Pontryagin class
In mathematics, the Pontryagin classes, named for Lev Pontryagin, 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 pk(E) is defined as
- pk(E) = pk(E, Z) = (−1)k c2k(E ⊗ C) ∈ H4k(M, Z),
where:
- 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 rational Pontryagin class pk(E, Q) is defined to be the image of pk(E) in H4k(M, Q), the 4k-cohomology group of M with rational coefficients.
Properties
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.
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, 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),](../I/m/398b2d853e577a27bd8eefa3fc8264df98c8a61a.svg)
where Ω 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 pk(M, Q) in H4k(M, 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
- k1, k2, ..., km such that k1+k2+...+km =n.
the Pontryagin number is defined by
![P_{k_1,k_2,\dots, k_m}=p_{k_1}\smile p_{k_2}\smile \cdots\smile p_{k_m}([M])](../I/m/176640f7fd7c32d30f974a94992542b49fcb4b7c.svg)
where pk denotes the k-th Pontryagin class and [M] the fundamental class of M.
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.
Generalizations
There is also a quaternionic Pontryagin class, for vector bundles with quaternion structure.
See also
References
- 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
- Hazewinkel, Michiel, ed. (2001), "Pontryagin class", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4