In mathematics, the Pontryagin classes, named after Lev Pontryagin, are certain characteristic classes of real vector bundles. The Pontryagin classes lie in cohomology groups 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 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
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 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 <math>E_{10}</math> over the 9-sphere. (The clutching function for <math>E_{10}</math> arises from the 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 Wu formula <math>w_9 = w_1 w_8 + Sq^1(w_8)</math>. Moreover, this vector bundle is stably nontrivial, i.e. the Whitney sum of <math>E_{10}</math> with any trivial bundle remains nontrivial.
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, the total Pontryagin class is expressed as
:<math>p=\left[1-\frac