Characteristic class

A characteristic class is a mathematical object from differential topology. It is a topological invariant beam of a vector and may be represented by a differential form. A characteristic class describes more or less the " craziness " of a bundle, the characteristic class of a trivial bundle corresponds mostly to the one- element.

Definition

Be or. Is a vector bundle with fiber and the Grassmann manifold, then you can define a unique up to homotopy mapping that is superimposed by a bundle map in the tautological bundle over.

Let be a commutative ring with unit element. For each cohomology class is defined by the characteristic

Motivation

An n -dimensional vector bundle is trivial if and only if its classifying map is null homotopic ( homotopic to a constant map ) is. This condition is, however, difficult to verify. Is easier to check whether the induced illustrations are trivial in homology or cohomology and this is what is measured by characteristic classes.

Examples

  • Boots -Whitney classes of real vector bundles
  • Euler class of oriented real vector bundles
  • Chern classes of complex vector bundles
  • Pontryagin classes of real vector bundles

Principal bundle

Generally, one can define characteristic classes of principal bundles. Each cohomology class of the classifying space of the Lie group corresponding to a characteristic class of principal bundles. This is defined by, where the classifying map of is.

In the case of or the characteristic classes of principal bundles correspond to the characteristic classes of the associated vector bundle.

Conversely, the frame bundle as a principal bundle ( with structure group or ) consider whose characteristic classes correspond to the characteristic classes of the vector bundle with a metric provided to each ( real or complex ) vector bundles.

Characteristic classes of principal bundles can be made using Chern -Weil theory calculated from the curvature form of a connection. In particular, the characteristic classes of flat bundles disappear. For this one can then define secondary characteristic classes.

Differentialform Stiefel–Whitney class Chern–Weil theory
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