Splitting principle

Remarks

One can take for $F(E)$ the bundle of complete flags in the fibers of $E$. It is built in steps. Over the projective bundle $P(E)$, the pullback of $E$ contains the tautological line bundle and splits it off. The Leray–Hirsch theorem shows that the pullback in cohomology is injective. Then one repeats with the complement, of rank $n - 1$.

In use: to prove an identity between characteristic classes that is natural and holds in the injective image, it suffices to prove it for sums of line bundles. For example, the Chern classes of a sum of line bundles are $c(L_1 \oplus \cdots \oplus L_n) = \prod_{i=1}^n (1 + c_1(L_i))$, so $c_k(E)$ pulls back to the $k$-th elementary symmetric polynomial in the classes $x_i = c_1(L_i)$. The Chern character and the Todd class are defined by symmetric power series in these $x_i$ for this reason.

Going deeper

For a student

The splitting principle reduces computations with Chern or Stiefel–Whitney classes of rank $n$ bundles to computations with $n$ first Chern or Stiefel–Whitney classes, the Chern roots. The Whitney sum formula, the formula for the Chern classes of a tensor product, and the formula $w(T\R P^n) = (1+a)^{n+1}$ all become polynomial identities in these roots. It is also behind the description of $H^*(BU(n); \Z)$ as the symmetric polynomials in $n$ variables of degree $2$, via the map $BU(1)^n \to BU(n)$.