Chern and Pontryagin classes

A rank-$r$ complex bundle has $c_j(E)\in H^{2j}(X;\mathbb Z)$, vanishing for $j>r$. Its total Chern class is natural and multiplicative under direct sum. The splitting principle permits computations after a pullback where the bundle splits into lines and the induced cohomology map is injective. The resulting line first classes are called formal Chern roots. They need not already exist as globally defined classes on the original base, so writing roots is a computational device rather than an assertion that the bundle actually splits there.

If the roots are $x_1,\ldots,x_r$, the Chern classes are their elementary symmetric polynomials. Tensoring a line with another line adds first classes; taking a dual negates each root. Thus $c_j(E^*)=(-1)^j c_j(E)$. These identities follow from the integral theory and its naturality. They remain meaningful on bases with torsion, where trying to replace every class by a smooth curvature form would discard information before the calculation even begins.

For a real bundle $V$, define $p_j(V)=(-1)^j c_{2j}(V\otimes_{\mathbb R}\mathbb C)$ in degree $4j$. This sign is part of the normalization. For a complex bundle $E$, its underlying real bundle satisfies $E_{\mathbb R}\otimes\mathbb C\cong E\oplus\overline E$. Since the conjugate bundle has negated Chern roots, the calculation gives $p(E_{\mathbb R})=\prod_j(1+x_j^2)$. In particular its first Pontryagin class is the difference between a square of the first Chern class and twice the second.

The complex orientation on $E_{\mathbb R}$ identifies its Euler class with the top Chern class, and its even Stiefel–Whitney classes are reductions of the Chern classes; its odd Stiefel–Whitney classes vanish. For a complex line, the underlying oriented plane therefore has Euler class equal to its first Chern class and first Pontryagin class equal to its square. Reversing the real orientation reverses the Euler class but leaves Pontryagin classes unchanged. These statements distinguish orientation-sensitive data from orientation-insensitive data.

Characteristic classes can vanish because the base has no cohomology in their degree, even when the bundle is nontrivial. A complex line on a two-sphere can have a nonzero first Chern class and Euler class while every Pontryagin class of its realification vanishes for dimensional reasons. That is not a contradiction or a failure of naturality. It simply says that a degree-four invariant cannot detect a degree-two obstruction on this base. The target cohomology group is as important as the formula.

Characteristic numbers require a closed oriented base whose dimension matches the degree of the product being integrated. For complex projective space use the complex orientation and the positive hyperplane class $u$, normalized so its top power integrates to one. Negative coefficients in a Chern class are allowed; these are signed cohomology classes, not probabilities or unsigned counts. Before interpreting a numerical result, specify the bundle orientation, the base orientation, the coefficient ring, and which tensor operation produced the bundle.