Marathe K. Topics in Physical Mathematics

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138 5 Characteristic Classes

generalized to Riemannian polyhedra by Allendoerﬀer and Weil and to arbi-

trary manifolds by Chern. In this latter generalization the Gaussian integral

curvature is replaced by an invariant formed from the Riemann curvature. It

forms the starting point of the theory of characteristic classes.

Gauss’ idea of studying the geometry of a surface intrinsically, without

leaving it (i.e., by means of measurements made on the surface itself), is

of fundamental importance in modern diﬀerential geometry and its applica-

tions to physical theories. We are similarly compelled to study the geome-

try of the three-dimensional physical world by the intrinsic method, i.e.,

without leaving it. This idea was already implicit in Riemann’s work, which

extended Gauss’ intrinsic method to the study of manifolds of arbitrary di-

mension. This work together with the work of Ricci and Levi-Civita provided

the foundation for Einstein’s theory of general relativity. The constructions

discussed in this chapter extend these ideas and provide important tools for

modern mathematical physics.

5.2 Classifying Spaces

Let G be a Lie group. The classiﬁcation of principal G-bundles over a manifold

M is achieved by the use of classifying spaces. A topological space B

(G)

is said to be k-classifying for G if the following conditions hold:

1. There exists a contractible space E

(G)onwhichG acts freely and B

(G)

is the quotient of E

(G) under this G-actionsuchthat

(G) → B

(G)

is a principal ﬁber bundle with structure group G.

2. Given a manifold M of dim ≤ k and a principal bundle P (M, G), there ex-

ists a continuous map f : M → B

(G) such that the pull-back f

∗

(G))

to M is a principal bundle with structure group G that is isomorphic to P .

It can be shown that homotopic maps give rise to equivalent bundles and

that all principal G-bundles over M arise in this way. Let [M, B

(G)] denote

the set of equivalence classes under homotopy, of maps from M to B

(G).

Then the classifying property may be stated as follows:

Theorem 5.1 (Classifying property) Let M be a compact, connected man-

ifold and G a compact, connected Lie group. Then there exists a one-to-one

correspondence between the set [M, B

(G)] of homotopy classes of maps and

the set of isomorphism classes of principal G-bundles over M.

The spaces E

(G)andB

(G) may be taken to be manifolds for a

ﬁxed k. However, classifying spaces can be constructed for arbitrary ﬁnite-

dimensional manifolds. They are denoted by E(G)andB(G)andareingen-

5.3 Characteristic Classes 139

eral inﬁnite-dimensional. The spaces E(G)andB(G) are called universal

classifying spaces for principal G-bundles.

Example 5.1 The complex Hopf ﬁbration S

2n+1

→ CP

is a principal

U(1)-bundle and is n-classifying. By forming a tower of these ﬁbrations by

inclusion, i.e., by considering two series of inclusions

⊂ S

⊂···

⊂ CP

⊂···,

we obtain a direct system of principal U(1)-bundles. Taking the direct limit

of this system we get the spaces S

∞

and CP

∞

such that S

∞

is a principal

U(1)-ﬁbration over CP

∞

.ThusB(U (1)) = CP

∞

and E(U (1)) = S

∞

.This

classiﬁcation is closely related to the electromagnetic ﬁeld as a U (1)-gauge

ﬁeld and, in particular, to the construction of the Dirac monopole and the

monopole quantization condition.

A similar argument applied to the quaternionic Hopf ﬁbration gives the

classifying spaces for principal SU(2)-bundles as indicated in the following

example.

Example 5.2 The quaternionic Hopf ﬁbration S

4n+3

→ HP

is a prin-

cipal SU(2)-bundle and is n-classifying. By considering the towers

⊂ S

⊂···

⊂ HP

⊂···,

we obtain the universal classifying spaces B(SU(2)) = HP

∞

and E(SU(2))

= S

∞

. This example is closely related to the classiﬁcation of Yang–Mills

ﬁelds with gauge group SU(2), and, in particular, to the construction and

classiﬁcation of instantons.

5.3 Characteristic Classes

The beginning of the theory of characteristic classes was made in the

1930s by Stiefel and Whitney. Stiefel studied certain homology classes of the

tangent bundle of a smooth manifold, while Whitney considered the case

of an arbitrary sphere bundle and introduced the concept of a characteristic

cohomology class. In the next decade, Pontryagin constructed important new

characteristic classes by studying the homology of real Grassmann manifolds,

and Chern deﬁned characteristic classes for complex vector bundles. Chern’s

study of the cohomology of complex Grassmann manifolds also led to a better

understanding of Pontryagin’s real characteristic classes.

We begin with a brief discussion of the Stiefel–Whitney classes in terms

of certain cohomology operations. Recall ﬁrst that the de Rham cohomology

140 5 Characteristic Classes

has a graded algebra structure induced by the exterior product. This induced

product in cohomology is in fact a special case of a cohomology operation

in algebraic topology called the cup product:

∪ : H

(M; P) ×H

(M; P) → H

i+j

(M; P), (α, β) → α ∪ β.

The cup product induces the structure of a graded ring on the cohomology

space H

∗

(M; P). The ring H

∗

(M; P) is called the cohomology ring of M .If

P is a ﬁeld, then the ring H

∗

(M; P)isaP -algebra called the cohomology

algebra of M . In the rest of this paragraph we take P = Z

and omit

its explicit indication. We also use the Steenrod squaring operations

,i≥ 0, which are characterized by the following four properties.

1. For each topological pair (X, A)and∀n ≥ 0themap

: H

(X, A) → H

n+i

(X, A)

is an additive homomorphism.

2. (Naturality)Iff :(X, A) → (Y,B) is a morphism of topological pairs,

then

◦ f

∗

= f

∗

◦ Sq

where f

∗

: H

∗

(Y,B) → H

∗

(X, A) is the homomorphism induced by f .

3. If α ∈ H

(X, A), then

(α)=α, Sq

(α)=α ∪α and Sq

(α)=0, ∀i>n.

4. (Cartan formula)Letα, β be such that α ∪ β is deﬁned. Then

(α ∪β)=



i+j=k

(α) ∪ Sq

(β).

Theorem 5.2 Let

be a real vector bundle of rank n (i.e., dim F = n) over the base manifold M

and let E

be the complement of the image of the zero section θ of E over

M.LetE

= E

∩ E

,x∈ M, then we have the following results.

5.3 Characteristic Classes 141



0 i = n,

i = n.

(E, E



0 i<n,

i−n

(M) i ≥ n.

3. There exists a unique class u ∈ H

(E,E

) such that ∀x ∈ M the class u

restricted to (F

) is the unique non-zero class in H

).Further-

more, the cup product by u deﬁnes an isomorphism c

: α → α ∪ u of H

(E) → H

k+n

(E,E

), ∀k. (5.3)

The class u introduced in the above theorem is called the Thom class.Now

the zero section θ embeds M as a deformation retract of E with retraction

map π : E → M. Thus, π

∗

: H

(M) → H

(E) is an isomorphism for all

k.TheThom isomorphism φ : H

(M) → H

k+n

(E,E

) is deﬁned as the

composition of the two isomorphisms π

∗

and c

(M)

∗

−→ H

(E)

∪u

−→ H

k+n

(E,E

),φ(α)=π

∗

(α) ∪u.

Composing the Steenrod squaring operation Sq

with φ

−1

results in a homo-

morphism

(E,E

)

−→ H

n+k

(E,E

)

−1

−→ H

(M), ∀k ≥ 0.

We deﬁne the kth Stiefel–Whitney class w

(ξ) ∈ H

(M) of the vector bundle

ξ =(E,π,M) as the image of the Thom class u under the above homomor-

phism, i.e.,

(ξ)=φ

−1

(Sq

(u)). (5.4)

We remark that the deﬁnitions of the Thom class and the Thom isomorphism

extend to the case of oriented vector bundles and their integral cohomology.

Theorem 5.2 also extends to this case with obvious modiﬁcations. The class

χ ∈ H

(M; Z), which corresponds to the Thom class under the canonical

isomorphism π

∗

: H

(M; Z) → H

(E; Z), is called the Euler class of the

oriented bundle ξ. Moreover, we have the following proposition.

Proposition 5.3 Let M be an n-dimensional manifold. The natural homo-

morphism H

(M; Z) → H

(M; Z

), induced by the reduction of coeﬃcients

mod 2, maps the Euler class χ(ξ) to the Stiefel–Whitney class w

(ξ) in the

top dimension.

The Stiefel–Whitney classes are characterized by the following four prop-

erties.

142 5 Characteristic Classes

(ξ) ∈ H

(M; Z

),i≥ 0.

In particular, w

(ξ)=1andw

(ξ)=0fori>rank(ξ).

2. (Naturality)Ifξ



=(E



,π



) is another vector bundle and f : M →



is covered by a bundle map from ξ to ξ



,then

(ξ)=f

∗

(ξ



)).

3. (Whitney product formula) With the above notation, if M = M



then

(ξ ⊕ ξ





i=0

(ξ) ∪ w

k−i

(ξ



). (5.5)

4. If γ is the canonical line bundle over the real projective space RP

∼

then w

(γ) =0.

We recall that the ring of formal series over H

∗

(M; Z

)istheset



∞



i=0

| a

∈ H

(M; Z

)



with termwise addition and with multiplication induced by the cup product.

One can show that the set of invertible elements in this ring consists of

elements with a

=1.Thetotal Stiefel–Whitney class is deﬁned as the

formal series

w(ξ)=1+w

(ξ)+···+ w

(ξ)+··· . (5.6)

Its inverse

¯w =1+

∞



i=1

¯w

can be computed by formal power series expansion. Each ¯w

is then expressed

in terms of w

,j≤ i. For example,

¯w

= w

, ¯w

= w

+ w

We deﬁne the ith Stiefel–Whitney class of a manifold M by

(M):=w

(TM).

If M can be immersed in R

m+k

then we have the following theorem.

Theorem 5.4 (Whitney duality) Let ν denote the normal bundle of M in

m+k

.Then

(ν)= ¯w

(M), ∀i ∈ N.

In particular, ¯w

(M)=0for i>k.

5.3 Characteristic Classes 143

Applying this theorem to projective spaces, we get the following best es-

timate result for immersions of compact manifolds in Euclidean spaces.

Theorem 5.5 Let m =2

,r∈ N.If RP

can be immersed in R

m+k

then k ≥ m − 1.

Let M be a compact m-dimensional manifold and let

=(r

,...,r

), such that



i=1

= m,

where r

, 1 ≤ i ≤ m, are non-negative integers. Deﬁne the monomial w

(M)

(M):=w

(M)

···w

(M)

The value of w

(M) on the fundamental homology class [M] is called the

Stiefe–Whitney number of M associated to the monomial w

(M). Any

M has a ﬁnite set of Stiefel–Whitney numbers. One important application of

these numbers is found in the following theorem of Pontryagin and Thom.

Theorem 5.6 Let M be a compact m-dimensional manifold. Then M can

be realized as the boundary of a compact (m+1)-dimensional manifold if and

only if all the Stiefel–Whitney numbers of M are zero.

This theorem is the starting point of Thom’s theory of cobordism.

Deﬁnition 5.1 Two closed m-dimensional manifolds M

are said to be

cobordant (M

∼

) if their disjoint union M

M

is the boundary of

acompact(m +1)-dimensional manifold. The relation ∼

is an equivalence

relation, and the equivalence class [M] of M under this relation is called

an unoriented cobordism class of M. The class of closed m-dimensional

manifolds together with unoriented cobordism as morphism is called the m-

dimensional cobordism category.

Theorem 5.6 and the above deﬁnition lead to the following result.

Theorem 5.7 [M

]=[M

] if and only if all Stiefel–Whitney numbers of M

and M

are the same.

Deﬁne the set

= {[M] | M is a closed m-dimensional manifold}. (5.7)

Disjoint union induces an operation on T

(denoted by + ) with which it

is an Abelian group. In view of Theorem 5.7 the group (T

, +) is ﬁnite. It

is called the m-dimensional unoriented cobordism group.Thomproved

that the unoriented cobordism group T

is canonically isomorphic to the

homotopy group π

m+k

of a certain universal space for a suﬃciently large

k.Wenotethat{T

} canbemadeintoagraded ring with the product

induced by Cartesian product of manifolds. The zero element of this ring is

144 5 Characteristic Classes

represented by the class of the empty manifold and the unity by the class of

the one-point manifold. It is called the unoriented cobordism ring.There

are several theories of cobordism for manifolds carrying additional structure

such as orientation, group action, etc. (see Stong [360]). The 2-dimensional

cobordism category was used by Segal in his axioms for conformal ﬁeld theory.

They form the starting point for the Atiyah–Segal axioms for (topological)

quantum ﬁeld theories. These axioms are discussed in Chapter 7.

Let ξ =(E,B,π,F) be a vector bundle with a Riemannian metric. Let

:= {v ∈ E

|v≤1},

n−1

:= {v ∈ E

|v =1}.

The space

D(ξ)=



x∈B

(resp., S(ξ)=



x∈B

n−1

)

has a natural structure of a manifold with which it becomes a bundle over

B, called the disk bundle of ξ (resp., the sphere bundle of ξ). The bun-

dles D(ξ)(resp.,S(ξ)) deﬁned by a diﬀerent choice of the metric on ξ are

equivalent (isomorphic as bundles). The quotient bundle T (ξ)=D(ξ)/S(ξ)

is called the Thom space of ξ. The cobordism groups T

are isomorphic

to π

m+k

(T (γ

)), where γ

is the universal k-plane bundle. Thom has shown

that these isomorphisms induce an isomorphism θ : {T

}→π

(T )ofgraded

rings. Let P (M, G) be a principal bundle over a manifold M with structure

group G. We will deﬁne a set of cohomology classes in H

∗

(M; R) (the coho-

mology ring of M with real coeﬃcients) associated to the principal bundle,

which characterize the bundle up to bundle isomorphism. They are called the

real characteristic classes of P (M,G). There are several diﬀerent ways of

deﬁning these characteristic classes (see Husemoller [198], Kamber and Ton-

deur [215], and Milnor and Stasheﬀ [286]). We deﬁne them by using connec-

tions in the bundle P and the Weil homomorphism deﬁned below.

Let ρ : G → GL(V ) be a representation of G on the real vector space V .

Let S

(V )denotethesetofk-linear symmetric real-valued functions on V

and let

S(V )=⊕

∞

k=0

(V )

be the symmetric algebra of V with product of f ∈ S

(V ),g∈ S

(V )

deﬁned by

(f · g)(v

,...,v

h+k

(h + k)!



σ∈S

h+k

f(v

σ(1)

,...,v

σ(h)

)g(v

σ(h+1)

,...,v

σ(h+k)

where S

,n∈ N, is the symmetric group of permutations on n numbers.

The group G acts on S

(V ) by an action induced by the representation ρ,

which we also denote by ρ,givenby

5.3 Characteristic Classes 145

(ρ(a)f)(v

,...,v

)=f(ρ(a

−1

,...ρ(a

−1

We note that from this deﬁnition it follows that

ρ(a)(ρ(b)f)=ρ(ab)f.

Let (e

,...,e

)beabasisofV .Therelation

,...,x

)=f(v,v,...,v),v=



i=1

allows us to establish a correspondence between functions in S

(V )andthe

space P

of homogeneous polynomials of degree k in n variables with the

coeﬃcient of x

···x

+ ···+ m

= k,givenby

! ···m

f(e

,...,e

  

times

,...,e

  

times

This correspondence can be extended to S(V ) and turns out to be an algebra

homomorphism. The inverse of the mapping f → P

is called polarization.

If P ∈P

,wedenotebyf

the element of S

(V ) obtained by polarization

of P . For example, if P ∈P

,then



P (x

+ y

,...,x

+ y

) −P (x

,...,x

) −P (y

,...,y

)



where v



i=1



i=1

. If P ∈P

,then



P (x

+ y

+ z

,...,x

+ y

+ z

)

− P (x

+ y

,...,x

+ y

) −P (x

+ z

,...,x

+ z

)

− P (y

+ z

,...,y

+ z

)+P (x

,...,x

)

+ P (y

,...,y

)+P (z

,...,z

)



where v



i=1



i=1



i=1

. We deﬁne

(V,ρ):={f ∈ S

(V ) | ρ(a)(f)=f, ∀a ∈ G}. (5.8)

In view of the isomorphism between symmetric functions and homo-

geneous polynomials described above, we call the space I

(V,ρ)thespace

of G-invariant symmetric polynomials of degree k on V , the action

of G by ρ being understood. We deﬁne the space I(V,ρ)ofallG-invariant

symmetric polynomials on V by

I(V, ρ):=

∞



k=0

(V,ρ). (5.9)

146 5 Characteristic Classes

It is easy to verify that I(V,ρ) is a subalgebra of S(V ). In what follows we

will take V to be the Lie algebra g of the Lie group G and ρ to be the adjoint

representation of G on g, and we denote I

(V,ρ)byI

(G)andI(V,ρ)by

I(G).

Let ω be the connection 1-form of a connection on P (M,G)andΩ the

curvature 2-form of ω.Forf ∈ I

(G), we deﬁne the 2k-form f

on P by

,...,X

(2k)!



σ∈S

sgn(σ)f(u

,...,u

), (5.10)

where u

= Ω(X

σ(2i−1)

σ(2i)

) ∈ g, 1 ≤ i ≤ k.

Since the curvature form Ω is tensorial with respect to the adjoint action of

G on g and f is G-invariant, the 2k-form f

on P descends to a 2k-form

on M; i.e., there exists a form

∈ Λ

(M) such that

∗

(

)=f

where π is the canonical bundle projection of P on M.Itcanbeshownthat

the form

is closed and hence deﬁnes a cohomology class [

] ∈ H

(M; R).

Furthermore, this cohomology class turns out to be independent of the choice

of a particular connection ω on P , i.e., if ω

,ω

are two connections on P

with respective curvature forms Ω

,Ω

then [

]=[

] ∈ H

(M; R). In

view of this result we can deﬁne a map

: I

(G) → H

(M; R)byw

(f):=[

]. (5.11)

The family of maps {w

} deﬁnes the map

w : I(G) → H

∗

(M; R). (5.12)

The map w is called the Weil homomorphism.Wenotethatw is an

algebra homomorphism of the algebra I(G)intothecohomology algebra

∗

(M; R). The image of w is an algebra called the real characteristic

algebra of the bundle P . The construction discussed above extends to the

algebra I

(G) of the complex valued G-invariant polynomials on g to give

us the complex algebra homomorphism

: I

(G) → H

∗

(M; C). (5.13)

This is called the Chern–Weil homomorphism. The image of w

is an

algebra called the complex characteristic algebra of the bundle P .

An element p ∈ w(I(G)) ⊂ H

∗

(M; R) is called a real characteristic

class of the bundle P . Similarly, an element c ∈ w

(G)) ⊂ H

∗

(M; C)is

called a complex characteristic class of the bundle P .

5.3 Characteristic Classes 147

Characteristic classes are topological invariants of the principal bundle P

and characterize P up to isomorphism. They can also be viewed as topo-

logical invariants of the vector bundle associated to P by the fundamental

or deﬁning representation of the structure group G. It is possible to give an

axiomatic formulation of characteristic classes of vector bundles directly (see

Kobayashi and Nomizu [226] and Milnor and Stasheﬀ [286]). In studying the

properties of characteristic classes we will use either of these formulations as

is convenient.

For the Lie groups that are commonly encountered in physical theories we

can show that the algebra of characteristic classes is ﬁnitely generated. We

also exhibits a basis for each such algebra. Before giving several examples

of this, we recall the correspondence between homogeneous polynomials of

degree k and symmetric functions in S

(V ). In the following examples V is a

vector space of matrices and the polynomial variables are the entries of these

matrices. In view of this remark, in the examples discussed below, we give

only the homogeneous polynomials p

of degree k for the construction of the

characteristic classes.

Example 5.3 Let G = GL(n, R); then the characteristic algebra I(G) is

generated by p

,...,p

,wherethep

are deﬁned by

det



λI

−

2π





i=0

n−i

,A∈ gl(n, R).

If Ω is the curvature form of some connection ω on P ,thekth Pontryagin

class of P is deﬁned to be the cohomology class p

represented by the unique

closed 4k-form β

on M such that π

∗

(β

)=(p

)

; i.e., the kth Pontryagin

class is given by

:= w(p

)=[β

The 4k-form (p

)

can be expressed as

)

(2π)

(2k)!



,...,j

,...,i

∧···∧Ω

, (5.14)

where the sum is over all ordered subsets (i

,...,i

) of 2k diﬀerent elements

in (1,...,n) and all permutations (j

,...,j

) of (i

,...,i

), δ

,...,j

,...,i

de-

notes the sign of the permutation and the Ω

are the components of Ω in

gl(n, R).

The reason for considering only p

with even index i is given in the following

example.

Example 5.4 Let G = O(n, R). The characteristic algebra I(G) is generated

by p

,... deﬁned as in Example 5.3. Observe that for A ∈ o(n, R)=

so(n, R) one has