Pontryagin class
From Academic Kids
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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Definition
Given a vector bundle <math>E<math> over <math>M<math> its k-th Pontryagin class <math>p_k(E)<math> can be defined as
- <math>p_k(E)=p_k(E,\mathbb{Z})=(-1)^kc_{2k}(E \otimes \mathbb{C})\in H^{4k}(M,\mathbb{Z}),<math>
here <math>c_{2k}(E \otimes \mathbb{C})<math> denotes times 2k-th Chern class of the complexification <math>E \otimes \mathbb{C}=E\oplus i E<math> of <math>E<math> and <math>H^{4k}(M,\mathbb{Z})<math>, the 4k-cohomology group of <math>M<math> with integer coefficients.
Rational Pontryagin class <math>p_k(E,{\mathbb Q})<math> is defined to be image of <math>p_k(E)<math> in <math>H^{4k}(M,\mathbb{Q})<math>, the 4k-cohomology group of <math>M<math> 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.
Properties
If all Pontryagin classes and Stiefel-Whitney classes of <math>E<math> vanish then the bundle is stably trivial, i.e. its Whitney sum with a trivial bundle is trivial. The total Pontryagin class <math>p(E)=1+p_1(E)+p_2(E)+...\in H^{*}(M,\mathbb{Z}),<math> is multiplicative with respect to Whitney sum of vector bundles, i.e <math>p(E\oplus F)=p(E)\cup p(F)<math> for two vector bundles <math>E<math> and <math>F<math> over <math>M<math>, i.e.
- <math>p_1(E\oplus F)=p_1(E)+p_1(F),<math>
- <math>p_2(E\oplus F)=p_2(E)+p_1(E)\cup p_1(F)+p_2(F) <math>
and so on. Given a 2k-dimensional vector bundle E we have
- <math>p_k(E)=e(E)\cup e(E),<math>
where <math>e(E)<math> denotes Euler class of E, and the notation is 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
- <math>p_n(E,\mathbb{Q})\in H^{4k}(M,\mathbb{Q})<math>
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
- <math> Tr(\Omega\wedge...\wedge\Omega)<math>
constructed with 2k copies of the curvature form <math>\Omega<math>. In particular the value
- <math> p_n(E,\mathbb{Q})=[Tr(\Omega\wedge...\wedge\Omega)]\in H^{4k}_{dR}(M)<math>
does not depend on the choice of connection. Here
- <math> H^{*}_{dR}(M)<math>
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's theorem states that if manifolds are homeomorphic then their rational Pontryagin classes
- <math>p_k(M,\mathbb{Q}) \in H^{4k}(M,\mathbb{Q})<math>
are the same.
If the dimension is at least five, there at most finitely many different smooth manifolds with given homotopy type and Pontryagin classes.
Generalizations
There is also a quaternionic Pontryagin class, for vector bundles with quaternion structure.
