en>EmilJ: relation to scattered topology

2013-08-16T19:14:52Z

relation to scattered topology

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In [[mathematics]], the '''splitting principle''' is a technique used to reduce questions about [[vector bundle]]s to the case of [[line bundle]]s.

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 <math>\xi\colon E\rightarrow X</math> be a vector bundle of rank <math>n</math> over a manifold <math>X</math>. There exists a space <math>Y=Fl(E)</math>, called the flag bundle associated to <math>E</math>, and a map <math>p\colon Y\rightarrow X</math> such that

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

The line bundles <math>L_i</math> or their first characteristic class are called '''Chern roots.'''

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

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 <math>Y</math> and then pushed down to <math>X</math>.

==Symmetric polynomial==
Under the splitting principle, characteristic classes correspond to [[symmetric polynomials]] (and for the [[Euler class]], [[alternating polynomials]]) in the class of line bundles.

The [[Chern class]]es and [[Pontryagin class]]es correspond to [[symmetric polynomials]]: they are symmetric polynomials in the corresponding classes of line bundles (<math>p_k</math> is the ''k''th symmetric polynomial in the <math>p_1</math> of the line bundles, and so forth).

The [[Euler class]] is an invariant of an ''oriented'' vector bundle, and thus the line bundles are ordered up to sign; the corresponding polynomial is the [[Vandermonde polynomial]], the basic [[alternating polynomial]]. Further, for an even dimensional manifold, its square is the top Pontryagin class, which corresponds to the square of the Vandermonde polynomial being the [[discriminant]].

==See also==

*[[K-theory]]
*[[Grothendieck splitting principle]] for holomorphic vector bundles on the complex projective line

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

[[Category:Characteristic classes]]
[[Category:Vector bundles]]
[[Category:Mathematical principles]]

Scattered order - Revision history

en>EmilJ: relation to scattered topology