Editing Pontryagin class

In [[mathematics]], the '''Pontryagin classes''', named after [[Lev Pontryagin]], are certain [[characteristic class]]es of real vector bundles. The Pontryagin classes lie in [[cohomology group]]s with degrees a multiple of four.

== Definition ==
Given a real vector bundle <math>E</math> over <math>M</math>, its <math>k</math>-th Pontryagin class <math>p_k(E)</math> is defined as 
:<math>p_k(E) = p_k(E, \Z) = (-1)^k c_{2k}(E\otimes \Complex) \in  H^{4k}(M, \Z),</math>
where:
*<math>c_{2k}(E\otimes \Complex)</math> denotes the <math>2k</math>-th [[Chern class]] of the [[complexification]] <math>E\otimes \Complex = E\oplus iE</math> of <math>E</math>, 
*<math>H^{4k}(M, \Z)</math> is the <math>4k</math>-[[cohomology]] group of <math>M</math> with [[integer]] coefficients.

The rational Pontryagin class <math>p_k(E, \Q)</math> is defined to be the image of <math>p_k(E)</math> in <math>H^{4k}(M, \Q)</math>, the <math>4k</math>-cohomology group of <math>M</math> with [[Rational number|rational]] coefficients.

== Properties ==
The '''total Pontryagin class''' 
:<math>p(E)=1+p_1(E)+p_2(E)+\cdots\in H^*(M,\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 <math>E</math> and <math>F</math> over <math>M</math>.  In terms of the individual Pontryagin classes <math>p_k</math>, 
:<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 <math>E_{10}</math> over the [[N-sphere|9-sphere]].  (The [[Clutching construction|clutching function]] for <math>E_{10}</math> arises from the [[Orthogonal group#Homotopy groups|homotopy group]] <math>\pi_8(\mathrm{O}(10)) = \Z/2\Z</math>.)  The Pontryagin classes and Stiefel-Whitney classes all vanish: the Pontryagin classes don't exist in degree 9, and the Stiefel–Whitney class <math>w_9</math> of <math>E_{10}</math> vanishes by the [[Stiefel-Whitney class#Relations over the Steenrod algebra|Wu formula]] <math>w_9 = w_1 w_8 + Sq^1(w_8)</math>.  Moreover, this vector bundle is stably nontrivial, i.e. the [[Glossary of differential geometry and topology#W|Whitney sum]] of <math>E_{10}</math> with any trivial bundle remains nontrivial. {{Harv|Hatcher|2009|p=76}}

Given a <math>2 k</math>-dimensional vector bundle <math>E</math> we have 
:<math>p_k(E)=e(E)\smile e(E),</math>
where <math>e(E)</math> denotes the [[Euler class]] of <math>E</math>, and <math>\smile</math> denotes 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_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]] <math>E</math> over a <math>n</math>-dimensional [[differentiable manifold]] <math>M</math> equipped with a [[connection form|connection]], the total Pontryagin class is expressed as 
:<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 <math>\Omega</math> denotes the [[curvature form]], and <math>H^*_{dR} (M)</math> denotes the [[de Rham cohomology]] groups.{{fact|date=November 2024}}

=== 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 two compact, oriented, smooth manifolds are [[homeomorphism|homeomorphic]] then their rational Pontryagin classes <math>p_k(M, \mathbf{Q})</math> in <math>H^{4k}(M, \mathbf{Q})</math> are the same.  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.<ref>{{cite journal |last1=Novikov |first1=S. P. |author-link1=Sergei Novikov (mathematician) |title=Homotopically equivalent smooth manifolds. I |journal=Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya |date=1964 |volume=28 |pages=365–474 |mr=162246}}</ref>

=== Pontryagin classes from Chern classes ===
The Pontryagin classes of a complex vector bundle <math>\pi: E \to X</math> is completely determined by its Chern classes. This follows from the fact that <math>E\otimes_{\mathbb{R}}\mathbb{C} \cong E\oplus \bar{E}</math>, the Whitney sum formula, and properties of Chern classes of its complex conjugate bundle. That is, <math>c_i(\bar{E}) = (-1)^ic_i(E)</math> and <math>c(E\oplus\bar{E}) = c(E)c(\bar{E})</math>. Then, given this relation, we can see<blockquote><math>
1 - p_1(E) + p_2(E) - \cdots + (-1)^np_n(E) = 
(1 + c_1(E) + \cdots + c_n(E)) \cdot 
(1 - c_1(E) + c_2(E) -\cdots + (-1)^nc_n(E))
</math><ref>{{Cite web|url=https://www.math.stonybrook.edu/~markmclean/MAT566/lecture13.pdf|title=Pontryagin Classes|last=Mclean|first=Mark|date=|website=|url-status=live|archive-url=https://web.archive.org/web/20161108093927/https://www.math.stonybrook.edu/~markmclean/MAT566/lecture13.pdf|archive-date=2016-11-08}}{{self-published inline|date=November 2024}}</ref>.</blockquote>For example, we can apply this formula to find the Pontryagin classes of a complex vector bundle on a curve and a surface. For a curve, we have<blockquote><math>(1-c_1(E))(1 + c_1(E)) = 1 + c_1(E)^2</math></blockquote>so all of the Pontryagin classes of complex vector bundles are trivial. 

In general, looking at first two terms of the product<blockquote><math>(1-c_1(E) + c_2(E) + \ldots + (-1)^n c_n(E))(1 + c_1(E) + c_2(E) +\ldots + c_n(E)) = 1 - c_1(E)^2 + 2c_2(E) + \ldots</math></blockquote>we can see that <math>p_1(E) = c_1(E)^2 - 2c_2(E)</math>. In particular, for line bundles this simplifies further since <math>c_2(L) = 0</math> by dimension reasons.

=== Pontryagin classes on a Quartic K3 Surface ===
Recall that a quartic polynomial whose vanishing locus in <math>\mathbb{CP}^3</math> is a smooth subvariety is a K3 surface. If we use the normal sequence<blockquote><math>0 \to \mathcal{T}_X \to \mathcal{T}_{\mathbb{CP}^3}|_X \to \mathcal{O}(4) \to 0</math></blockquote>we can find<blockquote><math>\begin{align}
c(\mathcal{T}_X) &= \frac{c(\mathcal{T}_{\mathbb{CP}^3}|_X)}{c(\mathcal{O}(4))} \\
&= \frac{(1+[H])^4}{(1+4[H])} \\
&= (1 + 4[H] + 6[H]^2)\cdot(1 - 4[H] + 16[H]^2) \\
&= 1 + 6[H]^2
\end{align}</math></blockquote>showing <math>c_1(X) = 0</math> and <math>c_2(X) = 6[H]^2</math>. Since <math>[H]^2</math> corresponds to four points, due to Bézout's lemma, we have the second chern number as <math>24</math>. Since <math>p_1(X) = -2c_2(X)</math> in this case, we have

<math>p_1(X) = -48</math>. This number can be used to compute the third stable homotopy group of spheres.<ref>{{Cite web|url=http://math.mit.edu/~guozhen/homotopy%20groups.pdf|title=A Survey of Computations of Homotopy Groups of Spheres and Cobordisms|last=|first=|date=|website=|page=16|url-status=live|archive-url=https://web.archive.org/web/20160122111116/http://math.mit.edu/~guozhen/homotopy%20groups.pdf|archive-date=2016-01-22|access-date=}}{{self-published inline|date=November 2024}}</ref>

== Pontryagin numbers ==
'''Pontryagin numbers''' are certain [[topological invariant]]s of a smooth [[manifold]]. Each Pontryagin number of a manifold <math>M</math> vanishes if the dimension of <math>M</math> is not divisible by 4. It is defined in terms of the Pontryagin classes of the [[manifold]] <math>M</math> as follows:

Given a smooth <math>4 n</math>-dimensional manifold <math>M</math> and a collection of natural numbers 
:<math>k_1, k_2, \ldots , k_m</math> such that <math>k_1+k_2+\cdots +k_m =n</math>,
the Pontryagin number <math>P_{k_1,k_2,\dots,k_m}</math> is defined by
:<math>P_{k_1,k_2,\dots, k_m}=p_{k_1}\smile p_{k_2}\smile \cdots\smile p_{k_m}([M])</math>
where <math>p_k</math> denotes the <math>k</math>-th Pontryagin class and <math>[M]</math> the [[fundamental class]] of <math>M</math>.

=== 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]]s (as well as Pontryagin classes) can be calculated as integrals of certain polynomials from the curvature tensor of a Riemannian manifold.
#Invariants such as [[Signature (topology)|signature]] and [[Â genus|<math>\hat A</math>-genus]] can be expressed through Pontryagin numbers. For the theorem describing the linear combination of Pontryagin numbers giving the signature see [[Hirzebruch signature theorem]].

== Generalizations ==
There is also a ''quaternionic'' Pontryagin class, for vector bundles with [[quaternion]] structure.

== See also ==
*[[Chern–Simons form]]
*[[Hirzebruch signature theorem]]

== References ==
{{Reflist}}
*{{cite book
  |author= Milnor John W.
  |author-link= John Milnor
  |author2=Stasheff, James D. |authorlink2=Jim Stasheff 
  |title= Characteristic classes
  |series= 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 book | last=Hatcher | first=Allen | author-link=Allen Hatcher  | title=Vector Bundles & K-Theory | edition=2.1 | year=2009 | url=http://pi.math.cornell.edu/~hatcher/VBKT/VBpage.html}}

==External links==
* {{springer|title=Pontryagin class|id=p/p073750|mode=cs1}}

[[Category:Characteristic classes]]
[[Category:Differential topology]]