set [(In, ∂In),(X, x0)].

The set πn(X, x0) is endowed by a binary operation defined as follows: For [f],[g] ∈ πn(X, x0), we

set [f]·[g] = [f ·g], where the mapf ·g ∈[(_In_{, ∂}

In),(X, x0)] is given by

(f ·g)(t1,· · ·, tn) =







f(2t1, t2,· · · , tn) if 0≤t1 ≤1/2,

g(2t1 −1, t2,· · · , tn) if 1/2≤t1 ≤1.

Proposition 1.1.1([1], page 340). The pair(πn(X, x0),·)is a group which is commutative forn >1.

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1.1.2 Action of

π

1

(

X, x

0

)

on

π

n

(

X, x

0

)

.

Since there exists a canonical homeomorphism between the pairs (_Dn_,

Sn−1) and (In, ∂In), the group

πn(X, x0) can be interpreted as the set of homotopy classes relative to Sn−1 of maps (Dn,Sn−1) →

(X, x0). By using this interpretation, now we define an action of π1(X, x0) on πn(X, x0).

Let X be a path connected topological space, and x0 ∈ X. For [γ] ∈ π1(X, x0), we define a map

γ∗ :πn(X, x0)→πn(X, x0) as follows. For [f]∈πn(X, x0) we set γ∗f :Dn →X by

(γ∗f)(u) =







f(2u) if 0≤ |u| ≤1/2

γ(2−2|u|), if 1/2≤ |u| ≤1,

where |u| denotes the norm ofu in _Rn+1_.

The homotopy class [γ∗f] is independent of the representatives of [γ] and [f]. Moreover, for all [γ] and [η] in π1(X, x0), and [f]∈πn(X, x0) we have

[x∗₀f] = [f], and [(γ·η)∗f] = [γ∗(η∗f)],

where [x0] represents the identity element of π1(X, x0). Then the map π1(X, x0) ×πn(X, x0) →

πn(X, x0) given by ([γ],[f])7→[γ∗f] defines an action of π1(X, x0) on the setπn(X, x0).

Remark 1.1.1. If the action of π1(X, x0) on πn(X, x0) is trivial, we call X ann-simple space. When

a space X is n-simple, each map _Sn −→ X determines an element of πn(X, x0). If a space X is

n-simple for alln, X is called a simple space. It follows from the definition of this action that if X

is a simple space, then π1(X) is abelian.

When the topological space X is a Lie group, or a quotient of Lie groups, we have the following results about their homotopy groups.

Theorem 1.1.2 ([21], 16.9). Let Gbe a Lie group. Then π1(G) acts trivially onπn(G) for alln≥0.

Therefore, G is a simple space.

Proposition 1.1.3([21], 16.11). If G¯ is a Lie group, and Gis a closed connected Lie subgroup, then ¯

G/G is simple. Hence, for any n, a map _Sn→G/G¯ represents an element of πn( ¯G/G).

1.2 CW-Complexes

In this section we present the definition ofCW-complex, and some results concerningCW-complexes. Definition 1.2.1. Let X be a topological space. A q-cell in X is a continuous map eq _:

Dq → X

such that eq_|

(Dq\Sq−1) is a homeomorphism onto its image. The map ∂eq = eq|Sq−1 : S

q−1 _→ _X _is

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1.2. CW-COMPLEXES 3 Definition 1.2.2. Let X be a topological space and let A⊂ X be a closed subspace. We say X is obtained from A by adjoining q-cells {e_iq|i∈I} inX if the following conditions hold true:

1. ∂eq_i(_Sq−1₎_{, i}_∈_I _{is contained in}_A _and _eq

i(Dq\Sq

−1₎_∩_eq

j(Dq\Sq

−1_), _i₆₌_j _{is empty.}

2. A subset B ⊂ X is closed if and only if B∩A closed in A and B∩eq_i(_Dq_\

Sq−1) is closed in

eq_i(_Dq_{) for all} _q_.

3. X =A∪ieqi(Dq\Sq

−1_).

Definition 1.2.3. LetX be a topological space andAbe a closed subspace ofX. The pair (X, A) is called a relative CW-complex if there is a sequence of closed topological subspaces {Xq}q≥−1 which

satisfy the following properties:

1. X−1 =A and Xq+1 is obtained from Xq by adjoining (q+ 1)-cells,

2. X =∪q≥−1Xq.

3. B ⊂X is closed in X if and only if B∩Xq is closed in Xq for all q.

The relative CW-complex (X, φ) is called a CW-complex. If X is a CW-complex and X = Xq for

someq, we callqthedimension ofX. ACW-complex is called afinite if has only finitely many cells. A subcomplex of a CW-complex X is a subspace A ⊂ X which is a union of cells of X. Moreover, we say that the relative CW-complex (X, A) is a CW-pair if A is a CW-subcomplex ofX.

Remark 1.2.1. Let B and F be finite CW-complexes, then the product B ×F is a CW-complex. Indeed, if ek _:

Dk→B is a k-cell inB, and el is a l-cell inF, then the product Dk×Dl→B×F of

these maps gives a (k+l)-cell _Dk+l _→_B_×_F _in_B_×_F _{which we denote by} _ek_×_el_.

Proposition 1.2.1. Let E, B and F CW-complexes, and let π : E → B be a fiber bundle with typical fiber F. Then E admits a CW-complex structure with cells ek_×_el_{, where} _ek _and _el _{are cells}

on B and F, respectively.

Proof. Let ek _:

Dk →B be a k-cell inB. Since Dk is contractible, the induced bundle (ek)∗E →Dk

is trivial. We choose a trivialization ϕk :Dk×F →(ek)∗E and consider the following commutative

diagram

Dk×F

ϕk //

pr1

(ek)∗E ¯ek // pr1 E π

Dk Id //Dk e

k

/

/_B.

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where ¯ek_{: (}_ek₎∗_E _→_E _{is the projection over} _E_{. Since} _ek_|

int(Dk) is a homeomorphism onto its image,

and the restriction ¯ek|_pr−1

1 (int(Dk)) is also a homeomorphism, for any l-cell e l _:

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a map ek+l _:

Dk ×Dl → E given by the composition Dk×Dl (id,e

l₎

−→ _Dk_×_F ϕk

→ (ek₎∗_E _→¯ek _E _whose restriction toint(_Dk₎_×_int₍

Dl) is a homeomorphism. Therefore ek+lis a (k+l)-cell of E. We denote

it by ek_×_el_{. Now, we show that the collection} _{_ek_×_el _: _ek _{is a cell in} _{B, e}l _{is a cell in}_F_} _{of all}

(k+l)-cells constructed in this way gives a CW-decomposition ofE. To do this, it is sufficient to show that E =∪k,l(ek×el)(int(Dk)×int(Dl)). Let x∈ E, then π(x) belongs to the interior of a unique

cell inB, so there exists a uniquey∈int(_Dt) for some non-negative integer tsuch thatet(y) =π(x). Moreover, since ¯et_◦_ϕ

t : Dt×F → E is a homeomorphism over int(Dt×F), there exists a unique z in the interior of some cell of F such that ¯et_◦_ϕ

t(y, z) =x. Therefore, there exists a nonnegative

integer s and u ∈ _Ds _{such that} _et _× _es₍_{y, z}_{) =} _x_{. Therefore,} _x _∈ ₍_et _×_es₎₍_int₍

Dt)× int(Ds)).

Finally, we need to prove that ∂(ek × el) ⊂ Ek+l−1, and to do this, it is enough to note that

∂(_Dk×_Dl_{) =} _∂

Dk×Dl∪Dk×∂Dl.

Definition 1.2.4. A continuous map f : X → Y between CW-complexes X and Y is called a cellular map if f takes eachk-skeleton Xk of X onto thek-skeleton Yk of Y for each k.

Theorem 1.2.2 ([3], page 44). Every continuous map between CW-complexes is homotopic to a cellular map.

1.2.1 Homology and Cohomology of

CW

-complexes.

In this section we present the definition and results concerning the homology and cohomology of

CW-complexes. For details we refer the reader to [1] and [16].

Singular homology.

Definition 1.2.5. The standard q-simplex ∆q _{is defined by}

∆q ={(t0,· · · , tq)∈Rq+1 :ti ≥0,

q

X

i=0

ti = 1}.

The face maps are the functions fm

q : ∆q →∆q+1 defined by

f_qm(t0,· · · , tq) = (t0,· · ·, tm,0, tm+1,· · ·, tq). (1.2)

A singular q-simplex of a topological space X is a continuous map σ : ∆q →X.

Let G be an abelian group. We denote by Sq(X;Z) the free abelian group generated by the

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1.2. CW-COMPLEXES 5 differential ∂q : Sq(X;G) → Sq−1(X;G) to be the homomorphism defined by its action on singular

simplices σ by

∂q(σ) = q

X

m=0

(−1)m_σ_◦_fm

q . (1.3)

Thus, on a chain

l

P

i=1

riσi ∈Sq(X;G), the differential∂ is given by

∂(

l

X

i=1

riσi) =

l X i=1 ri( q X m=0

(−1)mσi◦f_qm). (1.4)

From this definition it follows that ∂2 = 0. ByS∗(X;G) we denote the chain complex (Sq(X;G), ∂), and it is called thesingular chain complex with coefficients inG. Given a continuous mapf :X →Y

one obtains an induced chain map of complexes f∗ : S∗(X;G) → S∗(Y;G) defined on a singular simplex σ byf∗(σ) = f◦σ.

Definition 1.2.6. The homologyH∗(X;G) of the singular chain complexS∗(X;G) is defined by

Hq(X;G) =

ker(∂ :Sq(X;G)→Sq−1(X;G))

Im(∂ :Sq+1(X;G)→Sq(X;G))

. (1.5)

The group H∗(X, G) is called the singular homology of X with coefficients in G. When G =Z, we

omit the reference to G.

Definition 1.2.7. Let X be a topological space and A ⊂ X be a subspace of X. The relative singular chain complex of the pair (X, A) with coefficients in G is defined by

Sq(X, A;G) =

Sq(X;G) Sq(A;G)

. (1.6)

Remark 1.2.2. Sq(X, A;G) is a free G-module. A basis element of Sq(X, A;G) is represented by a

singular q-simplex in X whose image is not contained in A. Moreover, one obtains a commutative diagram with exact rows

0 //Sq(A;G) //

∂

Sq(X;G) //

∂

Sq(X, A;G) //

∂

0

0 //Sq−1(A;G) //Sq−1(X;G) //Sq−1(X, A;G) //0

, (1.7)

where ∂ :Sq(X, A;G)→Sq−1(X, A;G) is the map induced by ∂ :Sq(X;G)→Sq−1(X;G).

The complex (Sq(X, A;G), ∂) is called thesingular chain complex for the pair (X, A) with

coeffi-cients in G. Its homology is defined by

Hq(X, A;G) =

ker(∂ :Sq(X, A;G)→Sq−1(X, A;G))

Im(∂ :Sq+1(X, A;G)→Sq(X, A;G))

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Lemma 1.2.3 ([1], page 137). If X is a CW-complex, then

1. Hq(Xn, Xn−1;G) = 0 for q 6= n, and it is free abelian for q = n, with a basis in one-to-one

correspondence with the n-cells of X.

2. Hq(Xn;G) = 0 for q > n. In particular, if X is finite-dimensional then Hq(X;G) = 0 for q >dimX.

3. The inclusion i:Xn →X induces an isomorphism i∗ :Hq(Xn;G)→Hq(X;G) if q < n.

Remark 1.2.3. (see [1]) From the long exact sequences for the pairs (Xq+1, Xq), (Xq, Xq−1) and

(Xq−1, Xq−2) one obtains the following sequence

· · · →Hq+1(Xq+1, Xq;G) dq+1

→ Hq(Xq, Xq−1;G)

dq

→Hq−1(Xq−1, Xq−2;G)

dq−1

→ · · · . (1.9) Here dq :Hq(Xq, Xq−1;G)→Hq−1(Xq−1, Xq−2;G) is given by the composition

Hq(Xq, Xq−1;G)

∂q

→Hq−1(Xq−1, Xq−2;G)

i

→Hq−1(Xq−1, Xq−2;G). (1.10)

where∂qis the connection homomorphism in the sequence in singular homology for the pair (Xq, Xq−1)

and i is induced by the inclusion (Xq−1, φ)→(Xq−1, Xq−2)

We set Cq(X) = Hq(Xq, Xq−1;G). Thus, the long exact sequence (1.9) is written as follows

· · · →Cq+1(X;G)

∂q+1

→ Cq(X;G) ∂q

→Cq−1(X;G)

∂q−1

→ · · · , (1.11) The pair(C∗(X;G), ∂) is a chain complex. This complex is called the cellular chain complex of X, and its homology groups are called the cellular homology groups of X. By HCW

q (X;G) we denote

the q-th cellular homology group of X.

Remark 1.2.4. Since Hq(Xq, Xq−1;G) is free with a basis in one-to-one correspondence with the q

-cells of X, each of its elements can be thought as a linear combination of q-cells of X. Thus, the connection map ∂q :Cq(X;G)→Cq−1(X;G) is given by the equation

∂q(σ q i) =

X

j dije

q−1

j , (1.12)

where the sum is over all (q−1)–cellse_jq−1 ofXanddij is the degree of the mapSq

−1

i →Xq−1 →Sq

−1

j ,

that is the composition of the attaching map of eq_i with the quotient map collapsing Xq−1\eq

−1

j to

a point.

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1.3. FIBER BUNDLES 7 The cellular cohomology groups of a CW-complex X are the algebraic duals of the cellular homology groups of X. The cellular cohomology chain complex ofX and its cohomology group are given by the following definition.

Definition 1.2.8. Let X be a CW-complex and (C∗(X;G), ∂) be the cellular cochain complex of

X. Define the cellular cohomology groups with coefficients in the abelian group G by the formula

Hq(X;G) = ker(δ :Hom(Cq(X;G), G)→Hom(Cq+1(X;G), G))

im(δ :Hom(Cq−1(X;G), G)→Hom(Cq(X;G), G)) ,

where δ is defined by δ(f)(σ) =f(∂σ). Remark 1.2.5. Let Cq₍_X_;_G_{) =} _Hom₍_C

q(X;G), G). From the definition of δ follows that δ2 = 0,

and so (C∗(X;G), δ) is a complex. Therefore, Hq(X;G) is well defined. This complex is called the cellular cochain complex of X.

1.3 Fiber Bundles

In this section we will consider fiber bundles and smooth maps. For details we refer the reader to [24], [6], [7], [26].

Definition 1.3.1. Let E, B, F be manifolds. A locally trivial fiber bundle over a manifoldM with fiber F is a map π : E → M with the property that for each point x ∈ M there exists an open neighborhood U of x in M and a diffeomorphism ϕ : π−1(U) → U ×F such that the following diagram commutes

U ×F pr1

{

U oo π π−1₍_U₎

ϕ

O

O (1.13)

Remark 1.3.1. Terminology and definitions

1. The space E is called the total space, M the base space, F the typical fiber, π the projection, and for eachx∈M, the set Ex =π−1(x) thefiber overx.

2. The mappr1 :M ×F →M is a fiber bundle with typical fiberF called the trivial bundle.

3. The mapϕ:π−1₍_U₎_→_U_×_F _{is called a}_{local trivialization. From the definition 1.3.1 it follows}

that ϕ has the form ϕ(p) = (π(p), ψ(p)), for some map ψ : π−1₍_U₎ _→ _F_{. If} _ϕ

α : π−1(Uα) → Uα ×F, and ϕβ : π−1(Uβ) → Uβ ×F are trivializations such that Uα ∩Uβ 6= φ, then the composition ϕα◦ϕ−β1 :Uα∩Uβ ×F →Uα∩Uβ ×F is given by

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where gαβ :Uα∩Uβ → Dif f(F) is a smooth map called the transition function. In addition,

{Uα} is a trivializing open cover of M and gαβ take values in a subgroup G⊂ Dif f(F), then

the collection {gαβ :Uα∩Uβ →G} of transition functions satisfies the following relations: gαβ(x)gβγ(x)gγα(x) = e, for all x∈Uα∩Uβ∩Uγ 6=φ. (1.14)

The relations (1.14) are called thecocycle conditions.

4. A section of a fiber bundle π : E → M is a smooth map s : M → E such that π◦s = idM.

Note that for each x∈M,s(x)∈π−1₍_x_).

Definition 1.3.2. A morphism between two fiber bundles π : E → M and π0 : E0 → M0 is a pair ( ˜f , f), where ˜f :E →E0 and f :M →M0 such that the following diagram commutes

E f˜ // π

E0 π0

M f //M0.

(1.15)

Remark 1.3.2. Other definitions.

1. From the definition it follows that any morphism of fiber bundles maps fibers into fibers, and the map f :M →M0 is determined by ˜f.

2. A morphism ( ˜f , f)0 of fiber bundles is called anisomorphism if there exists another morphism (˜g, g) such that ˜f ◦˜g =idE0,g˜◦f˜=id_E.

3. A bundle π : E → M with typical fiber F is called trivial if it is isomorphic to the trivial bundle pr1 :M ×F →M.

4. A fiber bundlep: ˜M →M with a discrete fiber Γ is called acovering. An coveringp:M×Γ→

M is called a trivial covering. Example 1.3.1. Vector bundles

Definition 1.3.3. A fiber bundle π : E → M with typical fiber V, where V is a finite dimensional vector space, is called avector bundle if for eachx∈M the fiberπ−1(x) is a vector space, and an atlas exists such that for all trivializationsϕ:π−1(U)→U×V, the restriction ϕ:Ex → {x} ×V, x∈U

is an isomorphism of vector spaces.

Remark 1.3.3. Terminology and examples

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2. LetM be a n-dimensional manifold and letT M =tx∈MTxM. The map π:T M →M defined

by π(x, v) = x, where v ∈ TxM, is a vector bundle of rank n called the tangent bundle of M. The sections of the tangent bundle are called vectorf ields on M. When the manifold is a Riemannian manifold, one can consider the subbundle of the tangent bundle T M consisting of all unit tangent vectors. This bundle is called the unit tangent bundle, and we denote it by

S1T M.

3. Any vector bundle π:E →M has the zero section.

1.3.1 Principal Bundles.

Another important example of locally trivial fiber bundles is the principal bundle which we study in what follows.

Definition 1.3.4. A locally trivial fiber bundle π : P → M is called a principal G-bundle if the typical fiber is a Lie group G that acts freely on the right on the total space P in such a way that the base space is the orbit space of this action and each local trivialization from the atlas

ϕ : π−1₍_U₎ _→ _U _×_G _{is a} _G_{-equivariant map, where the action of} _G _on _U _×_G _{is given by right}

multiplication on G; (x, g1)·g2 = (x, g1g2).

Remark 1.3.4. 1. For each x∈ M, the fiber Px is diffeomorphic to thestructure group G via the

map θ :G→Px given by θ(g) = p·g, where π(p) = x.

2. We will also use the notation P(M, G) to refer to a principal G-bundle π : P → M with structure group G.

Definition 1.3.5. LetPi(Mi, Gi), i= 1,2, be principal bundles. A morphism ( ˜f , f) of fiber bundles

joint to a homomorphism of Lie groups ϕ : G1 → G2 is called a morphism of principal bundles if

˜

f(p·g) = ˜f(p)·ϕ(g).

Definition 1.3.6. Let ˜f :P1 →P2be a morphism of two principal bundlesP1(M, G1) andP2(M, G2)

with a common base space M. We say that the bundle P1(M1, G1) is a reduction of the structure

group G2 of P2(M2, G2) to G1 if f :P1 →P2 is an embedding, and f :M →M is the identity map.

Proposition 1.3.1. Letf˜:P1 →P2be a morphism of two principal bundlesP1(M, G)andP2(M, G).

If ϕ=idG and f =idM, then f is an isomorphism.

Proof. Let ˜f : P1 → P2 be a morphism of principal G-bundles with the same base space M. First

of all, we prove that ˜f is a bijection on the fibers. Let x1 ∈ M and p1, p2 ∈ π−11(x1) such that

˜

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p2 =p1·g. So, ˜f(p2) = ˜f(p1)·g = ˜f(p2)·g, and since the action of G onP2 is free, we obtain that

g = 1. Thusp1 =p2, and ˜f is injective. To prove the surjectivity let usp2 ∈π−21(x) for some x∈M.

Letp1 ∈π1−1(x) be any element. Since π2◦f =π1, one obtain that ˜f(p1)∈π2−1(x). Since the action

is transitive on the fibers, there exists g ∈ G such that ˜f(p1)·g = p2, so ˜f(p1 ·g) = p2, and ˜f is

surjective. Thus, we obtain that ˜f is bijective on the fibers, and therefore ˜f is bijective.

On the other hand, since π2 ◦f˜ = π1 and π1, π2 are submersions, for each point p ∈ P1, the

differential (df˜)p is injective, and since P1 and P2 have the same dimension, one obtain that (df˜)p is

an isomorphism. Thus, the inverse function theorem implies that ˜f is a local diffeomorphism. Since ˜

f is bijective and a local diffeomorphism, the map ˜f is a diffeomorphism, and hence the morphism ( ˜f , ϕ) is an isomorphism of principalG-bundles.

Remark 1.3.5. Proposition 1.3.1 holds true for any principal Gi-bundles Pi(Mi, Gi) whenever ϕ : G1 →G2 is an isomorphism of groups and the induced map f :M1 →M2 is a diffeomorphism.

Proposition 1.3.2. A principal G-bundle P(M, G) is trivial if and only if P admits a section. Proof. Suppose that P is isomorphic to a trivial bundle as a G-bundle, and let f :P →M ×G be an isomorphism. We define s:M →P by

s(x) =f−1(x, e) (1.16)

The map sis a section of P. In fact, sinceπ=pr1◦f, where pr1 :M×G→M is the projection on

M, we have that (π◦s)(x) = (π◦f−1₎₍_{x, e}_{) = (}_pr

1◦f◦f−1)(x, e) = x. Moreover, by definition, s is

smooth. Conversely, if s :M →P is a section of P, we set f :M ×G→P by the equation

f(x, g) = s(x)·g. (1.17)

As the action ofGonP is smooth, ands is also smooth, we have thatf is a smooth map. Moreover, it is easy to prove thatf is aG-equivariant map. So, by proposition 1.3.1, we only need to prove that

f is a morphism of principalG-bundles, but it follows from the definition off. In fact, if π:P →M

is the projection map, and pr1 :M ×G→M is the projection over M, then

(π◦f)(x, g) = π(s(x)·g) =π(s(x)) =x=pr1(x, g). (1.18)

Therefore,π◦f =pr1 and f is a morphism of principal G-bundles.

Remark 1.3.6. From the proof of proposition 1.3.2 it follows that any local trivializationϕ:π−1₍_U₎_→

U ×G determines a local section s : U → P|U which is defined by s(x) = ϕ−1(x, e). Moreover, is sα :Uα →P|Uα and sβ :Uβ →P|Uβ are local trivializations such that Uα∩Uβ 6=φ, then

sβ(x) =sα(x)·gαβ(x), (1.19)

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1.3. FIBER BUNDLES 11

1.3.2 Associated and induced bundles.

Let P(M, G) be a principal G-bundle. Suppose that G acts on the left on a manifold Y. Then G

acts on the left on the product manifold P ×Y as follows

g·(p, y) = (p·g−1, g·y).

The quotient space (P×Y)/G=:P×GY has a structure of differentiable manifold and the projection

πG : P ×GY → M given by πG([(p, y)]) = π(p) is a fiber bundle with fiber Y ([24], page 54). This

bundle is called the fiber bundle with fiber Y associated to P. In particular, since G acts on the left onG/H, where H is a closed Lie subgroup of G, one obtains the bundle P ×GG/H.

Proposition 1.3.3([24], page 57). LetP(M, G)be a principal bundle andH be a closed Lie subgroup of G. Then the space P/H is diffeomorphic to the space P ×GG/H.

Proposition 1.3.4 ([24], page 52). Let M be a manifold, {Uα} an open covering of M and G a Lie group. Then for each family of functions {gαβ : Uα ∩Uβ −→ G} which satisfies the relations 1.14

there exists a principal bundle P(M, G) whose transition functions are the functions gαβ.

Let P(M, G) be a principal bundle, {Uα} an open covering of M such that P|Uα is trivial with trivializations ψα and transition functions gαβ. LetN be a manifold and f :N −→M be a smooth

map. Then{f−1(Uα)}is an open covering ofN and the family of functions{gαβ◦f :f−1(Uα∩Uβ)−→

G} satisfies the relations 1.14. Then, by proposition 1.3.4 there exists a principal G-bundle over N

with transition functions gαβ ◦f. This bundle is called the pullback of P with respect to f. We

denote it by f∗P. The total space f∗P of the pullback of P(M, G) with respect to f can be written as follows

f∗P ={(x, p)∈N ×P :f(x) = π(p)}.

Moreover, if π1 and π2 are the restrictions of the projections of N ×P over N and P to f∗P, then

the following diagram

f∗P π2 // π1

P

π

N f //M

is commutative.

Proposition 1.3.5 ([24], page 60). The projection π1 : f∗P → N is a principal G-bundle, and

π2 : f∗P → P is a morphism of G-bundles. Moreover, the bundle f∗P is the unique bundle which

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Remark 1.3.7. Any section s :M → P of the principal G-bundle π : P → M determines a section

s∗ :N →f∗P of the induced bundle f∗P →N. Indeed, this section is given by

s∗(x) = (x, s(f(x))). (1.20) Theorem 1.3.6 ([24], page 57). Let P be a principal G-bundle and let H be a closed Lie subgroup of

G. Then Ghas a reduction to a principal H-bundle if and only if the associated bundle P/H admits

a section.

Proof. Suppose that P(M, G) is reducible to a closed subgroup H and let Q(M, H) be a reduced bundle with injection f : Q→ P. Let µ:P → P/H be the natural projection. First of all, we will prove that the mapµ◦f :Q→P/H is constant on the fibers ofQ. Ifu, v ∈π_Q−1(x) for somex∈M, we have v =u·h for some h∈H. So

(µ◦f)(v) = (µ◦f)(u·h) =µ(f(u)·h) = (µ)(f(u)) = (µ◦f)(u).

Since πQ:Q→Q/H =M is a quotient map, µ◦f induces a maps:M →P/H =E such that the

following diagram is commutative

Q µ◦f

x

x πQ

P/H oo s Q/H =M.

Then s◦πQ = µ◦f. Now we will show that s is a section of πE : P/H = E → M. Since µ◦f

and πQ are smooth, s is also smooth. Indeed, if σ : U ⊂ M → Q is a smooth local section of the

bundle πQ : Q → Q/H, then s|U = s◦idU = s◦(πQ ◦σ) = (s◦πQ)◦σ = (µ◦f)◦σ is smooth.

Moreover, since πQ is surjective, for any x ∈ M exists q ∈ Q such that πQ(q) = x we obtain that s(x) = s(πQ(q)) = (s◦πQ)(q) = (µ◦f)(q) and (πE◦s)(x) =πE((µ◦f)(q)) = (π◦f)(q) = π(f(q)) = πQ(q) = x. Conversely, given a section s : M → E, we consider the bundle ψ = (µ : P → P/H),

then the pullback bundles∗P →M is a principal H-bundle and the map ˜s :s∗P →P is a morphism of principal bundles. Therefore, s∗P is a reduced bundle of P.

1.4 The Bundle of Linear Frames.

Let M be a n-dimensional manifold. Let us consider the set

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1.5. THE EXTENSION PROBLEM. 13 Recall that for any isomorphism A : _Rn _→

Rn the composition u ◦A : Rn → TxM is also an

isomorphism for all u ∈L(M). This defines a right action of Aut(_Rn_{) on} _L₍_M_{). It is easy to verify}

that this action is free. Now, since each u∈L(M) is completely determined by its values on a basis of _Rn, we can identify each element of L(M) with an ordered basis ofTxM. Therefore, we can write

L(M) ={(X1,· · · , Xn)x | (X1,· · · , Xn)xis a basis of TxM atx, x∈M}.

Now, we have a natural projection π : L(M) → M given by π((X1,· · ·, Xn)x) = x. By using the

canonical isomorphism Aut(_Rn)∼GL(n) we can see eachA∈Aut(_Rn) as an invertible matrix (aj_i). Therefore, the action above described can be written as follows:

(X1,· · · , Xn)x·(aji) = (Y1,· · · , Yn)x,

where Yi =ajiXj.

Proposition 1.4.1 ([24], page 55). The projection π:L(M)→M is a principal GL(n)-bundle. The bundle π :L(M)→ M is called the bundle of linear frames onM. Some properties of this bundle are the following.

1. The set of reductions of π : L(M) → M to the orthogonal group O(n) is in one to one correspondence with the set of Riemannian metric on M. We denote by O(M) the total space of the reduced bundle, and we call it the bundle of orthogonal frames.

2. If M is an oriented Riemannian n-dimensional manifold, the bundle π : L(M) → M reduces to the special orthogonal group SO(n). In this case the reduced bundle is called thebundle of positively oriented orthonormal frames, and it is denoted by SO(M).

3. Let us consider the standard left action of the group GL(n) on the manifold _Rn. The bundle with fiber _Rn _{associated to the bundle} _L₍_M_{) is isomorphic to the tangent bundle of} _M_{. This}

isomorphism ˜f :L(M)×GL(n)T M is given by

˜

f((X1,· · · , Xn)x,(a1,· · ·, an)) =a1X1+· · ·+anXn.

1.5 The extension problem.

This section deals with the extension problem for sections of locally trivial fiber bundles over a finite

CW-complexB, and whose fiber is a manifold F which is simple as a topological space in the sense of observation 1.1.1.

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1.5.1 The extension problem for maps.

Let X and Y be topological spaces, A ⊂ X, and f : A → Y be a continuous map. The extension problem consists in determine if there exists a continuous map ˜f : X → Y such that the following diagram

X

˜

f

~

Y oo f A i

O

O .

is commutative. The map ˜f is called an extension of f.

This problem is closely related to the higher homotopy groups of topological spaces. An important result about this relation is the following

Proposition 1.5.1. Let X be arc connected space, and h : _Sn−1 → X be a continuous map. Then the following conditions are equivalent:

1. h is homotopic to a constant map, i.e, the class [f] is the zero element of the groupπn−1(X).

2. h extends to a continuous map k :_Dn _→_X_.

Proof. Suppose that h is homotopic to a constant map. Let c : _Sn−1 → X be a constant map homotopic to h and let H : _Sn−1 ×I → X be a homotopy between h and c. Consider the map

π :_Sn−1_×_I _→

Dn defined by

π(x, t) = (1−t)x. (1.21)

Thenπ is a continuous map, andπ is a closed map because_Sn−1×I is compact and_Dnis Hausdorff. Moreover, π is surjective because π(x/|x|,1− |x|) = x for every x 6= 0 and π(x,1) = 0. Therefore,

π is a quotient map. Note that π−1_{(0) =}

Sn−1 ×1, and π is otherwise injective. Since H|Sn−1×1 is

constant, then H is constant on the fibers of π, so H induces a continuous map k : _Dn _→ _X _such

that k◦π=H [see [17], page 142]. Hence,

k|_Sn−1 = (k◦π)|

Sn−1×0 =H|Sn−1×0 =h.

then k is a continuous extension of h.

Conversely, suppose that h extends to a map k : _Dn_→ _X_{, that is} _k_◦_i₌_h_{, where} _i _:

Sn−1 →Dn is

the inclusion, and let π be the map defined by the equation (1.21). We consider the mapH =k◦π :

Sn−1×I →X. Then

H|_Sn−1_×{₁_} = (k◦π)|

Sn−1×{1} =k◦i=h and H|Sn−1×{0} = (k◦π)|Sn−1×{1} =k|{0},

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1.5. THE EXTENSION PROBLEM. 15 If the pair (X, A) is a relative CW-complex, and Y is a simple path-connected space there exists an algorithm that allows us to solve the extension problem. To solve this problem one can use induction on the dimension of cells of X. Since the 0-skeleton is X0 ₌_A_{∪ {}_e0

i}, the map f can be

extended from A toX0 because any function is continuous at isolated points. Now we suppose that

f :A →Y has been extended to a map fk−1 :Xk−1 → Y for some k ≥1, and we wish to extend it

to a map fk :Xk → Y. Let ek : (Dk,Sk−1) → (Xk, Xk−1) be a k-cell of X. Note that f is defined

overek₍

Sk−1)⊂Xk−1, so the problem reduces to extendf toek(Dk\Sk−1). To visualize this problem

consider the following diagram

Sk−1 e

k / / i

ek(_Sk−1) f //

i Y id

Dk e

k

/

/_ek₍

Dk)

˜

f _/_/ Y.

(1.22)

From this diagram we can deduce that such an extension ˜f : ek₍

Dk) → Y exists if and only if the

composition f ◦ek_|

Sk−1 :S

k−1 _→ek _ek₍

Sk−1)→f Y extends to a map Dk e

k →ek₍

Dk)

¯

f

→Y, for each k-cell

ek_{. Since}_Y _{is a simple space, the composition} _f_◦_ek_|

Sk−1 defines an element of πk−1(Y) (see remark

1.1.1) that we denote byα(ek, f). Note thatα(ek, f) is zero if and only if the map f can be extended to a map fk : Xk → Y. The elements α(ek, f) allow us to define a cellular cochain which is an

obstruction to the extension of f.

Definition 1.5.1. The cellular cochain αf ∈Ck(X, A, πk−1(Y)) defined by the equation

αf(ek_i) =α(ek_i, f) (1.23) is called an obstruction cochain.

The most important properties of the cochain αf defined by the equation (1.23) are given in the

following theorem.

Theorem 1.5.2 ([16], page 169). Let (X, A) be a relative CW-complex, let k ≥ 1, and let Y be a connected simple space. Let f :Xk−1 →Y be a continuous map. Then

1. The cochain αf is a cocycle.

2. The cellular cocycle αf ∈ Ck(X, A, πk−1(Y)) vanishes if and only if fk−1 extends to a map

fk:Xk →Y.

3. The cohomology class [αf]∈ Hk(X, A;πk−1(Y)) vanishes if and only if the restriction f|Xk−2 : Xk−2 →Y extends to a map fk :Xk→Y.

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1.5.2 The extension problem for sections of fiber bundles.

In this subsection we apply the mean results of the section 1.5.1 to sections of locally trivial fiber bundles. A most detailled treatment about this topic can be found in [21].

LetB a manifold with a structure ofCW-complex,Gbe a connected Lie group, and letπ:E →B

be a fiber bundle with structure group G, and fiber F, where F is a manifold which is simple as a topological space. Let L ⊂ B be a subcomplex of B such that (B, L) is a relative CW-complex, and s : L → E be a section of E over L. The extension problem consists in determine if s can be extended to a section on B. As we did in section 1.5.1, s can be extended to a map s0 : B0 → E

by defining s(x), x ∈ B0 _\_L _{to be an arbitrary element of the fiber} _E

x. This map is a section

that extends s. Now we suppose that s have been extended to a section sk−1 :Bk−1 → E, and let

ek : (_Dk,_Sk−1) → (Bk, Bk−1) be a k-cell of B. Since sk−1 : Bk−1 → E is defined on ek(Sk−1), the

problem reduces to extend sk−1 from ek(Sk−1) to ek(Dk). Let us consider the following commutative

diagram

(ek₎∗₍_E_|

Bk) //E|_Bk

Sk−1

s∗_k−₁ 88

i _/_/

Dk e

k

/

s∗_k

O O Bk sk O O

Bk−1

i

o

sk−1

c

c (1.24)

where s∗_k₋₁ :_Sk−1 _→₍_ek₎∗₍_E_|

Bk) is the section of the pullback bundle (ek)∗E induced bys_k₋₁. Proposition 1.5.3. Let π, sk−1, ek, s∗k−1 be as above. Then s

∗

k−1 can be extended to a section s

∗

k :

Dk → (ek)∗(E) for each cell ek if and only if there exists a section sk : Bk → E which extends sk−1 :Bk−1 →E.

Proof. For a section sk of E which extends sk−1, we can define the section s∗k :Dk → (ek)

∗₍_E_{) (see} the equation (1.20)) by

s∗_k(y) = [y, sk(ek(y))], (1.25) and s∗_k clearly extends s∗_k₋₁. Conversely, if for all the k-cells ek_, _s∗

k : Dk → (ek)

∗₍_E_{) is a section of} (ek₎∗₍_E_{) which extends}_s∗

k, then we define a map sk:Bk →E by

sk(x) =







sk−1(x) if x∈Bk−1

(π2◦s∗k◦(ek)

−1₎₍_x₎ _if _x_∈_ek₍

Dk\Sk−1).

where π2 : (ek)∗E ⊂ Dk×E → E is the restriction of the projection of Dk×E onto E to (ek)∗E.

By using the equalityπ◦π2 =ek◦π1, whereπ1 : (ek)∗E →Dk is the restriction of the projection of Dk×E ontoDk to (ek)∗E, it is easy to prove thatsk is a section of E which extends sk−1.

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1.5. THE EXTENSION PROBLEM. 17 Let ϕ: (ek₎∗₍_E₎_→

Dk×F be a trivialization of (ek)∗(E), and pr2 : Dk×F → F the projection

over F. The composition map

S :_Sk−1 s ∗ k−1

−→(ek)∗(E)−→ϕ _Dk_×_F _−→pr2 _F _(1.26)

is a representative of an element in πk−1(F) for each k-cell ek. By proposition 1.5.1, this element is

zero in πk−1(F) if and only if the section s∗k−1 extends to a sections

∗

k.

Remark 1.5.1. We denote by α(s, ek_{) the element of} _π

k−1(F) defined by equation (1.26).

Lemma 1.5.4. For each k-cell ek, the local section s∗_k₋₁ extends from _Sk−1 to _Dk if and only if

α(ek_{, s}_{) = 0.}

Proof. Suppose that s∗_k₋₁ extends to a map s∗_k : _Dk → (ek)∗E. By proposition 1.5.1, s∗_k₋₁ is null-homotopic, and so the composition S := π2◦ϕ◦s∗k−1 (see equation (1.26)) is also nullhomotopic.

Hence α(ek_{, s}_{) = 0. Conversely, if} _α₍_ek_{, s}_{) = 0, then the map} _S _:

Sk−1 → F given by the equation

(1.26) is nullhomotopic, and by proposition 1.5.1 it can be extended to a map ˜S : _Dk _→ _F_{. Since} S =π2◦ϕ◦s∗k−1, the map s

∗

k−1 can be extended to Dk.

Remark 1.5.2. Let π : E → B be a fiber bundle, B be a path-connected manifold with typical fiber F, and x0, x1 ∈ B. Any path γ : I → B from x0 to x1 determines a homotopy equivalence

γ∗ : Ex0 → Ex1. In particular, we have a homomorphism between π1(B, x0) and the group of

homotopy classes of homotopy equivalences of the fiber Ex0 [see [16], page 117]. Now, since each

homotopy equivalence γ∗ : F → F induces a bijection [Sn, F] → [Sn, F], and [Sn, F] is isomorphic to πn(F) because F is simple, we obtain a homomorphism π1(B) → Aut(πn(F)). In other words,

we have an action of π1(B) on πn(F) for all n [see [16], page 159]. When this action is trivial,

there exists a canonical isomorphism between πn(F) and πn(Fx0) for all n, and all x0 ∈ B. Under

this condition we have that αs(eki) ∈ πk−1(F) for any k-cell eki, and so we have a well defined map αs :Ck(B,Z)→πk−1(F).

In what follows we suppose that this action is trivial. Thus we have the following definition Definition 1.5.2. We define a cellular k-cochain αs ∈Ck(Bk, Bk−1, πk−1(F)) by the equation

αs(ek) =α(s, ek). (1.27)

Proposition 1.5.5. LetB be a manifold with a structure of a finite CW-complex and letπ:E →B

be a fiber bundle with fiber F, where F is a manifold which is simple as a topological space. A local

section s:Bk−1 _→_E _of _E _{can be extended to a section over} _Bk _{if and only if the cellular} _k_-cochain αs is zero.

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Proof. It follows from Lemma 1.5.4 and Proposition 1.5.3.

Proposition 1.5.6 ([21], page 167). The cochain αs is a cocycle.

Remark 1.5.3. The cohomology class [αs] is an obstruction to the extension of s∗k−1 fromBk

−1 _to_Bk_.

i.e., [αs] = 0 if and only if s∗k−1 can be extended.

Remark 1.5.4. Proposition 1.5.3 and observation 1.5.3 imply that the cohomology class [αs] is also

an obstruction to the extension of s fromBk−1 to Bk.

In what follows we will show that [αs] is independent of the chosen trivializations, and the homotopy class of the section s.

Proposition 1.5.7. Let X be a topological space with a left action of a connected Lie group G, and let f1, f2 :Sk−1 →X and g :Dk →G be continuous maps. If for all x∈Sk−1 holds

f2(x) = g(x)·f1(x), x∈Sk−1

then f1 and f2 are homotopic.

Proof. First of all, we note that _Dk is contractible, so the map g : _Dk → G is nullhomotopic. Let

gt:Dk→G be a homotopy between the constant mapc:Dk →G, x7→e and g. Sinceg|Sk−1 is also

homotopic to c, the map Ht:Sk−1 →X defined by

Ht(x) =gt(x)·f1(x)

is a homotopy between f1 and f2.

Lemma 1.5.8. The cohomology class [αs] in Hk(Bk, Bk−1, πk−1(F)) is independent of the chosen

trivialization.

Proof. Let ek _{: (}

Dk,Sk−1) → (Bk, Bk−1) be a k-cell on B, and ϕi : (ek)∗(E)→ Dk×F, i = 1,2 be

trivializations of the bundle_Dk _→₍_ek₎∗₍_E_{). Let [}_α

s,1] and [αs,2] be the obstruction cohomology classes

corresponding to these trivializations. We will show that [αs,1] = [αs,2]. To do it, by the construction

of the obstruction cochain is sufficient to show that the maps Si =π2◦ϕi◦sk∗−1 :Sk−1 →F, i= 1,2

are homotopic.

Now, if g :_Dk →G is the transition function of E from ϕ1 to ϕ2, then S1 and S2 are related by

the equation

S2(x) = g(x)·S1(x), x∈Sk−1.

Since _Dk is contractible, by proposition 1.5.7, S1 and S2 are homotopic.

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1.5. THE EXTENSION PROBLEM. 19 Proof. Suppose thats1 and s2 are homotopic. To show that [αs1] = [αs2] it is sufficient to show that

for each cell ek _{the sections}_s∗

1k and s

∗

2k of (ek)

∗_E _{are homotopic. Let} _H

t:Bk−1 →E be a homotopy

between s1 and s2, we define a map ˜Ht:Sk−1 →(ek)∗E by

˜

Ht(x) = (x, Ht(ek(x))).

Since ˜H0(x) = (x, H0(ek(x))) = (x, s1(ek(x))) = s1∗k and ˜H1(x) = (x, H1(ek(x))) = (x, s2(ek(x))) =

s∗₂_k, the map ˜Ht is a homotopy between s∗1k and s

∗

2k. Then for any local trivializationϕofE we have

that Si =π2◦ϕ◦s∗ik, i= 1,2 are homotopic for each k-cell ek, and so [αs1] = [αs2].

The principal results of this section are summarized in the following theorem.

Theorem 1.5.10. Let B be a CW-complex, F be a manifold which is simple as a topological space and π :E →B be a fiber bundle with fiber F, and let s:Bk−1 _→_E _{be a local section of} _E _{over the}

(k−1)-skeleton of B. Then there exists a cellular cocycleαs ∈Ck(Bk, Bk−1, πk−1(F))which vanishes

if and only if s extends to a section s˜:Bk_→_E_.

Remark 1.5.5. The cohomology class [αs]∈Hk(Bk, Bk−1, πk−1(F)) vanishes if and only if the

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Chapter 2 Principal

G-Bundles with Singularities.

In this chapter we give the definition of principal G-bundle with singularities and show that the collection of all principal G-bundles with singularities over a base space M and their morphisms form a category. In addition, we prove that the pullback of a principal G-bundle with singularities under a map transversal to singularity submanifold is a principalG-bundle with singularities. Next, by using the obstruction theory for fiber bundles, we construct a collection of characteristic classes for principal G-bundles with singularities.

2.1 The Category of Principal

G

-bundles with singularities.

In this section we define principal G-bundles with singularities and their morphisms and prove that they form a category.

Definition 2.1.1. Let ¯G be a Lie group, let G ⊂ G¯ a Lie subgroup of G. Let M be a manifold, and Σ be a closed subset of M. A principal ¯G-bundle ¯P(M,G¯) is called a a principal G-bundle with singularities Σ if the structure group ¯G of ¯P|M\Σ reduces to G.

Remark 2.1.1. If ¯P(M,G¯) is a principalG-bundle with singularities and Σ is the set of singularities, then the reduction of ¯P|M\Σ toG induces a section s :M \Σ→P /G¯ . When Σ consists of isolated

points, we call the elements of Σ isolated singularities of the section s and the principal bundle ¯P a principal G-bundle with isolated singularities.

Remark 2.1.2. If ¯P(M,G¯) is a principal {e}-bundle with singularities Σ, then ¯P|M\Σ reduces to

a principal bundle P(M \Σ,{e}). From remark 2.1.1 it follows that the bundle ¯P /{e}, which is isomorphic to the principal bundle ¯P|M\Σ, admits a section. Hence, ¯P|M\Σ is trivial.

Example 2.1.1. Let X be a vector field on a manifold M of dimension n, Σ⊂M be the set of zeros of X, and G be the group of affine transformations on _Rn−1. Then this vector field determines a

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structure of principal G-bundle with singularities on the frame bundle L(M) on M: over M \Σ the bundle L(M) reduces to the principal G-bundle consisting of frames with the first vector equal to the value of X at the corresponding point.

Example 2.1.2. Let M be a Riemannian manifold of dimension n. The principal GL(n)-bundle of frames L(M) is a principal O(n)-bundle with an empty set of singularities.

Example 2.1.3. Let M be a Riemannian two-dimensional manifold, X be a vector field on M, and Σ ⊂ M be the set of zeros of X. Then X induces a trivialization of the bundle of orthonormal frames O(M) onM \Σ to the subgroup_Z2 ={−I, I} ofO(2). Therefore, X determines a structure

of principal_Z2-bundle with singularities onO(M), where Σ is the set of singularities. If, in addition,

the manifoldM is oriented, thenX determines a structure of principal{e}-bundle with singularities. Example 2.1.4.

Let M be a two dimensional connected Riemannian manifold, and let ∆1, ∆2 be one dimensional

singular distributions on M. Suppose that these distributions have the same singular set Σ, and at each regular point x∈ M \Σ the lines ∆1(x) and ∆2(x) are perpendicular. Let us consider the

following set

P ={(e1, e2)x ∈O(M) :ei is a generator of ∆j(x), x∈M\Σ, 1≤i, j ≤2}. (2.1)

The projection π :P →M \Σ gives a reduction of the bundle of orthonormal frames on M \Σ to the subgroupG=_Z2n Z4. Therefore,SO(M)→M is a principal Z2n Z4-bundle with singularities.

If, in addition, the manifold M is oriented, the bundle SO(M)→ M is a principal _Z4-bundle with

singularities.

Example 2.1.5. Let S be an oriented surface in 3-dimensional Euclidean space _R3 _{with isolated}

umbilical points, and h be the shape operator of S. Since h is a symmetric operator with respect to the first fundamental form, for each point p∈S the tangent space TpS has an orthonormal basis

consisting of eigenvectors of h. These eigenvectors are called principal directions of S at p. The pointsp inS where the eigenvalues ofh corresponding to the principal directions coincide are called umbilical points. Let Σ be the set of umbilical points on S. Let us consider the space

P ={(e1, e2)x ∈SO(S) :h(ei) =kiei, where k1 > k2, x∈S\Σ} (2.2)

The bundle πP :P \Σ→S given by πP((e1, e2)x) =x is a principalZ2-bundle which is a reduction

of the bundle of positively oriented orthonormal frames over S\Σ. Therefore, SO(S) is a principal

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2.1. THE CATEGORY OF PRINCIPALG-BUNDLES WITH SINGULARITIES. 23 Definition 2.1.2. Let ¯Pi(Mi,G¯i), i= 1,2 be principalGi-bundles with singularities, and Σ1 and Σ2

be the corresponding sets of singularities. Amorphism between ¯P1 and ¯P2 is a morphism ˜f : ¯P1 →P¯2

(see definition 1.3.5) of principal bundles such that the induced mapf :M1 →M2 maps Σ1 onto Σ2.

Proposition 2.1.1. The collection of all principal G-bundles with singularities and morphisms be-tween them form a category.

Proof. Let ¯Pi(Mi,G¯i), i= 1,2,3 be principalGi-bundles with singularities Σi, i= 1,2,3, ˜f :P1 →P2

and ˜g : P2 → P3 be morphisms of principal bundles with singularities such that ˜g ◦f˜: P1 → P3

is defined. Since ˜f and ˜g are morphisms of principal bundles, ˜g◦f˜is also a morphism of principal bundles. Moreover, f(Σ1) = Σ2 and g(Σ2) = Σ3 imply (g ◦f)(Σ1) = Σ3. Therefore, ˜g ◦f˜is a

morphism of principal bundles with singularities. It is clear that the class of principal G-bundles with singularities and morphisms satisfy all the axioms of category.

Definition 2.1.3. Let f : X → Y be a smooth map and W be a submanifold of Y. The map f is called transversal to W at the point p∈X if it satisfies one of the following properties:

1. f(p)∈/ W.

2. f(p)∈W and Tf(p)Y = (df)p(TpX) +Tf(p)W.

If f is transversal to W at pfor all p∈X, we say that f is transversal to W.

Proposition 2.1.2 ([2], page 174). Let f :X →Y be a smooth map and W be a submanifold of Y. If f is transversal to W, then f−1₍_W₎ _{is a submanifold of} _X _and _codim₍_f−1₍_W_{)) =}_codim₍_W_).

Proposition 2.1.3. LetM be a compact manifold of dimension n, and let P¯ be a principal G-bundle over M with singularities Σ. If f : N → M is a map transversal to Σ, then the pullback bundle

f∗( ¯P) is a principal G-bundle over N with singularities f−1(Σ).

Proof. Let ¯P be a principal G-bundle with singularities Σ. We need to prove that the associated fiber bundle f∗( ¯P)/Gadmits a section f∗s :N \f−1_(Σ)_→ _f∗_{( ¯}_P₎_/G_{. First of all, note that} _f−1_(Σ)

is a closed submanifold of N ( see proposition 2.1.2). Now, the following commutative diagram

f∗( ¯P) f¯ //

f∗π

¯

P

π

N \f−1(Σ) f //M \Σ.

(2.3)

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f∗( ¯P)/G f¯ // f∗_π

¯

P /G=E π

N \f−1_(Σ) f //_M _\_Σ_.

(2.4)

where we denote byπ, f∗π,f¯the quotient maps. Now, since ¯P|M\Σ reduces toG, the associated fiber

bundle π:E →M \Σ admits a section s:M \Σ→E, then the pullback bundle f∗π :f∗( ¯P /G)→

N \f−1(Σ) also admits a section f∗s : N \f−1(Σ). Furthermore, since the bundles f∗( ¯P /G) and

f∗( ¯P)/G are isomorphic, the bundle f∗( ¯P)/G admits a section ˜s : N \f−1(Σ). More exactly, if Ψ : f∗( ¯P /G) → f∗( ¯P)/G is such an isomorphism, then we can define a section ˜s by ˜s = Ψ◦f∗s. Hence, f∗( ¯P) is a principal G-bundle with singularities f−1_(Σ).

Remark 2.1.3. When the manifoldsM and N have the same dimension, and ¯P is a principal bundle with isolated singularities, the induced bundle f∗P¯ is also a principal bundle with isolated singular-ities.

2.2 Singularities and Indexes.

In this section we apply the obstruction theory for sections of fiber bundles to principal G-bundles with singularities in order to construct characteristic classes which take their values in homotopy groups of certain quotient of Lie groups. An important fact for the given construction is that the quotient ¯G/G, where ¯G is a Lie group and G is a closed connected Lie subgroup of ¯G, is simple as topological space ( proposition 1.1.3). In what follows, we assume that π1(M) acts trivially on all

homotopy groups πi( ¯G/G), so all of them are canonically identified in the sense of Remark 1.5.2.

2.2.1 Isolated Singularities and Indexes.

Let ¯G be a connected Lie group, G a closed connected Lie subgroup of ¯G, let π :P →M be a prin-cipal G-bundle with singularities over an n-dimensional manifold M, and let T be a closed tubular neighborhood of Σ in M. We choose a CW-structure onT, and we extend it to a CW-structure on

M such that the pair (M, M\T◦) is a relativeCW-complex. From section 1.5.2 it follows that there exists a cohomology class [αs]∈H∗(M, M\

◦

T , π∗−1( ¯G/G)), which is an obstruction to the extension

of the section s:M \Σ→P /G¯ corresponding to the reduction of the bundle ¯P|M\Σ ( see Theorem

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2.2. SINGULARITIES AND INDEXES. 25 In what follows we assume that ¯P is a principal G-bundle with isolated singularities and that eachn-cell en

i of M contains at most one singular point xi ∈Σ. In this case we can assume that the

closed tubular neighborhood T of Σ is the disjoint union of n-cells of M such that each of its cells contains exactly one singular point. In these conditions, the obstruction cohomology class [αs] is an element of the group Hn(M, M \

◦

T , πn−1( ¯G/G)).

Definition 2.2.1. Let xi ∈ Σ and en_i be the unique n-cell of M whose image contains xi. The element α(s, en_i) (see remark 1.5.1) in πn−1( ¯G/G) is called the index of the section s at the point

xi ∈Σ.

Remark 2.2.1. Since the cohomology class [αs] is independent of the chosen CW-complex structure

onM, the index is well defined.

Example 2.2.1. LetX be a vector field on a two-dimensional oriented compact manifoldM, Σ the set of zeros of X, and let ¯P be the principal {e}-bundle with singularities Σ introduced in the example 2.1.3. Since SO(2)/{e} is isomorphic to SO(2), SO(2) is homeomorphic to _S1_, _π

1(S1) = Z, and

π1(SO(2)/{e}) = Z. Therefore, the index of each singular pointx ∈Σ is represented by an integer

number. This index coincides with the Poincar´e’s index of zeros of vector fields.

Example 2.2.2. Let S be an oriented surface in the 3-dimensional Euclidean space _R3_{, Σ the set of}

umbilical points ofS, and ¯P be the principal_Z2-bundle with singularities Σ given in the example 2.1.5.

Then the index of each singular pointx∈Σ is an element of the fundamental groupπ1(SO(2)/Z2) =

Z. This index is twice the index of each one dimensional distribution on S generated by principal

vectors of TpS onU \ {x} , whereU is an open neighborhood of xsuch that U ∩Σ ={x}.

Proposition 2.2.1. Let M be an oriented compact manifold of dimension n, let P¯ be a principal

G-bundle with singularities Σ and let s : M \Σ → P /G¯ be the section induced by the reduction of ¯

PM\Σ to G. Then the following assumption are equivalent

1. the sum of the indices of s at the singular points is zero;

2. the obstruction class [αs] is zero.

Proof. We suppose that [αs] = 0. Then by evaluation at the fundamental class [M] =

P

ie n i we

obtain

X

i

αs(en_i) =αs([M]) = 0.

Conversely, suppose that the sum of indices of singular points is zero. That is, P

iαs(e n

i) = 0.

Since the map ρ[M] : Hn(M, πn−1( ¯G/G)) → H0(M, πn−1( ¯G/G)) given by ρ[M](ϕ) = ϕ([M]) is an

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2.2.2 Singularities of higher dimension and indexes.

In this section M denotes an n-dimensional oriented compact manifold, Σ is a k-dimensional closed submanifold ofM, ¯P is a principalG-bundle with singularities Σ, ands:M\Σ→P /G¯ is the section of ¯P /G corresponding to the reduction of ¯P|M\Σ to G. Let us consider the bundle π : T →Σ with

fiber _Dn−k, where T ⊂ M is a closed tubular neighborhood. It follows from proposition 1.2.1 that

CW-complex structures on Σ and _Dn−k _{determine a} _CW_{-complex structure on} _T_{. In particular,}

using a CW-decomposition of Σ and the CW-decomposition of _Dn−k _{consisting of a 0-cell} _e0_{, a}

(n−k−1)-cellen−k _{and one (}_n₋_k₋_1)-cell_en−k_{, we see the}_CW_{-decomposition of} _T _{contains only}

cells of the following types:

ej(Σ)×e0, ej(Σ)×en−k−1, ej(Σ)×en−k, j = 0,· · · , k, (2.5) where by ej_{(Σ) we denote a} _j_{-cell in Σ.}

Lemma 2.2.2. Let P¯ be a principal G-bundle with singularities Σ, and π:T →Σ be a bundle over Σ, where T is a closed tubular neighborhood of Σ. Suppose that T has the CW-complex structure constructed above. Then first obstruction to the extension of the section s : M \int(T) → P /G¯ to

M appears in the dimension n−k.

Proof. Let us consider the CW-decomposition of T given by (2.5). Since the cells e0 and en−k−1

belong to the boundary of _Dn−k_{, the cells} _ej_(Σ) _×_e0 _and _ej_(Σ) _×_en−k−1 _{belong to the boundary}

of T. Moreover, since the section s : M \int(T) → P /G¯ is defined on the boundary of T, then [αs](ej(Σ)×e0) = 0, and [αs](ej(Σ)×en−k−1) = 0, the possible first obstruction to the extension of s from M \int(T) to M appears in dimensionn−k.

Proposition 2.2.3. Let Σ be a k-dimensional closed submanifold of M, π : ¯P →M be a principal

G-bundle with singularities Σ, s :M \Σ→P /G¯ the section of P /G¯ corresponding to the reduction of P¯|M\Σ toG and T a closed tubular neighborhood of Σ. If s is defined over (M\int(T))∪Tn−k−1,

then(n−k+i)-cocycle α(si) which is an obstruction to the extension of s to(M\T)∪Tn−k+i satisfies

the following equality

α(_si)(∂ei+1(Σ)×en−k) = 0, (2.6) for every (i+ 1)-cell ei+1_(Σ) _of _Σ.

Proof. For any (n−k+i+ 1)-cell ei+1_(Σ)_×_en−k _of _T _{we have}

Topological invariants of principal G-bundles with singularities

Topological Invariants of Principal

G-bundles with Singularities

Advisor:

Mikhail Malakhaltsev

Universidad de Los Andes

Facultad de Ciencias

Departamento de Matem´

aticas

Bogot´

a

2016

Topological Invariants of Principal

G-bundles with Singularities

Dedicatory

Acknowledgement

Abstract

Contents

Notations

Introduction.

Chapter 1

Preliminaries.

1.1

Homotopy Groups.

1.1.1

Definition and examples

1.1.2

Action of

π

(

X, x

)

on

π

(

X, x

)

.

1.2

CW-Complexes

1.2.1

Homology and Cohomology of

CW

-complexes.

1.3

Fiber Bundles

1.3.1

Principal Bundles.

1.3.2

Associated and induced bundles.

1.4

The Bundle of Linear Frames.

1.5

The extension problem.

1.5.1

The extension problem for maps.

1.5.2

The extension problem for sections of fiber bundles.

Chapter 2

Principal

G-Bundles with Singularities.

2.1

The Category of Principal

G

-bundles with singularities.

2.2

Singularities and Indexes.

2.2.1

Isolated Singularities and Indexes.

2.2.2

Singularities of higher dimension and indexes.