# Intuition/idea behind a proof of the splitting principle?

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?

• Grassmanians map to products of projective spaces, where the universal bundles split. – Alex Degtyarev Feb 4 '16 at 21:38

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.

• 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. – Denis Nardin Feb 4 '16 at 22:15
• Good point. I don't have an immediate intuition for that but I'll think about it – David E Speyer Feb 4 '16 at 22:17
• Well, in the flag formulation, $F(E)$ is an iterated projective bundle, which explains the injectivity on cohomology... – Daniel Litt Feb 4 '16 at 22:31
• @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 – Denis Nardin Feb 5 '16 at 0:02

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.