mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the splitting principle is a technique used to reduce questions about

vector bundle In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every p ...

s 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. The theorem above holds for complex vector bundles and integer coefficients or for real vector bundles with

\mathbb_2

coefficients. In the complex case, the line bundles

L_i

or their first

characteristic class In mathematics, a characteristic class is a way of associating to each principal bundle of ''X'' a cohomology class of ''X''. The cohomology class measures the extent the bundle is "twisted" and whether it possesses sections. Characteristic classes ...

es are called Chern roots. 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 pushed down to

X

. Since vector bundles on

X

are used to define the

K-theory In mathematics, K-theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is a cohomology theory known as topological K-theory. In algebra and algebraic geometr ...

group

K(X)

, it is important to note that

p^*\colon K(X)\rightarrow K(Y)

is also injective for the map

p

in the above theorem. The splitting principle admits many variations. The following, in particular, concerns real vector bundles and their complexifications: H. Blane Lawson and Marie-Louise Michelsohn, ''Spin Geometry'', Proposition 11.2.

Symmetric polynomial

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 class In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles. They have since found applications in physics, Calabi–Yau ...

es.

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 Raoul Bott (September 24, 1923 – December 20, 2005) was a Hungarian-American mathematician known for numerous basic contributions to geometry in its broad sense. He is best known for his Bott periodicity theorem, the Morse–Bott functions whi ...

and Loring Tu. ''Differential Forms in Algebraic Topology'', section 21. Characteristic classes Vector bundles Mathematical principles

Symmetric polynomial

See also

References