The splitting principle is as follows.
Given a vector bundle $E \to X$ with $X$ compact Hausdorff, there is a compact Hausdorff space $F(E)$ and a map $p: F(E) \to X$ such that the induced map $p^*: K^*(X) \to K^*(F(E))$ is injective and $p^*(E)$ splits as the sum of line bundles.
My question is, what is the idea/intuition behind the proof of the splitting principle?
Best Answer
In the topological language you are using, $F(E)$ is the space of "orthogonal splittings". That is to say, $p^{-1}(x)$ is the space of all ways to write the fiber $E_x$ as an orthogonal sum of one dimensional spaces. Since it is the "space of splittings", there is a tautological splitting over it. "$\square$"
Remark on alternative versions you may have seen: It is more common to describe $F(E)$ as the space of flags. A (complete) flag $F_{\bullet}$ in a vector space $V$ is a chain of subspaces $F_1 \subset F_2 \subset \cdots \subset F_d = E$ where $\dim F_k = k$. When $E$ is equipped with a positive definite symmetric or Hermitian form, this is the same as a splitting; the summands of the splitting are $F_k \cap F_{k-1}^{\perp}$.
The flag formulation works better when working with holomorphic vector bundles, in which case the statement is that the vector bundle has a filtration with one dimensional filtered pieces, not necessarily a splitting.