In 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 ev ...

s to the case of

line bundle In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example, a curve in the plane having a tangent line at each point determines a varying line: the ''tangent bundle'' is a way of organisin ...

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. 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 class ...

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 geom ...

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 mathematics, a symmetric polynomial is a polynomial in variables, such that if any of the variables are interchanged, one obtains the same polynomial. Formally, is a ''symmetric polynomial'' if for any permutation of the subscripts one has ...

in the first Chern classes of complex line bundles; these are the Chern classes.

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

Symmetric polynomial

See also

References