In mathematics, the Pontryagin classes, named after Lev Pontryagin, are certain characteristic classes of real vector bundles. The Pontryagin classes lie in cohomology groups with degrees a multiple of four.
Given a real vector bundle E over M, its k-th Pontryagin class is defined as
- denotes the -th Chern class of the complexification of E,
- is the -cohomology group of M with integer coefficients.
The rational Pontryagin class is defined to be the image of in , the -cohomology group of M with rational 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 over the 9-sphere. (The clutching function for arises from the homotopy group .) 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
Pontryagin classes and curvatureEdit
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 manifoldEdit
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 classes from Chern classesEdit
The Pontryagin classes of a complex vector bundle can be completely determined by its Chern classes. This follows from the fact that , the Whitney sum formula, and properties of Chern classes of its complex conjugate bundle. That is, and . Then, this given the relation
for example, we can apply this formula to find the Pontryagin classes of a vector bundle on a curve and a surface. For a curve, we have
so all of the Pontryagin classes of complex vector bundles are trivial. On a surface, we have
showing . On line bundles this simplifies further since by dimension reasons.
Pontryagin classes on a Quartic K3 SurfaceEdit
Recall that a quartic polynomial whose vanishing locus in is a smooth subvariety is a K3 surface. If we use the normal sequence
we can find
showing and . Since corresponds to four points, due to Bezout's lemma, we have the second chern number as . Since in this case, we have
. This number can be used to compute the third stable homotopy group of spheres.
Pontryagin numbers are certain topological invariants of a smooth manifold. Each Pontryagin number of a manifold M vanishes if the dimension of M is not divisible by 4. It is defined in terms of the Pontryagin classes of the manifold M as follows:
Given a smooth -dimensional manifold M and a collection of natural numbers
- such that ,
the Pontryagin number is defined by
where 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 manifolds (as well as Pontryagin classes) can be calculated as integrals of certain polynomials from the curvature tensor of a Riemannian manifold.
- Invariants such as signature and -genus can be expressed through Pontryagin numbers. For the theorem describing the linear combination of Pontryagin numbers giving the signature see Hirzebruch signature theorem.
There is also a quaternionic Pontryagin class, for vector bundles with quaternion structure.
- Mclean, Mark. "Pontryagin Classes" (PDF). Archived (PDF) from the original on 2016-11-08.
- "A Survey of Computations of Homotopy Groups of Spheres and Cobordisms" (PDF). p. 16. Archived (PDF) from the original on 2016-01-22.
- 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.). Cite journal requires
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