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  "body": "\n## Statement\n\nLet $E \\to B$ be a complex vector bundle of rank $n$ over a paracompact space $B$.\nThen there are a space $F(E)$ and a continuous map $p \\colon F(E) \\to B$ such that:\n\n1. The pullback $p^* E$ is isomorphic to a [[Whitney sum|BE9NJRVDR3]] $L_1 \\oplus \\cdots \\oplus L_n$ of complex line bundles over $F(E)$.\n2. The induced map $p^* \\colon H^*(B; \\Z) \\to H^*(F(E); \\Z)$ in [[singular cohomology|6N7BC70CX0]] is injective.\n\nThe same holds for a real vector bundle, with real line bundles and with $\\Z/2\\Z$ coefficients in place of $\\Z$.\n\n## Remarks\n\nOne can take for $F(E)$ the bundle of complete flags in the fibers of $E$.\nIt is built in steps.\nOver the projective bundle $P(E)$, the pullback of $E$ contains the tautological line bundle and splits it off.\nThe Leray–Hirsch theorem shows that the pullback in cohomology is injective.\nThen one repeats with the complement, of rank $n - 1$.\n\nIn use: to prove an identity between [[characteristic classes|7HACJC0JE2]] that is natural and holds in the injective image, it suffices to prove it for sums of line bundles.\nFor 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 $x_i = c_1(L_i)$.\nThe Chern character and the Todd class are defined by symmetric power series in these $x_i$ for this reason.\n\n## Explanations\n\n### L1 human\n\nA vector bundle can be complicated, but a sum of line bundles is easy to handle.\nThe splitting principle says that any vector bundle becomes such a sum after pulling it back to a suitable bigger space.\nThe pullback loses no cohomological information, because the map on cohomology is injective.\nSo a formula about characteristic classes can be checked as if every bundle were a sum of lines.\n\n### L2 human\n\nThe 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.\nThe 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.\nIt is also behind the description of $H^*(BU(n); \\Z)$ as the symmetric polynomials in $n$ variables of degree $2$, via the map $BU(1)^n \\to BU(n)$.\n",
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