Pontryagin class

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 ${\displaystyle E}$ over ${\displaystyle M}$, its ${\displaystyle k}$-th Pontryagin class ${\displaystyle p_{k}(E)}$ is defined as

${\displaystyle p_{k}(E)=p_{k}(E,\mathbb {Z} )=(-1)^{k}c_{2k}(E\otimes \mathbb {C} )\in H^{4k}(M,\mathbb {Z} ),}$

where:

• ${\displaystyle c_{2k}(E\otimes \mathbb {C} )}$ denotes the ${\displaystyle 2k}$-th Chern class of the complexification ${\displaystyle E\otimes \mathbb {C} =E\oplus iE}$ of ${\displaystyle E}$,
• ${\displaystyle H^{4k}(M,\mathbb {Z} )}$ is the ${\displaystyle 4k}$-cohomology group of ${\displaystyle M}$ with integer coefficients.

The rational Pontryagin class ${\displaystyle p_{k}(E,\mathbb {Q} )}$ is defined to be the image of ${\displaystyle p_{k}(E)}$ in ${\displaystyle H^{4k}(M,\mathbb {Q} )}$, the ${\displaystyle 4k}$-cohomology group of ${\displaystyle M}$ with rational coefficients.

Properties

The total Pontryagin class

${\displaystyle p(E)=1+p_{1}(E)+p_{2}(E)+\cdots \in H^{*}(M,\mathbb {Z} ),}$

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

${\displaystyle 2p(E\oplus F)=2p(E)\smile p(F)}$

for two vector bundles ${\displaystyle E}$ and ${\displaystyle F}$ over ${\displaystyle M}$. In terms of the individual Pontryagin classes ${\displaystyle p_{k}}$,

${\displaystyle 2p_{1}(E\oplus F)=2p_{1}(E)+2p_{1}(F),}$

${\displaystyle 2p_{2}(E\oplus F)=2p_{2}(E)+2p_{1}(E)\smile p_{1}(F)+2p_{2}(F)}$

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 ${\displaystyle E_{10}}$ over the 9-sphere. (The clutching function for ${\displaystyle E_{10}}$ arises from the homotopy group ${\displaystyle \pi _{8}(\mathrm {O} (10))=\mathbb {Z} /2\mathbb {Z} }$.) The Pontryagin classes and Stiefel-Whitney classes all vanish: the Pontryagin classes don't exist in degree 9, and the Stiefel–Whitney class ${\displaystyle w_{9}}$ of ${\displaystyle E_{10}}$ vanishes by the Wu formula ${\displaystyle w_{9}=w_{1}w_{8}+Sq^{1}(w_{8})}$. Moreover, this vector bundle is stably nontrivial, i.e. the Whitney sum of ${\displaystyle E_{10}}$ with any trivial bundle remains nontrivial. ( Hatcher 2009 , p. 76)

Given a ${\displaystyle 2k}$-dimensional vector bundle ${\displaystyle E}$ we have

${\displaystyle p_{k}(E)=e(E)\smile e(E),}$

where ${\displaystyle e(E)}$ denotes the Euler class of ${\displaystyle E}$, and ${\displaystyle \smile }$ 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

${\displaystyle p_{k}(E,\mathbf {Q} )\in H^{4k}(M,\mathbf {Q} )}$

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.

For a vector bundle ${\displaystyle E}$ over a ${\displaystyle n}$-dimensional differentiable manifold ${\displaystyle M}$ equipped with a connection, the total Pontryagin class is expressed as

${\displaystyle p=\left[1-{\frac {{\rm {Tr}}(\Omega ^{2})}{8\pi ^{2}}}+{\frac {{\rm {Tr}}(\Omega ^{2})^{2}-2{\rm {Tr}}(\Omega ^{4})}{128\pi ^{4}}}-{\frac {{\rm {Tr}}(\Omega ^{2})^{3}-6{\rm {Tr}}(\Omega ^{2}){\rm {Tr}}(\Omega ^{4})+8{\rm {Tr}}(\Omega ^{6})}{3072\pi ^{6}}}+\cdots \right]\in H_{dR}^{*}(M),}$

where ${\displaystyle \Omega }$ denotes the curvature form, and ${\displaystyle H_{dR}^{*}(M)}$ 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 ${\displaystyle p_{k}(M,\mathbf {Q} )}$ in ${\displaystyle H^{4k}(M,\mathbf {Q} )}$ 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 ${\displaystyle \pi :E\to X}$ is completely determined by its Chern classes. This follows from the fact that ${\displaystyle E\otimes _{\mathbb {R} }\mathbb {C} \cong E\oplus {\bar {E}}}$, the Whitney sum formula, and properties of Chern classes of its complex conjugate bundle. That is, ${\displaystyle c_{i}({\bar {E}})=(-1)^{i}c_{i}(E)}$ and ${\displaystyle c(E\oplus {\bar {E}})=c(E)c({\bar {E}})}$. Then, this given the relation

${\displaystyle 1-p_{1}(E)+p_{2}(E)-\cdots +(-1)^{n}p_{n}(E)=(1+c_{1}(E)+\cdots +c_{n}(E))\cdot (1-c_{1}(E)+c_{2}(E)-\cdots +(-1)^{n}c_{n}(E))}$ ^[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

${\displaystyle (1-c_{1}(E))(1+c_{1}(E))=1+c_{1}(E)^{2}}$

so all of the Pontryagin classes of complex vector bundles are trivial. On a surface, we have

${\displaystyle (1-c_{1}(E)+c_{2}(E))(1+c_{1}(E)+c_{2}(E))=1-c_{1}(E)^{2}+2c_{2}(E)}$

showing ${\displaystyle p_{1}(E)=c_{1}(E)^{2}-2c_{2}(E)}$. On line bundles this simplifies further since ${\displaystyle c_{2}(L)=0}$ by dimension reasons.

Pontryagin classes on a Quartic K3 Surface

Recall that a quartic polynomial whose vanishing locus in ${\displaystyle \mathbb {CP} ^{3}}$ is a smooth subvariety is a K3 surface. If we use the normal sequence

${\displaystyle 0\to {\mathcal {T}}_{X}\to {\mathcal {T}}_{\mathbb {CP} ^{3}}|_{X}\to {\mathcal {O}}(4)\to 0}$

we can find

${\displaystyle {\begin{aligned}c({\mathcal {T}}_{X})&={\frac {c({\mathcal {T}}_{\mathbb {CP} ^{3}}|_{X})}{c({\mathcal {O}}(4))}}\\&={\frac {(1+[H])^{4}}{(1+4[H])}}\\&=(1+4[H]+6[H]^{2})\cdot (1-4[H]+16[H]^{2})\\&=1+6[H]^{2}\end{aligned}}}$

showing ${\displaystyle c_{1}(X)=0}$ and ${\displaystyle c_{2}(X)=6[H]^{2}}$. Since ${\displaystyle [H]^{2}}$ corresponds to four points, due to Bezout's lemma, we have the second chern number as ${\displaystyle 24}$. Since ${\displaystyle p_{1}(X)=-2c_{2}(X)}$ in this case, we have

${\displaystyle p_{1}(X)=-48}$. 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 ${\displaystyle M}$ vanishes if the dimension of ${\displaystyle M}$ is not divisible by 4. It is defined in terms of the Pontryagin classes of the manifold ${\displaystyle M}$ as follows:

Given a smooth ${\displaystyle 4n}$-dimensional manifold ${\displaystyle M}$ and a collection of natural numbers

${\displaystyle k_{1},k_{2},\ldots ,k_{m}}$ such that ${\displaystyle k_{1}+k_{2}+\cdots +k_{m}=n}$,

the Pontryagin number ${\displaystyle P_{k_{1},k_{2},\dots ,k_{m}}}$ is defined by

${\displaystyle P_{k_{1},k_{2},\dots ,k_{m}}=p_{k_{1}}\smile p_{k_{2}}\smile \cdots \smile p_{k_{m}}([M])}$

where ${\displaystyle p_{k}}$ denotes the ${\displaystyle k}$-th Pontryagin class and ${\displaystyle [M]}$ the fundamental class of ${\displaystyle M}$.

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 ${\displaystyle {\hat {A}}}$-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

• Chern–Simons form
• Hirzebruch signature theorem

References

1. "De Rham Cohomology - an overview | ScienceDirect Topics". www.sciencedirect.com. Retrieved 2022-02-02.
2. Mclean, Mark. "Pontryagin Classes" (PDF). Archived (PDF) from the original on 2016-11-08.
3. "A Survey of Computations of Homotopy Groups of Spheres and Cobordisms" (PDF). p. 16. Archived (PDF) from the original on 2016-01-22.

• Milnor John W.; Stasheff, James D. (1974). Characteristic classes. Annals of Mathematics Studies. Princeton, New Jersey; Tokyo: Princeton University Press / University of Tokyo Press. ISBN   0-691-08122-0.
• Hatcher, Allen (2009). Vector Bundles & K-Theory (2.1 ed.).

External links

• "Pontryagin class". Encyclopedia of Mathematics . EMS Press. 2001 [1994].