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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.
Theorem — Let ξ : E → X {\displaystyle \xi \colon E\rightarrow X} be a vector bundle of rank n {\displaystyle n} over a paracompact space X {\displaystyle X}. There exists a space Y = F l {\displaystyle Y=Fl} , called the flag bundle associated to E {\displaystyle E} , and a map p : Y → X {\displaystyle p\colon Y\rightarrow X} such that
The theorem above holds for complex vector bundles and integer coefficients or for real vector bundles with Z 2 {\displaystyle \mathbb {Z} _{2}} coefficients. In the complex case, the line bundles L i {\displaystyle L_{i}} or their first characteristic classes are called Chern roots.