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| == Polo Ralph Lauren Australia Stores Hi Everyone ==
| | In [[mathematics]], the '''Pontryagin classes''' 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 |
| <li>[http://www.histoirepassion.eu/spip.php?article1761/ http://www.histoirepassion.eu/spip.php?article1761/]</li>
| | :''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: |
| <li>[http://cerisier.info/spip.php?article25/ http://cerisier.info/spip.php?article25/]</li>
| | *''c''<sub>2''k''</sub>(''E'' ⊗ '''C''') denotes the 2''k''-th [[Chern class]] of the [[complexification]] ''E'' ⊗ '''C''' = ''E'' ⊕ ''iE'' of ''E'', |
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| | *''H''<sup>4''k''</sup>(''M'', '''Z''') is the 4''k''-[[cohomology]] group of ''M'' with [[integer]] coefficients. |
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| == Nike Shox Oz Mens Well ==
| | 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 == |
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| | The '''total Pontryagin class''' |
| <li>[http://www.zxcqtl.com/news/html/?402019.html http://www.zxcqtl.com/news/html/?402019.html]</li>
| | :<math>p(E)=1+p_1(E)+p_2(E)+\cdots\in H^*(M,\mathbf{Z}),</math> |
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| | is (modulo 2-torsion) multiplicative with respect to |
| <li>[http://www.hanzhuangpifa.org/forum.php?mod=viewthread&tid=1359815 http://www.hanzhuangpifa.org/forum.php?mod=viewthread&tid=1359815]</li>
| | [[Glossary of differential geometry and topology#W|Whitney sum]] of vector bundles, i.e., |
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| | :<math>2p(E\oplus F)=2p(E)\smile p(F)</math> |
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| | for two vector bundles ''E'' and ''F'' over ''M''. In terms of the individual Pontryagin classes ''p<sub>k</sub>'', |
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| | :<math>2p_1(E\oplus F)=2p_1(E)+2p_1(F),</math> |
| <li>[http://midvalleytreasurehunters.info//read.php?12,288137 http://midvalleytreasurehunters.info//read.php?12,288137]</li>
| | :<math>2p_2(E\oplus F)=2p_2(E)+2p_1(E)\smile p_1(F)+2p_2(F)</math> |
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| | and so on. |
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| == Nike Free ESPN analyst Ron Jaworski == | | 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}} |
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| | Given a 2''k''-dimensional vector bundle ''E'' we have |
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| | :<math>p_k(E)=e(E)\smile e(E),</math> |
| <li>[http://www.sebalo.info/spip/spip.php?article13 http://www.sebalo.info/spip/spip.php?article13]</li>
| | where ''e''(''E'') denotes the [[Euler class]] of ''E'', and <math>\smile</math> denotes the [[cup product]] of cohomology classes; under the [[splitting principle]], this corresponds to the square of the [[Vandermonde polynomial]] being the [[discriminant]]. |
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| == Nike Air Pegasus 29 Womens == | | === Pontryagin classes and curvature === |
| | 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. |
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| | 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 |
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| | :<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> |
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| == Nike Air Max 97 Camo Standard Poor's ==
| | where Ω denotes the [[curvature form]], and ''H*''<sub>dR</sub>(''M'') denotes the [[de Rham cohomology]] groups.{{Citation needed|date=July 2009}} |
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| | === Pontryagin classes of a manifold === |
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| | The '''Pontryagin classes of a smooth manifold''' are defined to be the Pontryagin classes of its [[tangent bundle]]. |
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| | [[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. |
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| | 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. |
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| == Nike Air Max Ltd Chile offers a cultured == | | == Pontryagin numbers == |
| | '''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: |
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| | Given a smooth 4''n''-dimensional manifold ''M'' and a collection of natural numbers |
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| | :''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''. |
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| | the Pontryagin number <math>P_{k_1,k_2,\dots,k_m}</math> is defined by |
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| | :<math>P_{k_1,k_2,\dots, k_m}=p_{k_1}\smile p_{k_2}\smile \cdots\smile p_{k_m}([M])</math> |
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| | where ''p<sub>k</sub>'' denotes the ''k''-th Pontryagin class and [''M''] the [[fundamental class]] of ''M''. |
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| == Nike Free Run 5.0 Womens or FTP transfer. Yes == | | === 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. |
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| | == Generalizations == |
|
| | There is also a ''quaternionic'' Pontryagin class, for vector bundles with [[quaternion]] structure. |
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| | == References == |
| | | *{{cite book |
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| | |author= [[John Milnor|Milnor John W.]] |
|
| | |coauthors= [[Jim Stasheff|Stasheff, James D.]] |
| <li>[http://verdamilio.net/tonio/spip.php?article1970/ http://verdamilio.net/tonio/spip.php?article1970/]</li>
| | |title= Characteristic classes |
|
| | |publisher= Annals of Mathematics Studies, No. 76. Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo |
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| | |year= 1974 |
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| | |isbn= 0-691-08122-0}} |
| <li>[http://verdamilio.net/tonio/spip.php?article1970/ http://verdamilio.net/tonio/spip.php?article1970/]</li>
| | * {{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}} |
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| | * {{springer|title=Pontryagin class|id=p/p073750}} |
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In mathematics, the Pontryagin classes are certain characteristic classes. The Pontryagin class lies in cohomology groups with degree a multiple of four. It applies to real vector bundles.
Definition
Given a real vector bundle E over M, its k-th Pontryagin class pk(E) is defined as
- pk(E) = pk(E, Z) = (−1)k c2k(E ⊗ C) ∈ H4k(M, Z),
where:
The rational Pontryagin class pk(E, Q) is defined to be the image of pk(E) in H4k(M, Q), the 4k-cohomology group of M with rational coefficients.
Properties
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 E10 over the 9-sphere. (The clutching function for E10 arises from the stable homotopy group π8(O(10)) = Z/2Z.) 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. Template:Harv
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; under the splitting principle, this corresponds to the square of the Vandermonde polynomial being the discriminant.
Pontryagin classes and curvature
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.Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park.
Pontryagin classes of a manifold
The Pontryagin classes of a smooth manifold are defined to be the Pontryagin classes of its tangent bundle.
Novikov proved in 1966 that if 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 numbers
Pontryagin numbers are certain topological invariants 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:
Given a smooth 4n-dimensional manifold M and a collection of natural numbers
- k1, k2, ..., km such that k1+k2+...+km =n.
the Pontryagin number is defined by
where pk 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 numbers 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 and -genus can be expressed through Pontryagin numbers.
Generalizations
There is also a quaternionic Pontryagin class, for vector bundles with quaternion structure.
See also
References
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