Pontryagin class

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

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[edit] Definition

Given a vector bundle E over M, its k-th Pontryagin class pk(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 image of pk(E) in H^{4k}(M,\mathbb{Q}), the 4k-cohomology group of M with rational coefficients.

Pontryagin classes have a meaning in real differential geometry — unlike the Chern class, which assumes a complex vector bundle at the outset.

[edit] Properties

If all Pontryagin classes and Stiefel-Whitney classes of E vanish then the bundle is stably trivial, i.e. its Whitney sum with a trivial bundle is trivial. The total Pontryagin class

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

is multiplicative with respect to Whitney sum of vector bundles, i.e.,

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

for two vector bundles E and F over M, i.e.

p_1(E\oplus F)=p_1(E)+p_1(F),
p_2(E\oplus F)=p_2(E)+p_1(E)\cup p_1(F)+p_2(F)

and so on. 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.

[edit] 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, its k-th Pontryagin class can be realized by the 4k-form

{\rm Tr}(\Omega\wedge \cdots\wedge\Omega)

constructed with 2k copies of the curvature form Ω. In particular the value

p_k(E,\mathbb{Q})=[{\rm Tr}(\Omega\wedge\cdots\wedge\Omega)]\in H^{4k}_{dR}(M)

does not depend on the choice of connection. Here

H^{*}_{dR}(M)

denotes the de Rham cohomology groups.

[edit] Pontryagin classes of a manifold

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

Novikov's theorem states 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 at most finitely many different smooth manifolds with given homotopy type and Pontryagin classes.

[edit] 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 pk denotes the k-th Pontryagin class and [M] the fundamental class of M.

[edit] 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.

[edit] Generalizations

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

[edit] See also

[edit] 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. 
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