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
over
, its
-th Pontryagin class
is defined as
pk(E)=pk(E,\Z)=(-1)kc2k(E ⊗ \Complex)\inH4k(M,\Z),
where:
denotes the
-th
Chern class of the
complexification
of
,
is the
-
cohomology group of
with
integer coefficients.
The rational Pontryagin class
is defined to be the image of
in
, the
-cohomology group of
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(E ⊕ F)=2p(E)\smilep(F)
for two vector bundles
and
over
. In terms of the individual Pontryagin classes
,
2p1(E ⊕ F)=2p1(E)+2p1(F),
2p2(E ⊕ F)=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
over the
9-sphere. (The
clutching function for
arises from the homotopy group
.) The Pontryagin classes and Stiefel-Whitney classes all vanish: the Pontryagin classes don't exist in degree 9, and the Stiefel–Whitney class
of
vanishes by the Wu formula
. Moreover, this vector bundle is stably nontrivial, i.e. the Whitney sum of
with any trivial bundle remains nontrivial.
Given a
-dimensional vector bundle
we have
where
denotes the
Euler class of
, and
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
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.
over a
-dimensional
differentiable manifold
equipped with a
connection, the total Pontryagin class is expressed as
2)}{8\pi
2)2-2{\rmTr}(\Omega4)}{128\pi
2)3-6{\rmTr}(\Omega2){\rmTr}(\Omega4)+8{\rmTr}(\Omega6)}{3072\pi6}+ … \right]\in
(M),
where
denotes the
curvature form, and
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
in
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
is completely determined by its Chern classes. This follows from the fact that
, the Whitney sum formula, and properties of Chern classes of its complex conjugate bundle. That is,
and
c(E ⊕ \bar{E})=c(E)c(\bar{E})
. Then, this given the relation
1-p1(E)+p2(E)- … +
=(1+c1(E)+ … +cn(E)) ⋅ (1-c1(E)+c2(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
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-
+2c2(E)
showing
. On line bundles this simplifies further since
by dimension reasons.
Pontryagin classes on a Quartic K3 Surface
Recall that a quartic polynomial whose vanishing locus in
is a smooth subvariety is a K3 surface. If we use the normal sequence
0\tol{T}X\to
|X\tol{O}(4)\to0
we can find
\begin{align}
c(l{T}X)&=
|X)}{c(l{O}(4))}\\
&=
\\
&=(1+4[H]+6[H]2) ⋅ (1-4[H]+16[H]2)\\
&=1+6[H]2
\end{align}
showing
and
. Since
corresponds to four points, due to Bézout's lemma, we have the second chern number as
. Since
in this case, we have
. 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
vanishes if the dimension of
is not divisible by 4. It is defined in terms of the Pontryagin classes of the
manifold
as follows:
Given a smooth
-dimensional manifold
and a collection of natural numbers
such that
,the Pontryagin number
is defined by
\smile
\smile … \smile
([M])
where
denotes the
-th Pontryagin class and
the
fundamental class of
.
Properties
- Pontryagin numbers are oriented cobordism invariant; and together with Stiefel-Whitney numbers they determine an oriented manifold's oriented cobordism class.
- 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.
- Invariants such as signature and
-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
- Book: Milnor John W. . John Milnor . Stasheff, James D. . Jim Stasheff . Characteristic classes . Annals of Mathematics Studies . 76 . Princeton University Press / University of Tokyo Press . Princeton, New Jersey; Tokyo . 1974 . 0-691-08122-0.
- Book: Hatcher, Allen . Allen Hatcher . Vector Bundles & K-Theory . 2.1 . 2009 .
Notes and References
- Web site: De Rham Cohomology - an overview ScienceDirect Topics. 2022-02-02. www.sciencedirect.com.
- 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.
- 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.