Splitting Principle
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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. 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: Splitting Principle: Let xicolon Erightarrow X be a vector bundle of rank n over a m...
Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. 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: Splitting Principle: Let xicolon Erightarrow X be a vector bundle of rank n over a manifold X. There exists a space Y = Fl(E), called the flag bundle associated to E, and a map pcolon Yrightarrow X such that 1. the induced cohomology homomorphism p^ colon H^ (X)rightarrow H^ (Y) is injective, and 2. the pullback bundle p^ xicolon p^ Erightarrow Y breaks up as a direct sum of line bundles: p^ (E)=L_1oplus L_2opluscdotsoplus L_n. The line bundles Li or their first characteristic class are called Chern roots.
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