TITLE: Where does the splitting principle come from and does it generalize
QUESTION [10 upvotes]: Basically, I'm aware of "splitting principles" for the following three objects (which are all isomorphic modulo torsion).  
1. The Chow group a la Fulton.
2. The classical Grothendieck group of vector bundles or coherent sheaves.
3. The $\gamma$-graded Grothendieck group.
I was just wondering where the idea of "the splitting principle" comes from. I'm guessing somewhere in topology when one wanted to define Chern classes and show some properties. But I don't know.
And above that, is there some more general way of looking at this? I know there is a theorem that connects higher K-groups with Chow groups in a sense. So I ask, is there a way of deducing the splitting principle for one of the above objects from the other? (It's easy if we want to do this modulo torsion, of course.)

REPLY [5 votes]: In algebraic topology, there are two slightly different ways of thinking about this: one bundle at a time (the more usual way) and universally (via bundles between classifying spaces).  The second approach is surprisingly simple and general, as shown in the very 
brief paper "A note on the splitting principal", http://www.math.uchicago.edu/~may/PAPERS/Split.pdf, #109 on my web page. A key point is to notice that the splitting principle concerns the reduction of the structural group of a pullback of a given (complex) vector bundle from $U(n)$ to its maximal torus $T^n$. Replacing $U(n)$ by other compact Lie groups leads to splitting principals for other kinds of bundles, e.g. symplectic and real with cohomology away from $2$.  Replacing $T^n$ by a maximal $2$-torus gives a splitting principal for real bundles and cohomology at $2$.