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.
The theorem above holds for complex vector bundles and integer coefficients or for real vector bundles with formula_1 coefficients. In the complex case, the line bundles formula_2 or their first characteristic classes are called Chern roots.
The fact that formula_3 is injective means that any equation which holds in formula_4 (say between various Chern classes) also holds in formula_5.
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 formula_6 and then pushed down to formula_7.
Since vector bundles on formula_7 are used to define the K-theory group formula_9, it is important to note that formula_10 is also injective for the map formula_11 in the above theorem.
Symmetric polynomial.
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.