splitting principle

One version of the splitting principle is captured in the following theorem. This theorem holds for complex vector bundles and cohomology with integer coefficients. It also holds for real vector bundles and cohomology with $\backslash mathbb\{Z\}\_2$ coefficients.

{{math_theorem|Let $\backslash xi\backslash colon\; E\backslash rightarrow\; X$ be a vector bundle of rank $n$ over a paracompact space $X$. There exists a space $Y=Fl(E)$, called the flag bundle associated to $E$, and a map $p\backslash colon\; Y\backslash rightarrow\; X$ such that

1. the induced cohomology homomorphism $p^*\backslash colon\; H^*(X)\backslash rightarrow\; H^*(Y)$ is injective, and
2. the pullback bundle $p^*\backslash xi\backslash colon\; p^*E\backslash rightarrow\; Y$ breaks up as a direct sum of line bundles: $p^*(E)=L\_1\backslash oplus\; L\_2\backslash oplus\backslash cdots\backslash oplus\; L\_n.$}}

In the complex case, the line bundles $L\_i$ or their first characteristic classes are called Chern roots.

Another version of the splitting principle concerns real vector bundles and their complexifications:H. Blane Lawson and Marie-Louise Michelsohn, Spin Geometry, Proposition 11.2.

{{math_theorem|Let $\backslash xi\backslash colon\; E\backslash rightarrow\; X$ be a real vector bundle of rank $2n$ over a paracompact space $X$. There exists a space $Y$ and a map $p\backslash colon\; Y\backslash rightarrow\; X$ such that

1. the induced cohomology homomorphism $p^*\backslash colon\; H^*(X)\backslash rightarrow\; H^*(Y)$ is injective, and
2. the pullback bundle $p^*\backslash xi\backslash colon\; p^*E\backslash rightarrow\; Y$ breaks up as a direct sum of line bundles and their conjugates: $p^*(E\backslash otimes\; \backslash mathbb\{C\})=L\_1\backslash oplus\; \backslash overline\{L\_1\}\backslash oplus\; \backslash cdots\backslash oplus\; L\_n\backslash oplus\; \backslash overline\{L\_n\}.$}}

The fact that $p^*\backslash colon\; H^*(X)\backslash rightarrow\; H^*(Y)$ is injective means that any equation which holds in $H^*(Y)$ — for example, among various Chern classes — also holds in $H^*(X)$. Often 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 pushed forward to $X$.

Since vector bundles on $X$ are used to define the K-theory group $K(X)$, it is important to note that $p^*\backslash colon\; K(X)\backslash rightarrow\; K(Y)$ is also injective for the map $p$ in the first theorem above.Oscar Randal-Williams, Characteristic classes and K-theory, Corollary 4.3.4, https://www.dpmms.cam.ac.uk/~or257/teaching/notes/Kthy.pdf

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.

• Grothendieck splitting principle for holomorphic vector bundles on the complex projective line

{{Reflist}}

• {{Citation | last=Hatcher | first=Allen | author-link=Allen Hatcher | title=Vector Bundles & K-Theory | url=http://pi.math.cornell.edu/~hatcher/VBKT/VBpage.html | edition=2.0 | year=2003}} section 3.1
• Raoul Bott and Loring Tu. Differential Forms in Algebraic Topology, section 21.

Category:Characteristic classes

Category:Vector bundles

Category:Mathematical principles