Splitting principle

In mathematics, the splitting principle is a technique used to reduce questions about vector bundles to the case of line bundles. In the theory of vector bundles, one often wishes to simplify computations, say of Chern classes. Often computations are well understood for line bundles and for direct sums of line bundles. In this case the splitting principle can be quite useful. Let be a vector bundle of rank over a paracompact space . There exists a space , called the flag bundle associated to , and a map such that the induced cohomology homomorphism is injective, and the pullback bundle breaks up as a direct sum of line bundles: The theorem above holds for complex vector bundles and integer coefficients or for real vector bundles with coefficients. In the complex case, the line bundles or their first characteristic classes are called Chern roots. The fact that is injective means that any equation which holds in (say between various Chern classes) also holds in . The point is that these equations are easier to understand for direct sums of line bundles than for arbitrary vector bundles, so equations should be understood in and then pushed down to . Since vector bundles on are used to define the K-theory group , it is important to note that is also injective for the map in the above theorem. The splitting principle admits many variations. The following, in particular, concerns real vector bundles and their complexifications: Let be a real vector bundle of rank over a paracompact space . There exists a space and a map such that the induced cohomology homomorphism is injective, and the pullback bundle breaks up as a direct sum of line bundles and their conjugates: Under the splitting principle, characteristic classes for complex vector bundles correspond to symmetric polynomials in the first Chern classes of complex line bundles; these are the Chern classes.