Pontryagin class

In mathematics, the Pontryagin classes are certain characteristic classes. The Pontryagin class lies in cohomology groups with degree a multiple of four. It applies to real vector bundles.

Definition

Given a real vector bundle E over M, its k-th Pontryagin class $p_k(E)$ is defined as

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

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

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

Properties

The total Pontryagin class

$p(E)=1+p_1(E)+p_2(E)+\cdots\in H^{*}(M,\mathbb{Z}),$

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

$2p(E\oplus F)=2p(E)\cup p(F)$

for two vector bundles E and F over M. In terms of the individual Pontryagin classes $p_k$,

$2p_1(E\oplus F)=2p_1(E)+2p_1(F),$

$2p_2(E\oplus F)=2p_2(E)+2p_1(E)\cup p_1(F)+2p_2(F)$

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 $E_{10}$ over the 9-sphere. (The clutching function for $E_{10}$ arises from the stable homotopy group $\pi_8(O(10))=\mathbb Z_2$.) The Pontryagin classes and Stiefel-Whitney classes all vanish: the Pontryagin classes don't exist in degree 9, and the Stiefel-Whitney class $w_9$ of $E_{10}$ vanishes by the Wu formula $w_9 = w_1 w_8 + Sq^1(w_8)$. Moreover, this vector bundle is stably nontrivial, i.e. the Whitney sum of $E_{10}$ with any trivial bundle remains nontrivial. (Hatcher 2009, p. 76)

Given a 2k-dimensional vector bundle E we have

$p_k(E)=e(E)\cup e(E),$

where $e(E)$ denotes the Euler class of E, and $\cup$ denotes the cup product of cohomology classes; under the splitting principle, this corresponds to the square of the Vandermonde polynomial being the discriminant.

Pontryagin classes and curvature

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

$p_k(E,\mathbb{Q})\in H^{4k}(M,\mathbb{Q})$

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 E over a n-dimensional differentiable manifold M equipped with a connection, the total Pontryagin class is expressed as

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

where $\Omega$ denotes the curvature form, and $H^{*}_{dR}(M)$ denotes the de Rham cohomology groups.

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 manifolds are homeomorphic then their rational Pontryagin classes: $p_k(M,\mathbb{Q}) \in H^{4k}(M,\mathbb{Q})$ 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 numbers

Pontryagin numbers are certain topological invariants of a smooth manifold. The Pontryagin number vanishes if the dimension of manifold is not divisible by 4. It is defined in terms of the Pontryagin classes of a manifold as follows:

Given a smooth 4n-dimensional manifold M and a collection of natural numbers

$k_1,k_2,\dots,k_m$ such that $k_1+k_2+\cdots+k_m=n$

the Pontryagin number $P_{k_1,k_2,\dots,k_m}$ is defined by

$P_{k_1,k_2,\dots, k_m}=p_{k_1}\cup p_{k_2}\cup \cdots\cup p_{k_m}([M])$

where $p_{k}$ 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 manifold (as well as Pontryagin classes) can be calculated as integrals of certain polynomial from curvature tensor of Riemannian manifold.
3. Such invariants as signature and $\hat A$-genus can be expressed through Pontryagin numbers.

Generalizations

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

See also

• Chern-Simons form

References

• Milnor John W.; Stasheff, James D. (1974). Characteristic classes. Annals of Mathematics Studies, No. 76. Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo. ISBN 0-691-08122-0. 
• Hatcher, Allen (2009). Vector Bundles & K-Theory (2.1 ed.). http://www.math.cornell.edu/~hatcher/VBKT/VBpage.html