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In [[mathematics]], the '''Pontryagin classes''', named for [[Lev Pontryagin]],  are certain [[characteristic class]]es. The Pontryagin class lies in [[cohomology group]]s with degree a multiple of four. It applies to real [[vector bundle]]s.
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== Definition ==
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Given a real vector bundle ''E'' over ''M'', its ''k''-th Pontryagin class ''p<sub>k</sub>''(''E'') is defined as
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:''p<sub>k</sub>''(''E'') = ''p<sub>k</sub>''(''E'', '''Z''') = (−1)<sup>''k''</sup> ''c''<sub>2''k''</sub>(''E'' ⊗ '''C''') ∈ ''H''<sup>4''k''</sup>(''M'', '''Z'''),
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where:
*''c''<sub>2''k''</sub>(''E'' ⊗ '''C''') denotes the 2''k''-th [[Chern class]] of the [[complexification]] ''E'' ⊗ '''C''' = ''E'' ⊕ ''iE'' of ''E'',
*''H''<sup>4''k''</sup>(''M'', '''Z''') is the 4''k''-[[cohomology]] group of ''M'' with [[integer]] coefficients.


The rational Pontryagin class ''p<sub>k</sub>''(''E'', '''Q''') is defined to be the image of ''p<sub>k</sub>''(''E'') in ''H''<sup>4''k''</sup>(''M'', '''Q'''), the 4''k''-[[cohomology]] group of ''M'' with [[Rational number|rational]] coefficients.
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== Properties ==
'''MathML'''
The '''total Pontryagin class'''  
:<math forcemathmode="mathml">E=mc^2</math>
:<math>p(E)=1+p_1(E)+p_2(E)+\cdots\in H^*(M,\mathbf{Z}),</math>
is (modulo 2-torsion) multiplicative with respect to
[[Glossary of differential geometry and topology#W|Whitney sum]] of vector bundles, i.e.,
:<math>2p(E\oplus F)=2p(E)\smile p(F)</math>
for two vector bundles ''E'' and ''F'' over ''M''.  In terms of the individual Pontryagin classes ''p<sub>k</sub>'',
:<math>2p_1(E\oplus F)=2p_1(E)+2p_1(F),</math>
:<math>2p_2(E\oplus F)=2p_2(E)+2p_1(E)\smile p_1(F)+2p_2(F)</math>
and so on.


The vanishing of the Pontryagin classes and [[Stiefel-Whitney class]]es of a vector bundle does not guarantee that the vector bundle is trivial.  For example, up to [[Vector bundle#Vector bundle morphisms|vector bundle isomorphism]], there is a unique nontrivial rank 10 vector bundle ''E''<sub>10</sub> over the [[N-sphere|9-sphere]].  (The [[clutching function]] for ''E''<sub>10</sub> arises from the [[Orthogonal group#Homotopy groups|stable homotopy group]] π<sub>8</sub>(O(10)) = '''Z'''/2'''Z'''.) The Pontryagin classes and Stiefel-Whitney classes all vanish: the Pontryagin classes don't exist in degree 9, and the [[Stiefel-Whitney class]] ''w''<sub>9</sub> of ''E''<sub>10</sub> vanishes by the [[Stiefel-Whitney class#Relations over the Steenrod algebra|Wu formula]] ''w''<sub>9</sub> = ''w''<sub>1</sub>''w''<sub>8</sub> + Sq<sup>1</sup>(''w''<sub>8</sub>).  Moreover, this vector bundle is stably nontrivial, i.e. the [[Glossary of differential geometry and topology#W|Whitney sum]] of ''E''<sub>10</sub> with any trivial bundle remains nontrivial. {{Harv|Hatcher|2009|p=76}}
<!--'''PNG''' (currently default in production)
:<math forcemathmode="png">E=mc^2</math>


Given a 2''k''-dimensional vector bundle ''E'' we have
'''source'''
:<math>p_k(E)=e(E)\smile e(E),</math>
:<math forcemathmode="source">E=mc^2</math> -->
where ''e''(''E'') denotes the [[Euler class]] of ''E'', and <math>\smile</math> denotes the [[cup product]] of cohomology classes.


=== Pontryagin classes and curvature ===
<span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples].
As was shown by [[Shiing-Shen Chern]] and [[André Weil]] around 1948, the rational Pontryagin classes
:<math>p_k(E,\mathbf{Q})\in H^{4k}(M,\mathbf{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 form|connection]], the total Pontryagin class is expressed as
==Demos==
:<math>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),</math>


where Ω denotes the [[curvature form]], and ''H*''<sub>dR</sub>(''M'') denotes the [[de Rham cohomology]] groups.{{Citation needed|date=July 2009}}
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:


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


[[Sergei Novikov (mathematician)|Novikov]] proved in 1966 that if manifolds are [[homeomorphism|homeomorphic]] then their rational Pontryagin classes ''p<sub>k</sub>''(''M'', '''Q''') in ''H''<sup>4''k''</sup>(''M'', '''Q''') are the same.
* accessibility:
** Safari + VoiceOver: [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv video only], [[File:Voiceover-mathml-example-1.wav|thumb|Voiceover-mathml-example-1]], [[File:Voiceover-mathml-example-2.wav|thumb|Voiceover-mathml-example-2]], [[File:Voiceover-mathml-example-3.wav|thumb|Voiceover-mathml-example-3]], [[File:Voiceover-mathml-example-4.wav|thumb|Voiceover-mathml-example-4]], [[File:Voiceover-mathml-example-5.wav|thumb|Voiceover-mathml-example-5]], [[File:Voiceover-mathml-example-6.wav|thumb|Voiceover-mathml-example-6]], [[File:Voiceover-mathml-example-7.wav|thumb|Voiceover-mathml-example-7]]
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** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]].
** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode.


If the dimension is at least five, there are at most finitely many different smooth manifolds with given [[Homotopy#Homotopy equivalence of spaces|homotopy type]] and Pontryagin classes.
==Test pages ==


== Pontryagin numbers ==
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:
'''Pontryagin numbers''' are certain [[topological invariant]]s 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:
*[[Displaystyle]]
*[[MathAxisAlignment]]
*[[Styling]]
*[[Linebreaking]]
*[[Unique Ids]]
*[[Help:Formula]]


Given a smooth 4''n''-dimensional manifold ''M'' and a collection of natural numbers
*[[Inputtypes|Inputtypes (private Wikis only)]]
:''k''<sub>1</sub>, ''k''<sub>2</sub>, ..., ''k<sub>m</sub>'' such that ''k''<sub>1</sub>+''k''<sub>2</sub>+...+''k<sub>m</sub>'' =''n''.
*[[Url2Image|Url2Image (private Wikis only)]]
the Pontryagin number <math>P_{k_1,k_2,\dots,k_m}</math> is defined by
==Bug reporting==
:<math>P_{k_1,k_2,\dots, k_m}=p_{k_1}\smile p_{k_2}\smile \cdots\smile p_{k_m}([M])</math>
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .
where ''p<sub>k</sub>'' 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 number]]s 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 (topology)|signature]] and [[Â genus|<math>\hat A</math>-genus]] can be expressed through Pontryagin numbers.
 
== Generalizations ==
There is also a ''quaternionic'' Pontryagin class, for vector bundles with [[quaternion]] structure.
 
== See also ==
*[[Chern–Simons form]]
 
== References ==
*{{cite book
  |author= [[John Milnor|Milnor John W.]]
  |author2=Stasheff, James D. |authorlink2=Jim Stasheff
  |title= Characteristic classes
  |work= Annals of Mathematics Studies
  |issue=76
  |publisher=Princeton University Press / University of Tokyo Press
  |location=Princeton, New Jersey; Tokyo
  |year= 1974
  |isbn= 0-691-08122-0}}
* {{Cite journal | last=Hatcher | first=Allen | author-link=Allen Hatcher  | title=Vector Bundles & K-Theory | edition=2.1 | year=2009 | ref=harv | postscript=<!--None--> | url=http://www.math.cornell.edu/~hatcher/VBKT/VBpage.html}}
 
==External links==
* {{springer|title=Pontryagin class|id=p/p073750}}
 
[[Category:Characteristic classes]]
[[Category:Differential topology]]

Latest revision as of 23:52, 15 September 2019

This is a preview for the new MathML rendering mode (with SVG fallback), which is availble in production for registered users.

If you would like use the MathML rendering mode, you need a wikipedia user account that can be registered here [[1]]

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Registered users will be able to choose between the following three rendering modes:

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