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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?

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    $\begingroup$ Grassmanians map to products of projective spaces, where the universal bundles split. $\endgroup$
    – Alex Degtyarev
    Commented Feb 4, 2016 at 21:38

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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.

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    $\begingroup$ I guess that the miracle here is less that there is an universal splitting (as you say it is quite trivial) and more that the map in K-theory (and integral cohomology) is injective. $\endgroup$
    – Denis Nardin
    Commented Feb 4, 2016 at 22:15
  • $\begingroup$ Good point. I don't have an immediate intuition for that but I'll think about it $\endgroup$
    – David E Speyer
    Commented Feb 4, 2016 at 22:17
  • $\begingroup$ Well, in the flag formulation, $F(E)$ is an iterated projective bundle, which explains the injectivity on cohomology... $\endgroup$
    – Daniel Litt
    Commented Feb 4, 2016 at 22:31
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    $\begingroup$ @DanielLitt I don't think it explains it. I know the proof but in the end it boils down to a special property of the cohomology of projective space (that it is a power series ring in one variable) plus the Leray-Hirsch theorem. I find this quite mysterious in fact but I'd welcome a good intuition $\endgroup$
    – Denis Nardin
    Commented Feb 5, 2016 at 0:02
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Perhaps my very short (4 pages plus bibliography) paper ``A note on the splitting principle'' http://www.math.uchicago.edu/~may/PAPERS/Split.pdf may be illuminating. It shows that the splitting principle can be viewed as a statement about the reduction of the structural group of a $G$-bundle $\xi$ from $G$ to a maximal torus $T$, where $G$ is a compact Lie group. It applies more generally than in just the usual examples. One starts with the bundle $BT\to BG$ with fiber $G/T$. For a $G$-bundle over $X$ classified by $f\colon X\to BG$, one has a pullback bundle $q\colon Y\to X$ with fiber $G/T$ together with a reduction of the structure group of $q^*\xi$ to $T$. When $H^*(BG;R)$ is concentrated in even degrees, $q^*\colon H^*(X;R)\to H^*(Y;R)$ is a monomorphism. That is easily seen to imply the splitting theorem as usually stated, and many variants thereof. As stated and explained briefly in the paper, the argument adapts to $K$-theory.

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