Splitting principle

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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: Splitting Principle: Let $\xi\colon E\rightarrow X$ be a vector bundle of rank $n$ over a manifold $X$ . There exists a space $Y = F l (E)$ , called the flag bundle associated to $E$ , and a map $p\colon Y\rightarrow X$ such that

the induced cohomology homomorphism $p^*\colon H^*(X)\rightarrow H^*(Y)$ is injective, and
the pullback bundle $p^*\xi\colon p^*E\rightarrow Y$ breaks up as a direct sum of line bundles: $p^*(E)=L_1\oplus L_2\oplus\cdots\oplus L_n$

The fact that $p^*\colon H^*(X)\rightarrow H^*(Y)$ is injective means that any equation which holds in $H * (Y)$ (say between various Chern classes) also holds in $H * (X)$ .

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 $Y$ and then pulled back to $X$ .

[edit] Reference

Bott and Tu. Differential Forms in Algebraic Topology, section 21.

Categories: Vector bundles

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Splitting principle

From Wikipedia, the free encyclopedia

[edit] Reference

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