Pontryagin class#Pontryagin numbers

Given a real vector bundle $E$ over $M$, its $k$-th Pontryagin class $p\_k(E)$ is defined as

:$p\_k(E)\; =\; p\_k(E,\; \backslash Z)\; =\; (-1)^k\; c\_\{2k\}(E\backslash otimes\; \backslash Complex)\; \backslash in\; H^\{4k\}(M,\; \backslash Z),$

where:

• $c\_\{2k\}(E\backslash otimes\; \backslash Complex)$ denotes the $2k$-th Chern class of the complexification $E\backslash otimes\; \backslash Complex\; =\; E\backslash oplus\; iE$ of $E$,
• $H^\{4k\}(M,\; \backslash Z)$ is the $4k$-cohomology group of $M$ with integer coefficients.

The rational Pontryagin class $p\_k(E,\; \backslash Q)$ is defined to be the image of $p\_k(E)$ in $H^\{4k\}(M,\; \backslash Q)$, the $4k$-cohomology group of $M$ with rational coefficients.

The total Pontryagin class

:$p(E)=1+p\_1(E)+p\_2(E)+\backslash cdots\backslash in\; H^*(M,\backslash Z),$

is (modulo 2-torsion) multiplicative with respect to

Whitney sum of vector bundles, i.e.,

:$2p(E\backslash oplus\; F)=2p(E)\backslash smile\; p(F)$

for two vector bundles $E$ and $F$ over $M$. In terms of the individual Pontryagin classes $p\_k$,

:$2p\_1(E\backslash oplus\; F)=2p\_1(E)+2p\_1(F),$

:$2p\_2(E\backslash oplus\; F)=2p\_2(E)+2p\_1(E)\backslash smile\; p\_1(F)+2p\_2(F)$

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 $E\_\{10\}$ over the 9-sphere. (The clutching function for $E\_\{10\}$ arises from the homotopy group $\backslash pi\_8(\backslash mathrm\{O\}(10))\; =\; \backslash Z/2\backslash Z$.) The Pontryagin classes and Stiefel-Whitney classes all vanish: the Pontryagin classes don't exist in degree 9, and the Stiefel–Whitney class $w\_9$ of $E\_\{10\}$ vanishes by the Wu formula $w\_9\; =\; w\_1\; w\_8\; +\; Sq^1(w\_8)$. Moreover, this vector bundle is stably nontrivial, i.e. the Whitney sum of $E\_\{10\}$ with any trivial bundle remains nontrivial. {{Harv|Hatcher|2009|p=76}}

Given a $2\; k$-dimensional vector bundle $E$ we have

:$p\_k(E)=e(E)\backslash smile\; e(E),$

where $e(E)$ denotes the Euler class of $E$, and $\backslash smile$ denotes the cup product of cohomology classes.

As was shown by Shiing-Shen Chern and André Weil around 1948, the rational Pontryagin classes

:$p\_k(E,\backslash mathbf\{Q\})\backslash in\; H^\{4k\}(M,\backslash mathbf\{Q\})$

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

:$p=\backslash left[1-\backslash frac\{\{\backslash rm\; Tr\}(\backslash Omega\; ^2)\}\{8\; \backslash pi\; ^2\}+\backslash frac\{\{\backslash rm\; Tr\}(\backslash Omega\; ^2)^2-2\; \{\backslash rm\; Tr\}(\backslash Omega\; ^4)\}\{128\; \backslash pi\; ^4\}-\backslash frac\{\{\backslash rm\; Tr\}(\backslash Omega\; ^2)^3-6\; \{\backslash rm\; Tr\}(\backslash Omega\; ^2)\; \{\backslash rm\; Tr\}(\backslash Omega\; ^4)+8\; \{\backslash rm\; Tr\}(\backslash Omega\; ^6)\}\{3072\; \backslash pi\; ^6\}+\backslash cdots\backslash right]\backslash in\; H^*\_\{dR\}(M),$

where $\backslash Omega$ denotes the curvature form, and $H^*\_\{dR\}\; (M)$ denotes the de Rham cohomology groups.{{fact|date=November 2024}}

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

Novikov proved in 1966 that if two compact, oriented, smooth manifolds are homeomorphic then their rational Pontryagin classes $p\_k(M,\; \backslash mathbf\{Q\})$ in $H^\{4k\}(M,\; \backslash mathbf\{Q\})$ 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.{{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}}

The Pontryagin classes of a complex vector bundle $\backslash pi:\; E\; \backslash to\; X$ is completely determined by its Chern classes. This follows from the fact that $E\backslash otimes\_\{\backslash mathbb\{R\}\}\backslash mathbb\{C\}\; \backslash cong\; E\backslash oplus\; \backslash bar\{E\}$, the Whitney sum formula, and properties of Chern classes of its complex conjugate bundle. That is, $c\_i(\backslash bar\{E\})\; =\; (-1)^ic\_i(E)$ and $c(E\backslash oplus\backslash bar\{E\})\; =\; c(E)c(\backslash bar\{E\})$. Then, given this relation, we can see

$$

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))

{{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}}.

$(1-c\_1(E))(1\; +\; c\_1(E))\; =\; 1\; +\; c\_1(E)^2$

In general, looking at first two terms of the product

$(1-c\_1(E)\; +\; c\_2(E)\; +\; \backslash ldots\; +\; (-1)^n\; c\_n(E))(1\; +\; c\_1(E)\; +\; c\_2(E)\; +\backslash ldots\; +\; c\_n(E))\; =\; 1\; -\; c\_1(E)^2\; +\; 2c\_2(E)\; +\; \backslash ldots$

Recall that a quartic polynomial whose vanishing locus in $\backslash mathbb\{CP\}^3$ is a smooth subvariety is a K3 surface. If we use the normal sequence

$0\; \backslash to\; \backslash mathcal\{T\}\_X\; \backslash to\; \backslash mathcal\{T\}\_\{\backslash mathbb\{CP\}^3\}|\_X\; \backslash to\; \backslash mathcal\{O\}(4)\; \backslash to\; 0$

$\backslash begin\{align\}$

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}

$p\_1(X)\; =\; -48$. This number can be used to compute the third stable homotopy group of spheres.{{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}}

Pontryagin numbers are certain topological invariants of a smooth manifold. Each Pontryagin number of a manifold $M$ vanishes if the dimension of $M$ is not divisible by 4. It is defined in terms of the Pontryagin classes of the manifold $M$ as follows:

Given a smooth $4\; n$-dimensional manifold $M$ and a collection of natural numbers

:$k\_1,\; k\_2,\; \backslash ldots\; ,\; k\_m$ such that $k\_1+k\_2+\backslash cdots\; +k\_m\; =n$,

the Pontryagin number $P\_\{k\_1,k\_2,\backslash dots,k\_m\}$ is defined by

:$P\_\{k\_1,k\_2,\backslash dots,\; k\_m\}=p\_\{k\_1\}\backslash smile\; p\_\{k\_2\}\backslash smile\; \backslash cdots\backslash smile\; p\_\{k\_m\}([M])$

where $p\_k$ denotes the $k$-th Pontryagin class and $[M]$ the fundamental class of $M$.

1. Pontryagin numbers are oriented cobordism invariant; and together with Stiefel-Whitney numbers they determine an oriented manifold's oriented cobordism class.
2. Pontryagin numbers of closed Riemannian manifolds (as well as Pontryagin classes) can be calculated as integrals of certain polynomials from the curvature tensor of a Riemannian manifold.
3. Invariants such as signature and $\backslash hat\; A$-genus can be expressed through Pontryagin numbers. For the theorem describing the linear combination of Pontryagin numbers giving the signature see Hirzebruch signature theorem.

There is also a quaternionic Pontryagin class, for vector bundles with quaternion structure.

• Chern–Simons form
• Hirzebruch signature theorem

{{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}}

• {{springer|title=Pontryagin class|id=p/p073750|mode=cs1}}

Category:Characteristic classes

Category:Differential topology