Given a real vector bundle E over M, its k-th Pontryagin class pk(E) is defined as
- pk(E) = pk(E, Z) = (−1)k c2k(E ⊗ C) ∈ H4k(M, Z),
- c2k(E ⊗ C) denotes the 2k-th Chern class of the complexification E ⊗ C = E ⊕ iE of E,
- H4k(M, Z) is the 4k-cohomology group of M with integer coefficients.
The total Pontryagin class
is (modulo 2-torsion) multiplicative with respect to Whitney sum of vector bundles, i.e.,
for two vector bundles E and F over M. In terms of the individual Pontryagin classes pk,
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 E10 over the 9-sphere. (The clutching function for E10 arises from the stable homotopy group π8(O(10)) = Z/2Z.) The Pontryagin classes and Stiefel-Whitney classes all vanish: the Pontryagin classes don't exist in degree 9, and the Stiefel-Whitney class w9 of E10 vanishes by the Wu formula w9 = w1w8 + Sq1(w8). Moreover, this vector bundle is stably nontrivial, i.e. the Whitney sum of E10 with any trivial bundle remains nontrivial. (Hatcher 2009, p. 76)
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.
Pontryagin classes and curvature
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.
Pontryagin classes of a manifold
The Pontryagin classes of a smooth manifold are defined to be the Pontryagin classes of its tangent bundle.
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 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
- k1, k2, ..., km such that k1+k2+...+km =n.
the Pontryagin number is defined by
where pk denotes the k-th Pontryagin class and [M] the fundamental class of M.
- 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.
There is also a quaternionic Pontryagin class, for vector bundles with quaternion structure.
- 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.)