Pontryagin class

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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.

DefinitionEdit

Given a real vector bundle E over M, its k-th Pontryagin class   is defined as

 

where:

  •   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.

PropertiesEdit

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

 

where e(E) denotes the Euler class of E, and   denotes the cup product of cohomology classes.

Pontryagin classes and curvatureEdit

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

 

where Ω denotes the curvature form, and H*dR(M) denotes the de Rham cohomology groups.[citation needed]

Pontryagin classes of a manifoldEdit

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

Novikov proved in 1966 that if two compact, oriented, smooth 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 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

 [1]

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[2].

Pontryagin numbersEdit

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.

PropertiesEdit

  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 manifolds (as well as Pontryagin classes) can be calculated as integrals of certain polynomials from the curvature tensor of a Riemannian manifold.
  3. 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.

GeneralizationsEdit

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

See alsoEdit

ReferencesEdit

  1. ^ Mclean, Mark. "Pontryagin Classes" (PDF). Archived (PDF) from the original on 2016-11-08.
  2. ^ "A Survey of Computations of Homotopy Groups of Spheres and Cobordisms" (PDF). p. 16. Archived (PDF) from the original on 2016-01-22.

External linksEdit