mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, 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 eve ...

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 organis ...

s. In the theory of vector bundles, one often wishes to simplify computations, for example of

Chern classes In mathematics, in particular in algebraic topology, differential geometry and topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundle, complex vector bundles. They ...

. Often computations are well understood for line bundles and for direct sums of line bundles. Then the splitting principle can be quite useful.

Statement

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

\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 to which the bundle is "twisted" and whether it possesses sections. Characterist ...

es are called Chern roots. Another version of the splitting principle concerns real vector bundles and their complexifications:

Consequences

The fact that

p^*\colon H^*(X)\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 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 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 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 h ...

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 become fundamental concepts in many branches ...

es.

References

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

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

Statement

Consequences

See also

References