A Note on the Splitting Principle

We offer a new perspective on the splitting principle. We give an easy proof that applies to all classical types of vector bundles and in fact to G-bundles for any compact connected Lie group G. The perspective gives precise calculational information and directly ties the splitting principle to the specification of characteristic classes in terms of classifying spaces. In the algebraic topology proseminar at Chicago, a student, Nils Barth, asked for the precise relationship between the splitting principle and the specification of Chern classes in terms of maximal tori. This note gives the quick and more general answer that popped to mind. It should be utterly standard, but it was new to me and to other faculty in the audience. Certainly I have not seen it in print. Let T = T n be a maximal torus in a compact connected Lie group G of rank n and let R be a commutative ring in which p is invertible for all primes p such that H∗(G;Z) has p-torsion. Classical results of Borel [3] determine these primes explicitly when G is simply connected and describe how to determine them in terms of the elementary abelian p-subgroups of G in general. Other classical results of Borel [1] imply that H∗(BG;R) is a polynomial ring over R on n even degree generators. The information relevant here is just that H∗(BG;R) is concentrated in even degrees. By the Bott-Samelson theorem [4], H∗(G/T ;Z) has no torsion and is also concentrated in even degrees, hence H∗(G/T ;R) is a free R-module. Let EG be a universal principal G-bundle and take BT = EG/T and BG = EG/G. Inclusion of orbits gives a G-bundle p : BT −→ BG with fiber G/T . Taking cohomology with coefficients in R henceforward, we see immediately that the Serre spectral sequence of this bundle collapses to give (1) H∗(BT ) ∼= H∗(BG) ⊗H∗(G/T ) as an H∗(BG)-module via p∗. In particular, H∗(BT ) is a free H∗(BG)-module. Now let ξ be a G-bundle over a space X . For convenience, we assume that X is path connected. Let ξ have classifying map f : X −→ BG. Of course, we can think of ξ as a principal G-bundle or as a G-bundle with fiber F for any G-space F . Construct the pullback diagram