Pontryagin class explained

In mathematics, the Pontryagin classes, named after Lev Pontryagin, are certain characteristic classes of real vector bundles. The Pontryagin classes lie in cohomology groups with degrees a multiple of four.

Definition

Given a real vector bundle

E

over

M

, its

k

-th Pontryagin class

pk(E)

is defined as

pk(E)=pk(E,\Z)=(-1)kc2k(E\Complex)\inH4k(M,\Z),

where:

c2k(E\Complex)

denotes the

2k

-th Chern class of the complexification

E\Complex=EiE

of

E

,

H4k(M,\Z)

is the

4k

-cohomology group of

M

with integer coefficients.

The rational Pontryagin class

pk(E,\Q)

is defined to be the image of

pk(E)

in

H4k(M,\Q)

, the

4k

-cohomology group of

M

with rational coefficients.

Properties

The total Pontryagin class

p(E)=1+p1(E)+p2(E)+ … \inH*(M,\Z),

is (modulo 2-torsion) multiplicative with respect to Whitney sum of vector bundles, i.e.,

2p(EF)=2p(E)\smilep(F)

for two vector bundles

E

and

F

over

M

. In terms of the individual Pontryagin classes

pk

,

2p1(EF)=2p1(E)+2p1(F),

2p2(EF)=2p2(E)+2p1(E)\smilep1(F)+2p2(F)

and so on.

The vanishing of the Pontryagin classes and Stiefel–Whitney classes of a vector bundle does not guarantee that the vector bundle is trivial. For example, up to vector bundle isomorphism, there is a unique nontrivial rank 10 vector bundle

E10

over the 9-sphere. (The clutching function for

E10

arises from the homotopy group

\pi8(O(10))=\Z/2\Z

.) The Pontryagin classes and Stiefel-Whitney classes all vanish: the Pontryagin classes don't exist in degree 9, and the Stiefel–Whitney class

w9

of

E10

vanishes by the Wu formula

w9=w1w8+

1(w
Sq
8)
. Moreover, this vector bundle is stably nontrivial, i.e. the Whitney sum of

E10

with any trivial bundle remains nontrivial.

Given a

2k

-dimensional vector bundle

E

we have

pk(E)=e(E)\smilee(E),

where

e(E)

denotes the Euler class of

E

, and

\smile

denotes the cup product of cohomology classes.

Pontryagin classes and curvature

As was shown by Shiing-Shen Chern and André Weil around 1948, the rational Pontryagin classes

pk(E,Q)\inH4k(M,Q)

can be presented as differential forms which depend polynomially on the curvature form of a vector bundle. This Chern–Weil theory revealed a major connection between algebraic topology and global differential geometry.

E

over a

n

-dimensional differentiable manifold

M

equipped with a connection, the total Pontryagin class is expressed as
p=\left[1-{\rmTr
(\Omega

2)}{8\pi

2}+{\rmTr
(\Omega

2)2-2{\rmTr}(\Omega4)}{128\pi

4}-{\rmTr
(\Omega

2)3-6{\rmTr}(\Omega2){\rmTr}(\Omega4)+8{\rmTr}(\Omega6)}{3072\pi6}+ … \right]\in

*
H
dR

(M),

where

\Omega

denotes the curvature form, and
*
H
dR

(M)

denotes the de Rham cohomology groups.[1]

Pontryagin classes of a manifold

The Pontryagin classes of a smooth manifold are defined to be the Pontryagin classes of its tangent bundle.

Novikov proved in 1966 that if two compact, oriented, smooth manifolds are homeomorphic then their rational Pontryagin classes

pk(M,Q)

in

H4k(M,Q)

are the same.

If the dimension is at least five, there are at most finitely many different smooth manifolds with given homotopy type and Pontryagin classes.

Pontryagin classes from Chern classes

The Pontryagin classes of a complex vector bundle

\pi:E\toX

is completely determined by its Chern classes. This follows from the fact that

ERC\congE\bar{E}

, the Whitney sum formula, and properties of Chern classes of its complex conjugate bundle. That is,

ci(\bar{E})=

ic
(-1)
i(E)
and

c(E\bar{E})=c(E)c(\bar{E})

. Then, this given the relation

1-p1(E)+p2(E)-+

np
(-1)
n(E)

=(1+c1(E)++cn(E))(1-c1(E)+c2(E)- … +

nc
(-1)
n(E))
[2]
for example, we can apply this formula to find the Pontryagin classes of a complex vector bundle on a curve and a surface. For a curve, we have

(1-c1(E))(1+c1(E))=1+

2
c
1(E)
so all of the Pontryagin classes of complex vector bundles are trivial. On a surface, we have

(1-c1(E)+c2(E))(1+c1(E)+c2(E))=1-

2
c
1(E)

+2c2(E)

showing

p1(E)=

2
c
1(E)

-2c2(E)

. On line bundles this simplifies further since

c2(L)=0

by dimension reasons.

Pontryagin classes on a Quartic K3 Surface

Recall that a quartic polynomial whose vanishing locus in

CP3

is a smooth subvariety is a K3 surface. If we use the normal sequence

0\tol{T}X\to

l{T}
CP3

|X\tol{O}(4)\to0

we can find

\begin{align} c(l{T}X)&=

c(l{T
CP3

|X)}{c(l{O}(4))}\\ &=

(1+[H])4
(1+4[H])

\\ &=(1+4[H]+6[H]2)(1-4[H]+16[H]2)\\ &=1+6[H]2 \end{align}

showing

c1(X)=0

and

c2(X)=6[H]2

. Since

[H]2

corresponds to four points, due to Bézout's lemma, we have the second chern number as

24

. Since

p1(X)=-2c2(X)

in this case, we have

p1(X)=-48

. This number can be used to compute the third stable homotopy group of spheres.[3]

Pontryagin numbers

Pontryagin numbers are certain topological invariants of a smooth manifold. Each Pontryagin number of a manifold

M

vanishes if the dimension of

M

is not divisible by 4. It is defined in terms of the Pontryagin classes of the manifold

M

as follows:

Given a smooth

4n

-dimensional manifold

M

and a collection of natural numbers

k1,k2,\ldots,km

such that

k1+k2+ … +km=n

,the Pontryagin number
P
k1,k2,...,km
is defined by
P
k1,k2,...,km
=p
k1

\smile

p
k2

\smile\smile

p
km

([M])

where

pk

denotes the

k

-th Pontryagin class and

[M]

the fundamental class of

M

.

Properties

  1. Pontryagin numbers are oriented cobordism invariant; and together with Stiefel-Whitney numbers they determine an oriented manifold's oriented cobordism class.
  2. Pontryagin numbers of closed Riemannian manifolds (as well as Pontryagin classes) can be calculated as integrals of certain polynomials from the curvature tensor of a Riemannian manifold.
  3. Invariants such as signature and

    \hatA

    -genus can be expressed through Pontryagin numbers. For the theorem describing the linear combination of Pontryagin numbers giving the signature see Hirzebruch signature theorem.

Generalizations

There is also a quaternionic Pontryagin class, for vector bundles with quaternion structure.

See also

References

Notes and References

  1. Web site: De Rham Cohomology - an overview ScienceDirect Topics. 2022-02-02. www.sciencedirect.com.
  2. Web site: Pontryagin Classes. Mclean. Mark. live. https://web.archive.org/web/20161108093927/https://www.math.stonybrook.edu/~markmclean/MAT566/lecture13.pdf. 2016-11-08.
  3. Web site: A Survey of Computations of Homotopy Groups of Spheres and Cobordisms. 16. live. https://web.archive.org/web/20160122111116/http://math.mit.edu/~guozhen/homotopy%20groups.pdf. 2016-01-22.