Pontryagin classes and curvature

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

${\displaystyle p_{k}(E,\mathbf {Q} )\in H^{4k}(M,\mathbf {Q} )}$

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

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

where Ω denotes the curvature form, and H*_dR(M) denotes the de Rham cohomology groups.^[1]

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 two compact, oriented, smooth manifolds are homeomorphic then their rational Pontryagin classes p_k(M, Q) in H^4k(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 classes from Chern classes

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

${\displaystyle 1-p_{1}(E)+p_{2}(E)-\cdots +(-1)^{n}p_{n}(E)=(1+c_{1}(E)+\cdots +c_{n}(E))\cdot (1-c_{1}(E)+c_{2}(E)-\cdots +(-1)^{n}c_{n}(E))}$^[2]

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

${\displaystyle (1-c_{1}(E))(1+c_{1}(E))=1+c_{1}(E)^{2}}$

so all of the Pontryagin classes of complex vector bundles are trivial. On a surface, we have

${\displaystyle (1-c_{1}(E)+c_{2}(E))(1+c_{1}(E)+c_{2}(E))=1-c_{1}(E)^{2}+2c_{2}(E)}$

showing ${\displaystyle p_{1}(E)=c_{1}(E)^{2}-2c_{2}(E)}$. On line bundles this simplifies further since ${\displaystyle c_{2}(L)=0}$ by dimension reasons.

Pontryagin classes on a Quartic K3 Surface

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

${\displaystyle 0\to {\mathcal {T}}_{X}\to {\mathcal {T}}_{\mathbb {CP} ^{3}}|_{X}\to {\mathcal {O}}(4)\to 0}$

we can find

${\displaystyle {\begin{aligned}c({\mathcal {T}}_{X})&={\frac {c({\mathcal {T}}_{\mathbb {CP} ^{3}}|_{X})}{c({\mathcal {O}}(4))}}\\&={\frac {(1+[H])^{4}}{(1+4[H])}}\\&=(1+4[H]+6[H]^{2})\cdot (1-4[H]+16[H]^{2})\\&=1+6[H]^{2}\end{aligned}}}$

showing ${\displaystyle c_{1}(X)=0}$ and ${\displaystyle c_{2}(X)=6[H]^{2}}$. Since ${\displaystyle [H]^{2}}$ corresponds to four points, due to Bezout's lemma, we have the second chern number as ${\displaystyle 24}$. Since ${\displaystyle p_{1}(X)=-2c_{2}(X)}$ in this case, we have

${\displaystyle p_{1}(X)=-48}$. This number can be used to compute the third stable homotopy group of spheres.^[3]

Properties

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 ${\displaystyle {\hat {A}}}$-genus can be expressed through Pontryagin numbers. For the theorem describing the linear combination of Pontryagin numbers giving the signature see Hirzebruch signature theorem.