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.
be a vector bundle of rank
over a paracompact space
, called the flag bundle associated to
such that The theorem above holds for complex vector bundles and integer coefficients or for real vector bundles with
In the complex case, the line bundles
or their first characteristic classes are called Chern roots.
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
are used to define the K-theory group
[1] The splitting principle admits many variations.
The following, in particular, concerns real vector bundles and their complexifications: [2] Theorem — Let
be a real vector bundle of rank
over a paracompact space
such that 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.