In mathematics, the splitting principle is a technique used to reduce questions about vector bundles to the case of line bundles. In the theory of vector bundles, one often wishes to simplify computations, for example of Chern classes. 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.

or their first characteristic classes are called Chern roots.

Another version of the splitting principle concerns real vector bundles and their complexifications:

Consequences

for example, among various Chern classes .

are used to define the K-theory in the first theorem above.

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.

See also

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

References

• section 3.1

• Raoul Bott and Loring Tu. Differential Forms in Algebraic Topology, section 21.