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
Here 
denotes the 2k-th Chern class of the complexification 
of E and 
, the 4k-cohomology group of M with integer coefficients.
The rational Pontryagin class 
is defined to be image of pk(E) in 
, 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
The total Pontryagin class
is multiplicative with respect to Whitney sum of vector bundles, i.e.,
for two vector bundles E and F over M, i.e.
and so on.
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. This corresponds to the fact that these classes generate the stable homotopy of BO(n).
Given a 2k-dimensional vector bundle E we have
where e(E) denotes the Euler class of E, and 
denotes the cup product of cohomology classes; under the splitting principle, this corresponds to the square of the Vandermonde polynomial being the discriminant.
[edit] 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.
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
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)](http://upload.wikimedia.org/math/d/d/2/dd2ae144be9d001cf3131d4b9ad8716d.png)
does not depend on the choice of connection. Here
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 proved in 1966 that if manifolds are homeomorphic then their rational Pontryagin classes :
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.
[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
such that 
the Pontryagin number 
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])](http://upload.wikimedia.org/math/2/1/6/21660a243635a265f7b1aca103950837.png)
where pk denotes the k-th Pontryagin class and [M] the fundamental class of M.
[edit] 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 manifold (as well as Pontryagin classes) can be calculated as integrals of certain polynomial from curvature tensor of Riemannian manifold.
- Such invariants as signature and
-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.
