characteristic classes of bundles and …€¦ · · 2012-12-04characteristic classes of bundles...

CHARACTERISTIC CLASSES OF BUNDLES ANDFOLIATIONS

M. MALAKHALTSEV

Lecture notes of mini-course at “First School in Applied Mathemat-ics” (December 3–5, 2012), Mathematical Department of The Univer-sity of the West Indians, Kingston, Jamaica

Contents

1. Introduction. Initial problem 21.1. Classification problem. 21.2. What do we know? 22. Euler characteristic 22.1. Combinatorial version 22.2. Euler characteristic with vector fields 33. Locally trivial bundles. Vector bundles. Principal bundles 33.1. Locally trivial bundle 33.2. Vector bundles 53.3. Principal bundles. Connection and curvature 63.4. Maxwell equations and principal bundles 74. Characteristic classes of vector bundle 104.1. Invariant polynomials 104.2. Chern-Weil homomorphism 114.3. Characteristic classes of vector bundles 124.4. Yang-Mills fields 144.5. Pontryagin characteristic classes 164.6. Euler characteristic class 165. Characteristic classes of foliation 175.1. Foliations 175.2. Partial connections and foliated bundles 185.3. Weil algebra 185.4. Chern-Weil homomorphism 195.5. Characteristic classes of foliations 206. Answers to some exercises 207. What to read? 20

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2 M. MALAKHALTSEV

1. Introduction. Initial problem

1.1. Classification problem. Classify manifolds up to diffeomorphisms.What invariants separate the diffeomorphism classes?

1.2. What do we know?

1.2.1. Classification in dimension 1. There are two manifolds: R andS1. The invariants which separate the classes: compactness; the fun-damental group π1(X), the homology group H1(X).

1.2.2. Classification of closed (compact and without boundary) mani-folds in dimension 2. Orientable manifolds. The sphere M0 = S2, thetorus M1 = T2, the connected sum of k tori Mk = T2#T2# · · ·#T2.The invariants which separate the classes: the fundamental groupπ1(X), the homology group H1(X).

1.2.3. Non-orientable manifolds. . The manifold Mk is the sphere S2

with k removed disks glued by Mobius bands. The invariants which sep-arate the classes: the fundamental group π1(X), the homology groupH1(X).

Exercise 1. The projective plane RP 2 is non-oriented, so is diffeomor-phic to some Nk. What is the k?

1.2.4. Classification of closed (compact and without boundary) mani-folds in dimension 3,4. Still unknown.

1.2.5. Classification of closed (compact and without boundary) mani-folds in dimension greater than 4. Impossible.

2. Euler characteristic

2.1. Combinatorial version. First version. Let P be a polyhedron,V be the number of vertices, E the number of edges, F the number offaces. Then the Euler characteristic of P is ξ(P ) = V − E + F .

Modern version. Let M be an m-dimensional closed manifold repre-sented as CW -complex, then

χ(M) = c0 − c1 + c2 − · · ·+ (−1)mcm, (1)

where ci is the number of i-dimensional cells.

Theorem 2.1. If M is a closed n-dimensional oriented manifold then

χ(M) =m∑i=0

(−1)i dimHi(M ;R) =m∑i=0

(−1)i dimH i(M ;R). (2)

Exercise 2. Find χ(Mk) and χ(Nk).

CHARACTERISTIC CLASSES OF BUNDLES AND FOLIATIONS 3

2.2. Euler characteristic with vector fields. Let X be a vectorfield on a closed manifold M . Assume that the set ΣX of zeros of X isdiscrete.

For a point p ∈ ΣV there exists a neighborhood U(p) such that foreach q ∈ U(p), q = p, we have X(q) = 0. We can assume that U(p)is the domain of a chart, so there is a diffeomorphism φ of U(p) to anopen neighborhood V (0) of 0 ∈ Rm, which maps the vector field X toa vector field Y on V (0).

Now the vector field Y takes zero value only at 0, and consider theunit vector field N = 1

∥Y ∥Y on V (0) \ 0 which can be considered as a

map N : V (0) → Sm−1. If we take a sphere Sm−1 ⊂ Rm around 0, therestriction of N to the sphere defines a map Sm−1 → Sm−1. The degreeof this map, call it I(X, p) is called the index of the vector field X atthe point p.

Theorem 2.2. Let M be a closed manifold, and X a vector field onM such that ΣX is discrete. Then

χ(M) =∑p∈ΣX

I(X, p). (3)

Corollary 2.2.1. Let M be a closed manifold. Then χ(M) = 0 if andonly if on M there exists a vector field X without zeros.

Exercise 3. For the vector field V on the plane R2 find the index ofthe zero (0, 0): a) V (x, y) = (x, y); b) V (x, y) = (y, x); c) V (x, y) =(−y, x).

Exercise 4. Construct a vector field on the sphere S2 with a uniquezero.

3. Locally trivial bundles. Vector bundles. Principalbundles

3.1. Locally trivial bundle.

3.1.1. Definition of bundle. Let E, M be manifolds, π : E → M bea submersion. Then the triple ξ = (E, π,M) is called a bundle, E iscalled the total space of ξ, M is the base of ξ, π the projection of ξ.The submanifold Ep = π−1(p) ⊂ E is called the fiber of π over p.

4 M. MALAKHALTSEV

3.1.2. Definition of bundle morphism. A bundle morphism α : ξ → ξ′,where ξ = (E, π,M) and ξ′ = (E ′, π′,M ′), is the commutative diagram:

EF //

π

E ′

π′

M

f // M ′

(4)

3.1.3. Section of a bundle. Let ξ = (E, π,M) be a bundle. A maps :M → E is called a section of ξ if π s = IdM .

3.1.4. Locally trivial bundle. A bundle ξ is called a locally trivial bundlewith the typical fiber Y if there exists a covering U = Uα of M suchthat for each Uα there is given a diffeomorphism φα : π−1(Uα) → Uα×Ywith the property that

π−1(Uα)φα //

π

##HHH

HHHH

HHUα × Y

pr1

wwwwwwwww

Uα

(5)

The covering U is called a trivializing covering for the bundle ξ.For any trivializing covering we can consider the maps

(Uα ∩ Uβ)× Yφ−1α //

pr1

((RRRRR

RRRRRR

RRπ−1(Uα ∩ Uβ)

π

φβ // (Uα ∩ Uβ)× Ypr1

vvmmmmmm

mmmmmm

m

Uα ∩ Uβ

(6)

From this diagram follows that φβ φ−1α (p, y) = (p, φβα(p, y)), and the

φβα(p, y) are called gluing maps. The gluing maps define the applica-tions

φβα : Uβ ∩ Uα → Diff(Y ) (7)

which satisfy the cocycle condition

φγα(p) = φγβ(p)φβα(p) (8)

for p ∈ Uα ∩ Uβ ∩ Uγ.If the gluing maps φβα take values in a Lie group G acting on Y (a

group of transformations of Y ), then the group G is called the structuregroup of ξ.

Exercise 5. A bundle isomorphic to the bundle pr1 : M × Y → Mis called a trivial bundle. What are the gluing functions of a trivialbundle?


Exercise 6. a) Find the gluing functions for the tangent bundle; b)What is the structure group of the tangent bundle?

Theorem 3.1. LetM be a manifold and U = Uα its covering. Givena cocycle φβα : Uα ∩ Uβ → G, where G is a Lie group acting on amanifold Y , one can construct a locally trivial bundle with typical fiberY whose gluing functions are φβα.

3.1.5. Pullback of locally trivial bundle. Let ξ = (E, π,M) be a vectorbundle and f : M → M ′ a map. If U = Uα is a trivializing coveringfor ξ, then one can take a covering f ∗U = f−1(Uα) and the gluingfunctions f ∗φβα = φβα f . The functions f ∗φβα satisfy the cocyclecondition, so, by Theorem 3.1, determine a vector bundle over M ′,which is called the pullback of ξ and is denoted by f ∗ξ.

Exercise 7. Assume a pullback of a bundle is a trivial bundle. Is ittrue that the initial bundle is trivial?

3.1.6. Characteristic class of a bundle. A characteristic class of a bun-dle is a rule which assigns to each bundle ξ = (E, π,M) a cohomologyclass c(ξ) ∈ H(M) such that c(f ∗ξ) = f ∗c(ξ).

The goal of subsequent sections is to give a construction of charac-teristic classes.

3.2. Vector bundles.

3.2.1. Definition of vector bundle and vector bundle morphism. Exam-ples.

Definition 3.1. A locally trivial bundle ξ = (E, π,M) with typicalfiber Rk and structure group G ⊂ GL(k) is called a vector bundle ofrank k.

Remark 3.1. Any fiber Ep = π−1(p) of a vector bundle has structureof a k-dimensional vector space.

Remark 3.2. If the gluing functions take values in G ⊂ GL(n), thegroup G is called the structure group of the vector bundle.

A morphism of vector bundles ξ1 and ξ2 is a morphism of bundles,i. e. a commutative diagram (4) where F is linear restricted to fibers.

Example 3.1.

1: Trivial vector bundle.2: The Mobius band.3: Tangent bundle.4: A distribution on a manifold.

Exercise 8. Prove that the Mobius band is a nontrivial vector bundle.

6 M. MALAKHALTSEV

3.2.2. Operations on vector bundles.The Whitney sum of vector bundles. Let ξ1 and ξ2 be vector bundlesover M , rank ξ1 = k1, rank ξ2 = k2. Then there exists a coveringU = Uα of M which trivializes both ξ1 and ξ2.

Exercise 9. Prove that this covering exists.

Let φ1βα and φ2

βα be the corresponding gluing functions for ξ and ξ′.Then one can construct the cocycle

ψβα =

(φ1βα 00 φ2

βα

)(9)

which takes values in GL(k1 + k2). The corresponding vector bundleis called the Whitney sum of ξ1 and ξ2, and is denoted as ξ1 ⊕ ξ2. Thefiber of ξ1⊕ ξ2 over a point p ∈M is the direct sum of the fibers of thebundles ξ1 and ξ2 over p.Functors applied to vector bundles. The other functors of linear algebrasuch that, e. g., ⊗, Hom, ∗ can be applied to vector bundles in the samemanner as we did for the functor ⊕.

Exercise 10. Is it possible that ξ1 is a trivial vector bundle, ξ2 is anontrivial vector bundle, and ξ1 ⊕ ξ2 is a trivial vector bundle?

3.2.3. Characteristic classes of manifolds. A characteristic class of amanifold is a characteristic class of its tangent bundle.

Example 3.2. The Euler characteristic is a characteristic class.

3.3. Principal bundles. Connection and curvature.

3.3.1. Definition of principal bundle. LetG be a Lie group and (P, π,M)be a locally trivial bundle with the typical fiber G and the structuregroup G which acts on itself by left translations. Such bundle is calledprincipal G-bundle.

Theorem 3.2. Let (P, π,M) be a principal G-bundle. Then the groupG acts on P and the orbits of this action are the fibers of the bundle.

3.3.2. Bundle of frames of vector bundle. Let ξ be a vector bundle withstructure group G ⊂ GL(k). Let us consider the bundle πL : L(ξ) →Mwhich consists of all the frames of all the fibres Ep of ξ, and πL is themap which assigns the point p to a frame of Ep. On the total spaceL(ξ) we have the free action of the group G ⊂ GL(k):

e′ = e · A, where e = ea , A = ∥Aaa′∥, ea′ = eaA

aa′ , (10)

whose orbits are the fibers π−1L (p). This bundle is called the frame

bundle of the vector bundle and is a G-principal bundle.


3.3.3. Fundamental vector fields. Let g be the Lie algebra of the LiegroupG. For each a ∈ g let exp(ta) be the corresponding one-parametersubgroup of G. Then we define the fundamental vector field as follows:

σ(a)p =d

dt

∣∣∣∣t=0

p · at. (11)

Thus we get a map σ : g → X(M).

Exercise 11. Find the fundamental vector fields of the group SO(2)acting on R2 by rotations.

Theorem 3.3. The map σ is a Lie algebra homomorphism, i. e. σ isa linear map and σ([a, b]) = [σ(a), σ(b)].

3.3.4. Connection in principal bundle. Curvature. Let (P, π,M) be aprincipal G-bundle, A connection form is a g-valued 1-form ω on Psuch that

1. ω(σ(a)) = a;2. ωpg(dRg(X)) = adg−1ωp(X).

A connection form ω in a principal bundle determines a horizontaldistribution H on P : H = kerω, that is a G-invariant distribution suchthat for each p ∈ P we have TpP = Hp ⊕ Vp, where V = ker dπ is thevertical distribution.

The curvature form of the connection ω is a g-valued 2-form on P :

Ω = dω +1

2[ω, ω]. (12)

Theorem 3.4. The curvature form of a connection has the properties:

(1) Ω(X, ∗) = 0 for any vertical vector X ∈ V (a form with thisproperty is called horizontal).

(2) Ωpg(dRgXp, dRgYp) = adg−1Ωp(Xp, Yp), where Xp, Yp ∈ TpP .

Exercise 12. Prove the properties of the curvature form.

Let us define covariant differential of Ω as where

DΩ(X, Y, Z) = dΩ(hX, hY, hZ), (13)

where h : TpP → Hp is the projection determined by the splittingTpP = Hp ⊕ Vp.

Theorem 3.5 (Bianchi’s identity).

DΩ = 0. (14)

3.4. Maxwell equations and principal bundles.

8 M. MALAKHALTSEV

3.4.1. Maxwell equations in terms of differential forms. The electro-magnetic field is described by two vector fields E, H satisfying theMaxwell equations.

1

c

∂

∂tH = −∇× E, ∇ · H = 0. (15)

1

c

∂

∂tE = ∇× H, ∇ · E = 0. (16)

With respect to the coordinate system x0 = ict, x1 = x, x2 = y, x3 = z,the fields E and H give the 2-form

F = c(Exdx+ Eydy + Ezdz) ∧ dt++ (Hxdy ∧ dz +Hydz ∧ dx+Hzdx ∧ dy). (17)

The first Maxwell equation is equivalent to dF = 0, and the secondone to d ⋆ F = 0, where ⋆ is the Hodge operator in R4.

Exercise 13. Verify that the Maxwell equations are equivalent to dF =0 and d ⋆ F = 0.

By the Poincare lemma a 1-form A exists such that dA = F (thepotential) and so we have

d ⋆ dA = 0. (18)

The potential is not unique, we can change it

A′ = A+ df (19)

where f is an arbitrary function, which is called a gauge transformation(a transformation of potential which leaves the field invariant).

In what follows we shall interprete F as a curvature form and A as aconnection form in a principal bundle. This reflects the important ideathat the curvature is a field (which can be measured) and a connectionis like a potential of the field, comp. a force and its potential energy.

3.4.2. Covariant derivative corresponding to a potential. Consider nowthe differential operator ∇i = ∂i + Ai, i = 0, 3 which we will apply tofields ψ with the transformation law

ψ′ = gψ, (20)

where g is a function which does not vanish, in the following way:

∇iψ = ∂iψ + Aiψ. (21)

Take the operator ∇′ which corresponds to the potential A′ = A −d log g:

∇′i = ∂i + A′

i = ∂i + (Ai − ∂i log g), (22)


then

∇′iψ

′ = ∂i(gψ) + A′i(gψ) =

= g ∂iψ + ψ ∂ig + Ai g ψ − g ∂i log g ψ = g (∂iψ + Aiψ) = g∇iψ. (23)

Therefore the field ∇ψ has the same transformation law as ψ has, underthe condition that the potential is changed by the gauge transformation(19) with f = − log g. Note also that

[∇i,∇j] = ∂iAj + ∂jAi = Fij. (24)

Now assume we have a vector bundle ξ = (E, π,M) of rank 1 (overR or C, and gβα : Uα∩Uβ → GL(1) are gluing functions. Then take theprincipal bundle (P, πP ,M) associated to the vector bundle ξ, that iswith the same gluing functions. A local trivialization of P over U ⊂Mgives us coordinates (x, a), x ∈ U , a ∈ GL(1), on P |U . With respect tothese coordinates the fundamental vector field σ(λ)(x,a) = λa ∂

∂a. Now

let us take the connection form on P |U

ω =1

ada+ A,

where A is a potential with the gauge transformation (19) where f =− log gαβ.

Note that, when we pass from one trivialization to another, thechange of the coordinates has the form (x, a′) = (x, gβαa). At thesame time

ω′ =1

a′da′ + A′ =

1

gβαad(gβαa) + A− d log gβα =

=1

gβαd(gβα) +

1

ada+ A− d log gβα =

1

ada+ A = ω. (25)

So the connection form ω is well defined. Further, as GL(1) is com-mutative, so the commutator [ω, ω] = 0, and the curvature form of ωis

Ω = dω = dA = F. (26)

A field ψ is a section of the bundle ξ and ∇iψ = ∂iψ + Aψ is thecovariant derivative of ψ with respect to the connection ω defined bythe potential A.

Exercise 14. Let M = R2 and F = dx ∧ dy. Find a potential of thisfield.

10 M. MALAKHALTSEV

3.4.3. Non commutative case. If ξ = (E, π,M) is a vector bundle ofrank k, then a trivialization of ξ over U gives us coordinates (x, y),where x ∈ U , y ∈ Rk, on EU . Then a field (a section of ξ) has localrepresentation y = ψ(x), and when we pass from one trivialization toanother, over Uα ∩ Uβ we have:

ψ′ = gβαψ, (27)

where gβα : Uα ∩ Uβ → G ⊂ GL(k) are the gluing maps.The on the total space L(E) of the principal bundle L(ξ) we have

local coordinates (x, a) over Uα and the change of the coordinates is

(x, a′) = (x, gβαa), x ∈ Uα ∩ Uβ, a ∈ G. (28)

A connection form is written with respect to these coordinates:

ω(x,a) = a−1(da+ Aa), (29)

where A : TM → g is a “potential field” whose gauge transformationshave the form

A′ = g−1βαdgβα + g−1

βαAgβα. (30)

The curvature form is expressed as follows:

Ω(x,a) = a−1Fa, where F = dA+1

2[A,A] = dA+ A ∧ A. (31)

4. Characteristic classes of vector bundle

4.1. Invariant polynomials. Let g be a Lie algebra of a Lie groupG. A tensor t(X1, · · · , Xm) on g is called invariant if

t(adgX1, · · · , adgXm) = t(X1, · · · , Xm) (32)

Now consider Sg, the space of symmetric tensors on g. All the invariantsymmetric tensors on g form an algebra with respect to the symmetrictensor product, call this algebra I(G).

Example 4.1. The ring I(GL(k)) of invariant polynomials is generatedby coefficients pi of the characteristic polynomial of a matrix, so

det(λI − A) = λk − p1(A)λk−1 + p2(A)λ

k−1 − · · ·+ (−1)kpk(A). (33)

In particular, p1(A) = trA, pk(A) = detA. In addition, pi are alge-braically independent.

Exercise 15. Prove that pk(A) are invariant polynomials.

Example 4.2. The ring I(U(k)) of invariant polynomials on the Liealgebra u(k) of the Lie group of unitary matrices U(k) is generated bycoefficients pi of the characteristic polynomial of a matrix, so

det(λI − iA) = λk − p1(A)λk−1 + p2(A)λ

k−1 − · · ·+ (−1)kpk(A), (34)


we take iA to have λ ∈ R (all the eigenvalues of a skew-hermitianmatrix are pure imaginary), and pi are algebraically independent.

Example 4.3. The ring I(O(k)) of invariant polynomials on the Liealgebra o(k) of the Lie group of orthogonal matrices O(k) is generatedby coefficients pi of the characteristic polynomial of a matrix, so

det(λI − A) = λk − p1(A)λk−2 + p2(A)λ

k−4 − · · ·+ (−1)kpk(A), (35)

The polynomials pi are algebraically independent.

Example 4.4. For the ring I(SO(k)) of invariant polynomials on theLie algebra o(k) of the Lie group SO(k) consider the polynomials pidefined by (35).

For this group we have two cases. If k = 2m+1, then the polynomialspi are algebraically independent and generate I(SO(2m + 1)). If k =2m, then the there exists a polynomial g such that fm = g2. Thepolynomials p1, · · · , pm−1, g are algebraically independent and generateI(SO(2m)).

Exercise 16. The polynomial g is called the Pfaffian. Find the pfaffianfor k = 2 and k = 4.

4.2. Chern-Weil homomorphism. Let (P, π,M) be a principal G-bundle, ω a connection in P , and Ω its curvature form. Then we havethe algebra homomorphism

k : I(g) → Ω(P ), k(t) = t(Ω), (36)

where

t(Ω)(X1, X2, · · · , X2k−1, X2k) =

=∑θ∈S2k

ε(θ)t(Ω(Xθ(1), Xθ(2)), · · · ,Ω(Xθ(2k−1), Xθ(2k))). (37)

is called the Chern-Weil homomorphism.

Theorem 4.1.

(1) For each t ∈ Im(g), k(t) = π∗t(Ω), where t(Ω) ∈ Ω2m(M) is aclosed form.

(2) The cohomology class ct = [t(Ω)] ∈ H2m(M) is independent ofthe connection.

Proof. First of all, by the Bianchi’s identity (Thm. 3.5), we have thatdt(Ω) = 0.

Then, Thm. 3.4 and the fact that t is invariant imply that thereexists a form t(Ω) on M such that t(Ω) = π∗t(Ω), and as dt(Ω) = 0,

we obtain that dt(Ω) = 0 as well.

12 M. MALAKHALTSEV

Now if ω and ω′ are two connections on P , we can take α = ω′ − ωand the 1-form α is horizontal and R∗

gα = ad(g)−1α.Consider the family of connections

ωs = ω + s(ω′ − ω) = ω + sα, s ∈ [0, 1]. (38)

If Ωs is the curvature form of ωs, we have

d

dsΩs = Dsα. (39)

which implies the following statement. Let t ∈ Im(g), and

Φ = m

∫ 1

0

t(α,Ωs, · · · ,Ωs)ds, (40)

then

dΦ = t(Ω1)− t(Ω0), (41)

which in turn imply the independence of connection.

Exercise 17. Take a connection in a complex vector bundle and provedirectly that the cohomology class of p1(Ω) does not depend on a choiceof connection.

4.3. Characteristic classes of vector bundles.

4.3.1. Chern classes. Let ξ be a complex vector bundle of rank k. Thenthe linear frame bundle L(ξ) is a principal GL(k;C)-bundle. Considerthe polynomials pi ∈ I(GL(k;C) defined by

det(λI − 1

2πiA) = λk + p1(A)λ

k−1 + · · ·+ (−1)kpk(A). (42)

Take a connection ω in L(ξ) and let Ω be its curvature form. Then,as we have seen in 4.2, there exist closed differential forms γi ∈ Ω2iMsuch that π∗γi = pi(Ω) and the cohomology classes of γi do not dependon a choice of connection.

Definition 4.1. The cohomology classes ci(ξ) = [γi] ∈ H2i(M ;R) arecalled Chern classes of ξ.

The class c(ξ) = 1+ c1(ξ)+ · · ·+ ck(ξ) ∈ H(M ;R) is called the totalChern class.

Theorem 4.2 (Properties of Chern classes).

Naturality: If f :M ′ →M is map, then c(f ∗ξ) = f ∗c(ξ).Whitney sum: c(ξ1 ⊕ ξ2) = c(ξ1)c(ξ2).


Chern class of the canonical bundle over CP 1: Let ξ be thecanonical bundle over CP 1. Then c1(ξ) is a generator of H

2(CP 1;R) ∼=R such that ∫

CP 1

c1(E) = −1. (43)

Proof. Let us calculate the Chern class of the canonical bundle of CP 1.Recall that the complex projective line is the set of all one-dimensionalcomplex subspaces L ⊂ C2. Each L has a basis consisting of one vectorz ∈ C2, and two vectors z, w span a same L if and only if w = λz,λ = 0. Therefore we can say that

CP 1 = (C2 \ 0)/(z ∼ λz). (44)

If z = (z0, z1), then we denote the equivalence class [z] by [z0 : z1]. So

CP 1 = [z0 : z1] | z0 = 0 or z1 = 0 (45)

and one can prove easily that CP 1 is a manifold diffeomorphic to the2-dimensional sphere S2.

The canonical bundle πE : E → CP 1 is the complex vector bundlewhose fiber over [z] is L = spanz. We can write explicitly

E =(L, w) ∈ CP 1 × C2 | w ∈ L

. (46)

and πE(L, z) = L. Note that, if L = [z0 : z1] and w ∈ L, then w ∈ L ifand only if z × w = z0w1 − z1w0 = 0.

If we the open covering of CP 1 by the open sets U0 = [z0 : z1] | z0 = 0,U1 = [z0 : z1] | z1 = 0, then we have trivializations:

ϕ0 : π−1E (U0) → U0 × C, (47)

ϕ0([z0 : z1], (w0, w1) = ([z0, z1], w0)

ϕ1 : π−1E (U1) → U1 × C, (48)

ϕ1([z0 : z1], (w0, w1) = ([z0, z1], w1).

and

ϕ−10 ([z0 : z1], λ) = ([z0 : z1], (λ,

z1z0λ)) (49)

Therefore, for [z0 : z1] ∈ U0 ∩ U1, we have

ϕ1 ϕ−10 ([z0 : z1], λ) = ([z0 : z1],

z1z0λ), (50)

and the gluing function ϕ10 is

ϕ10 : U0 ∩ U1 → GL(1;C) = C∗, ϕ10([z0 : z1]) =z1z0. (51)

Thus the principal GL(1,C)-bundle has the same trivializing coveringand the same gluing map. In fact the total space of this bundle is

14 M. MALAKHALTSEV

exactly P = C2 \ 0 with natural group action C∗ and projection πP :P → CP 1, πP (z) = [z].

The fundamental vector fields have the form:

σ(µ)z = µz where µ ∈ C = g(C∗). (52)

Take the connection form

ω = (z0dz0 + z1dz1)/(z0z0 + z1z1) (53)

This is a C-valued 1-form. Let us find the curvature form

Ω = dω = (dz0 ∧ dz0 + dz1 ∧ dz1)/(z0z0 + z1z1)+

+ (z0dz0 + z1dz1) ∧z0dz0 + z0dz0 + z1dz1 + z1dz1

(z0z0 + z1z1)2. (54)

Now consider the trivialization of P :

ψ0 : π−1P (U0) → U0 × C∗ ∼= C× C∗, ψ0(z0, z1) = (z1/z0, z0). (55)

So new coordinates are w = z1/z0, λ = z0, then z0 = λ and z1 = λw.With respect to these coordinates, we write

ω =1

λdλ+

w

1 + wwdw, (56)

and then

Ω = dω =1

1 + wwdw ∧ dw. (57)

Now the Chern class c1(ξ) = [ 12πi

Ω], and one can find that∫CP 1

Ω =

∫U0

Ω =

∫R2

2idx ∧ dy

(1 + x2 + y2)2=

= 2i

∫ 2π

0

∫ ∞

0

rdr ∧ dθ(1 + r2)2

= 2πi. (58)

Exercise 18. Find c1(TS2), where the tangent bundle of S2 is consid-ered as a complex vector bundle of rank 1 (multiplication by i is therotation on the angle π/2).

4.4. Yang-Mills fields. A gauge field is a 2-form on R4 (the space-time) with values in the matrix Lie algebra g of a matrix Lie groupG:

F = dA+ A ∧ A, (59)

and A is a matrix valued 1-form called a potential of the gauge field.An analog of the Maxwell equation (15) is the Bianchi equation

dF = F ∧ A− A ∧ F (60)


and an analog of the other Maxwell equation (16) is the Yang-Millsequation

D ⋆ F = d ⋆ F − ⋆F ∧ A+ A ∧ ⋆F = 0. (61)

A Yang-Mills field is a gauge field F satisfying (60) and (61) (classicallythe term Yang-Mills field was used in case G = SU(2)).

By physical reasons ∥F (x)∥ = 0 in a neighborhood of ∞. So wecan take S4 = R4 ∪ ∞ as the one-point compactification of R4. Thiscompactification is given by the stereographic projection p : S4 → R4

with the property that p(−e4) = 0, and e4 is the pole of the projection,where e4 = (0, 0, 0, 1).

Take U0 = S4 \ e4 and U1 = S4 \ −e4. The gauge field F on U0

has potential A and on U1 a potential B which are related by a gaugetransformation (30) on U0 ∩ U1

∼= R4 \ 0:

A = g−1Bg + g−1dg. (62)

The potential B vanishes in a neighborhood of e4, we need this to makeF vanish in a neighborhood of e4 = ∞.

The map g : U0 ∩ U1∼= R4 \ 0 → G can be considered as a gluing

map of a principal G-bundle π : P → S4, then the potentials A andB define a connection ω in this bundle and the curvature form Ω isdetermined by F (see (31)) over U0

∼= R4, and Ω = 0 in π−1(V ), whereV is a neighborhood of e4.

So a Yang-Mills field is a curvature form Ω of a connection in aprincipal G-bundle over S4 which satisfies the Yang-Mills equation D ∗Ω = 0.

Remark 4.1. The Yang-Mills fields are stationary points of the action

S(F ) =

∫M

1

4∥F∥2θ, (63)

where the norm is taken with respect to some metric on the base andθ is a volume form.

If ⋆Ω = ±Ω, then the gauge field is called (anti-)self-dual. In thiscase the equation (61) holds true due to the Bianchi equation.

In what follows let us consider G = SU(2). As we have seen a Yang-Mills field defines a principal SU(2)-bundle over S4 and a connectionω with curvature Ω in this bundle such that D ⋆ Ω = 0. Any bundleover S4 is completely caracterized up to an isomorphism by the degreeof the gluing map g : S3 → SU(2) ∼= S3. This number deg g ∈ Z iscalled topological charge of the Yang-Mills field.

16 M. MALAKHALTSEV

Theorem 4.3. The topological charge k can be calculated as

k =

∫S4Tr(F ∧ F ). (64)

Note that p(A) = Tr(A2), A ∈ su(2), is an invariant polinomialof second degree on su(2), therefore Tr(F ∧ F ) represents the secondChern class of the vector bundle ξ with fiber C2 associated to theprincipal bundle.

As H4(S4) ∼= R, the cohomology class c2(ξ) ∈ H4(S4 is completelycaracterized by the integral over S4, therefore we can say that the topo-logical charge is the second Chern class. Note that the topologicalcharge is an integer, this reflects the fact that really ck(ξ) ∈ H(M ;Z).

4.5. Pontryagin characteristic classes. Let ξ be a real vector bun-dle, and ξC its complexification. The Pontryagin characteristic classesare defined as follows:

pk(ξ) = (−1)kck(ξC). (65)

4.6. Euler characteristic class. Let ξ be an oriented real vector bun-dle with metric, rank ξ = 2m. Consider the principal SO(2m)-bundleSO(ξ) = (P, π,M) of oriented orthonormal frames of ξ. Let Ω be thecurvature form of a connection ω in SO(ξ), then Ω ∈ Ω2(P )⊗ o(2m).We take the Pfaffian g (see Example 4.4 and Exercise 16), then theEuler characteristic class, call it χ(ξ), is [γ] ∈ H2m(M), where

π∗γ =(−1)m

22mπmm!

∑εi1i2...i2mΩ

i1i2Ωi3

i4· · ·Ωi2m−1

i2m. (66)

If ξ has odd rank then the Euler class χ(ξ) = 0.

Theorem 4.4. LetM be a 2m-dimensional oriented compact manifold.Then the Euler characteristic ξ(M) can be expressed as follows:

χ(M) = χ(TM) =

∫M

γ, (67)

where γ is given by (66).

If dimM is odd, then we know that χ(M) = 0.

Theorem 4.5 (Properties of the Euler class). Naturality:: Let ξbe a vector bundle over a manifold M , then, if f : M ′ → M isa map, we have χ(f ∗(ξ)) = f ∗(χ(ξ)).

Whitney sum: Let ξ1, ξ2 be vector bundles overM , and rank ξ1 =rank ξ2 = 2. Then ξ(χ1 ⊕ χ2) = ξ(χ1)ξ(χ2).

Normalization: The Euler class of the canonical bundle overCP 1 coincides with the first Chern class.


Example 4.5. The Gauss-Bonnet formula is a partial case of (66).Let Σ be a compact oriented surface with a metric g, and let

Ω =

(0 −αα 0

)(68)

be the curvature form of the Levi-Civita connection of g.Then α = π∗KdA, where K is the Gauss curvature of g and dA is

the area form of g. Therefore, from (66) with m = 1 we get

(−1)

4π(Ω1

2 − Ω21) =

1

2πα = π∗(

1

2πKdA). (69)

Thus γ = 12πKdA and Theorem 4.4 gives us

χ(M) =1

2π

∫Σ

KdA (70)

which is exactly the Gauss-Bonnet formula.For example, the standard metric on the sphere S2 of radius R has

area 4πR2 and curvature R2, therefore χ(S2) = 2.

Exercise 19. Prove that α = π∗(KdA).

5. Characteristic classes of foliation

5.1. Foliations. A foliation F is an integrable distribution, called TFon a manifold M , the maximal integral manifolds of the distributionare called leaves of foliation. The vector bundle NF = TM/F is calledthe normal bundle of F .

A foliation determines equivalence relation on a manifold M : twopoints of M are equivalent if they belong to the same leaf of the folia-tion, by M/F we denote the quotient space.

Example 5.1. The total space M of a bundle π : M → B has thefoliation whose leaves are the fibers of the bundle. The quotient spaceM/F ∼= B is a manifold.

Example 5.2. The set of lines with the same slope α on the torusT2 = R2/Z2 forms a Kronecker foliation. If θ ∈ R \ Q, then all theleaves are dense and T2/F is not Hausdorff and does not admit astructure of manifold.

A manifold with foliation admits an atlas with coordinates (xi, yα)and coordinate changes of the form

xi′= f i′(xj) yα

′= gα

′(xj, yβ). (71)

The connected components of intersection of leaves of foliation withcoordinate charts are given by xi = const, and the distribution TF is

18 M. MALAKHALTSEV

given by equations dxi = 0. The xi are called basic coordinates and yα

transversal coordinates.The basic coordinates play role of coordinates on the quotient space

M/F (which is not a manifold in general). One can consider basicfunctions f = f(xi), basic forms fi1...imdx

i1 ∧ · · · ∧ dxim , basic vectorfields V = V i(xj) ∂

∂xi which are sections of the normal bundle, etc, soone can study geometry of the base space M/F . In what follows wewill construct “bundles over M/F”.

5.2. Partial connections and foliated bundles. Let π : P → Mbe a principal G-bundle. A partial connection in P is a G-invariantdistribution H on P such that H ∩ V = 0. A connection H on P is

called adapted to H if H ⊂ H.

Theorem 5.1. Let (M,F) be a foliation, and π : P →M be a principalG-bundle. The following statements are equivalent:

1) P can be endowed with a partial connection H such that

(a) H is an integrable distribution (H is a flat partial connec-tion);

(b) dπ(H) = TF .

In this case we have a foliation F on P which is called the liftof the foliation F .

2) There exists an atlas of the bundle P with gluing functions de-pending only on the base coordinates (the gluing functions areconstant along the leaves).

5.3. Weil algebra. Let g be a Lie algebra. The Weil algebra W (g) ofg is

W (g) = Λ(g)⊗ S(g), (72)

where Λ(g) is the algebra of skew-symmetric tensors on g, and S(g) isthe algebra of symmetric tensors on g. This algebra is graded:

W i,2j = Λi(g)⊗ Sj(g). (73)

We have g∗ = Λ1(g) = S1(g). We will denote α ∈ S1(g) = g∗ by α.For each x ∈ g define the interior derivation ix determined by

ixα = α(x), ixα = 0, (74)

and the Lie derivative

Lxα(y) = −α([x, y]), Lxα(y) = −α([x, y]). (75)


The differential dW : W (g) → W (g),

dW = d′ + d′′ (76)

d′α(x, y) = −α([x, y]), d′α =∑j

ej ⊗ Lej α. (77)

d′′α = α, d′′α = 0. (78)

So (W (g), dW ) is a differential graded algebra (DG-algebra).Note also that if g is the Lie algebra of a Lie group G, then G acts

g by the adjoint representation and on W (g) we have:

gt(x1, · · · , xm) = t(adg−1x1, · · · , adg−1xm). (79)

Theorem 5.2.

H(W (g), dW ) ∼= R. (80)

The Weil algebra admits a filtration F 2pW (g) = SpgW (g). Let usset

W (g)k = W (g)/F 2(k+1)W (g). (81)

Let H be a Lie subgroup of the Lie group G. Then

W (g, H) = t ∈ W (g) | ht = t,∀h ∈ H; iyt = 0,∀y ∈ h . (82)

is a DG-subalgebra of (W (g), dW ) (i. e. closed with respect to dW ).The filtration FW (g) defines a filtration of W (g, H), so we also have

W (g)k = W (g, H)/F 2(k+1)W (g, H). (83)

Exercise 20. Find H(W (g, GL(1))), where g is the Lie algebra ofaffine transformations of R.

5.4. Chern-Weil homomorphism. Let π : P → M be a principalG-bundle, G0 ⊂ G is a Lie subgroup, and P0 ⊂ P a principal G0-subbundle. Let s : M → P/G0 be the section corresponding to thereduction.

Take a connection H in the bundle P . The Chern-Weil homomor-phism

kω : W (g, G0) → Ω(P ), kω(t) = t(ω, · · · , ω,Ω, · · · ,Ω) (84)

induces the map

∆ = s∗ kω : HW (g, G0) → HM (85)

20 M. MALAKHALTSEV

5.5. Characteristic classes of foliations. Let (M,F) be a folia-tion of codimension q, NF its normal bundle, L(N(F)) the line framebundle of NF . Any vector bundle admits a metric, so the principalGL(q)-bundle L(NF) has principal O(q)-subbundle SO(NF). So wehave the map ∆ defined by (85).

The cohomology class ∆(a) ∈ HM , a ∈ H(W (gl(q), O(q))q) is calleda characteristic class of F .

Example 5.3 (The Godbillon-Vey characteristic class). Let (M,F) bea co-oriented foliation, dimM = 3, codimF = 1.

In this case q = 1, so GL+(1) ∼= R, SO(1) is the trivial group. Nowtake the element c = e⊗ e, and we have d(e⊗ e) = e⊗ e ∈ W 4(gl(1)).So dW c = 0 in W (gl(1))1, and there is defined [c] ∈ H3(W (gl(1))1).

The foliation F can be given by a 1-form θ such that dθ = φ ∧ θ.Then the 3-form α = φ ∧ dφ is closed, and the cohomology class of αdoes not depend on the choice of θ. The class GV (F) = [α] ∈ H3(M)is called the Godbillon-Vey characteristic class of the foliation F .

Theorem 5.3. ∆([c]) = GV (F).

Exercise 21. Prove that [φ ∧ dφ] does not depend on a choice of θ.

6. Answers to some exercises

1. k = 1.2. χ(Mk) = 2− 2k, χ(Nk) = 2− k.3. a) 1; b) 1; c) −1.4. Take a constant vector field on the plane and use stereographic

projection.5. There exist cα : Uα → Diff(Y ) such that φβα = cβc

−1α .

6. a) vi′= ∂xi′

∂xi vi; b) GL(n).

7. No.9. Take a covering subordinate to both coverings.10. Yes.11. λ(−y∂x + x∂y).14. xdy.16. Represent the determinant of a skew-symmetric matrix as the

square of a polynomial of its entries.20. R[x].

7. What to read?

1. Bott R., Tu L., Differential Forms in Algebraic Topology, Springer-Verlag, 1982.


2. Guillemin V., Sternberg S., Supersymmetry and Equivariant deRham Theory, Springer-Verlag, 1999.

3. Kamber F., Tondeur Ph., Foliated Bundles and CharacteristicClasses, Springer-Verlag, 1975.

4. Kobayashi S., Nomizu K., Foundation of Differential Geometry,Jhon Wiley and Sons, 1963.

5. Milnor J., Stasheff J. D., Characteristic Classes, Annals of Math-ematics Studies, No. 76. Princeton, NJ: Princeton University Press;Tokyo: University of Tokyo Press, 1974.

6. Morita S., Geometry of Differential Forms, American Mathemat-ical Society, 2001.

7. Warner. F., Foundations of Differentiable Manifolds and LieGroups, Springer-Verlag, 1971.

Universidad de los AndesE-mail address: [email protected]

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