Pontryagin class

From Academic Kids

(Redirected from Pontrjagin class)

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.

Contents

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.

See also

Navigation

Academic Kids Menu

  • Art and Cultures
    • Art (http://www.academickids.com/encyclopedia/index.php/Art)
    • Architecture (http://www.academickids.com/encyclopedia/index.php/Architecture)
    • Cultures (http://www.academickids.com/encyclopedia/index.php/Cultures)
    • Music (http://www.academickids.com/encyclopedia/index.php/Music)
    • Musical Instruments (http://academickids.com/encyclopedia/index.php/List_of_musical_instruments)
  • Biographies (http://www.academickids.com/encyclopedia/index.php/Biographies)
  • Clipart (http://www.academickids.com/encyclopedia/index.php/Clipart)
  • Geography (http://www.academickids.com/encyclopedia/index.php/Geography)
    • Countries of the World (http://www.academickids.com/encyclopedia/index.php/Countries)
    • Maps (http://www.academickids.com/encyclopedia/index.php/Maps)
    • Flags (http://www.academickids.com/encyclopedia/index.php/Flags)
    • Continents (http://www.academickids.com/encyclopedia/index.php/Continents)
  • History (http://www.academickids.com/encyclopedia/index.php/History)
    • Ancient Civilizations (http://www.academickids.com/encyclopedia/index.php/Ancient_Civilizations)
    • Industrial Revolution (http://www.academickids.com/encyclopedia/index.php/Industrial_Revolution)
    • Middle Ages (http://www.academickids.com/encyclopedia/index.php/Middle_Ages)
    • Prehistory (http://www.academickids.com/encyclopedia/index.php/Prehistory)
    • Renaissance (http://www.academickids.com/encyclopedia/index.php/Renaissance)
    • Timelines (http://www.academickids.com/encyclopedia/index.php/Timelines)
    • United States (http://www.academickids.com/encyclopedia/index.php/United_States)
    • Wars (http://www.academickids.com/encyclopedia/index.php/Wars)
    • World History (http://www.academickids.com/encyclopedia/index.php/History_of_the_world)
  • Human Body (http://www.academickids.com/encyclopedia/index.php/Human_Body)
  • Mathematics (http://www.academickids.com/encyclopedia/index.php/Mathematics)
  • Reference (http://www.academickids.com/encyclopedia/index.php/Reference)
  • Science (http://www.academickids.com/encyclopedia/index.php/Science)
    • Animals (http://www.academickids.com/encyclopedia/index.php/Animals)
    • Aviation (http://www.academickids.com/encyclopedia/index.php/Aviation)
    • Dinosaurs (http://www.academickids.com/encyclopedia/index.php/Dinosaurs)
    • Earth (http://www.academickids.com/encyclopedia/index.php/Earth)
    • Inventions (http://www.academickids.com/encyclopedia/index.php/Inventions)
    • Physical Science (http://www.academickids.com/encyclopedia/index.php/Physical_Science)
    • Plants (http://www.academickids.com/encyclopedia/index.php/Plants)
    • Scientists (http://www.academickids.com/encyclopedia/index.php/Scientists)
  • Social Studies (http://www.academickids.com/encyclopedia/index.php/Social_Studies)
    • Anthropology (http://www.academickids.com/encyclopedia/index.php/Anthropology)
    • Economics (http://www.academickids.com/encyclopedia/index.php/Economics)
    • Government (http://www.academickids.com/encyclopedia/index.php/Government)
    • Religion (http://www.academickids.com/encyclopedia/index.php/Religion)
    • Holidays (http://www.academickids.com/encyclopedia/index.php/Holidays)
  • Space and Astronomy
    • Solar System (http://www.academickids.com/encyclopedia/index.php/Solar_System)
    • Planets (http://www.academickids.com/encyclopedia/index.php/Planets)
  • Sports (http://www.academickids.com/encyclopedia/index.php/Sports)
  • Timelines (http://www.academickids.com/encyclopedia/index.php/Timelines)
  • Weather (http://www.academickids.com/encyclopedia/index.php/Weather)
  • US States (http://www.academickids.com/encyclopedia/index.php/US_States)

Information

  • Home Page (http://academickids.com/encyclopedia/index.php)
  • Contact Us (http://www.academickids.com/encyclopedia/index.php/Contactus)

  • Clip Art (http://classroomclipart.com)
Toolbox
Personal tools