Euler class

In mathematics, specifically in algebraic topology and in differential geometry and topology, the Euler class is a special type of characteristic classes, the orientable real vector bundle is assigned. It is named after Leonhard Euler, because it determines in the case of the tangent bundle of a manifold whose Euler characteristic.

It can be defined in different (equivalent) ways: as an obstacle to the existence of a section without zeros, as a pull -back of the orientation class at an intersection or as a picture of the Pfaffian determinant under the Chern -Weil isomorphism. In the case of flat bundles, there are other equivalent definitions.

Basic idea and motivation

The Euler class is a characteristic class, which is a topological invariant of oriented vector bundles: two isomorphic oriented vector bundles have the same Euler classes. In the case of differentiable manifolds, the Euler class of the tangent bundle determines the Euler characteristic of the manifold.

Euler class provides an obstacle to the existence of a section without zeros. In particular Euler characteristic of a closed orientable differentiable manifold provides a barrier to the existence of a vector field without singularities.

For a defined on a subset of the base - space zero missing section can define a relative Euler class, this provides a barrier to the cut continuability without zeros on the entire base.

Axioms

The (relative) Euler class is defined by the following axioms.

Each oriented - dimensional real vector bundle with a nowhere vanishing section on a ( possibly empty) subset is an element

( and if) assigned so that

• Valid for every continuous map
• For the tautological complex line bundle, regarded as a 2- dimensional real vector bundle, is a producer of.

Is called the Euler class of the bundle, ie the relative Euler class relative to the average.

Definition as obstruction class

For one-dimensional oriented vector bundle over the geometric realization of a simplicial complex obtained by obstruction theory, the obstruction class

For the continuation of a section in the vector bundle associated to the skeleton of.

The coefficient group

Is ( by orientation ) canonically isomorphic to and this isomorphism maps to the Euler class.

Definition using orientation class

For an oriented -dimensional vector bundles and the complement of the zero section, we consider the image of the orientation class ( Thom class)

In. Because is contractible, is a homotopy equivalence and

An isomorphism. Euler - class is defined by

Equivalent can be obtained by

For any section (for example, the zero -section) define.

If a section has no zeros, ie, it follows from it.

Relative Euler class: If a section is given without zeroing on a subset, then you can cut it to a (possibly with zeros ) to continue and then defines

Definition via Chern -Weil theory

We consider vector bundles over a differentiable manifold. The construction by Chern -Weil theory provides (only ) the image of the Euler class or the relative Euler class in particular, it provides the zero class for vector bundles of odd dimension.

For an oriented vector bundle of dimension one considers the associated principal bundle ( the framework bundle).

For a principal bundle with a connection form of the Euler class is the image of by

Defined Pfaffian determinant under the Chern -Weil homomorphism

Thus, as defined by the using the curvature form of the principal bundle differential form

Represented de Rham cohomology class. It can be shown that the Euler - class does not depend on the choice of connection and shape that it is in the picture.

The agreement of the so-defined Euler class with the above topologically defined the content of the 1945 by Chern ( and for Untermanigfaltigkeiten of Euclidean space as early as 1940 by Allendorf and Weil) proven set of generalized Gauss-Bonnet.

Relative Euler class: There is a section without zeros on a submanifold. ( We assume that the interface can be continued on an open neighborhood of. ) Then there is a connection form, satisfies the curvature form. Particular, it defines a relative cohomology class.

Euler class of SL ( n, R)- principal bundles

Under the isomorphisms

Corresponds to the Pfaffian determinant of a cohomology class in the cohomology of the classifying space of the Euler class of the universal bundle. For each bundle so you can be classified by means of the figure, the Euler class

Define. This agrees with the Euler class of the associated vector bundle.

Euler class of spheres bundles

The Euler class can be defined for arbitrary spheres bundle, in the case of the unit sphere bundle of a Riemannian vector bundle obtained the above-defined Euler class of the vector bundle.

Properties

• The canonical forms homomorphism Euler class from the n-th boot -Whitney class.
• The cup product is the highest Pontryagin class.
• For closed, orientable, differentiable manifolds with tangent bundle and the fundamental class of the Euler characteristic is.
• It is the vector bundles with inverted orientation, then.
• In particular odd dimension for Vektorbūndel. For closed, orientable, differentiable manifolds of odd dimension, the Euler characteristic vanishes.
• For the Whitney sum and the Cartesian product of vector bundles, the cup-product and the cross product referred to.
• For a generic section of a -dimensional oriented vector bundle over a closed orientable manifold -dimensional is the image of the fundamental class of the zero set in the Poincaré dual of. In the case of the tangent bundle, this results in the set of Poincaré - Hopf.
• If the normal bundle of a closed orientable submanifold, then the self- intersection number is from.
• If a section without zeros, then for all.
• Gysin sequence: For one-dimensional oriented vector bundle ( with the amount of non-zero vectors) gives the cup product with the Euler class exact sequence

Euler class of flat bundles

Simplicial definition

It should be a flat vector bundle over the geometric realization of a simplicial complex with simplices - .. Because simplices are contractible, the bundle is trivial over each simplex. At any chosen one can construct a section so by affine continuation. For this generic interface has no zeros on the skeleton, at most one zero per simplex and is transverse to the zero section. Then we define a simplicial - cocycle by

One can show that a cocycle is and its value does not depend on Zykeln on the selected interface. The cohomology class is represented by the Euler class of the flat bundle.

Flat SL (2, R)- bundle

Because you have the universal covering

This is a central extension and is therefore represented by a cohomology class. This is the universal Euler class for flat bundles, ie for a flat bundle with holonomy representation is obtained

The classifying map of the universal covering is.

Flat circle bundle

Denote the group of orientation- preserving homeomorphisms of the circle. Your universal covering is. The integers acting by translations on and you get an exact sequence

The corresponding cohomology class is the universal Euler class for flat bundles.

An explicit formula is specified by Jekel: universal Euler class is represented by the so-called orientation cocycle:

The orientation cocycle representing made ​​for all subgroups the universal Euler class for flat bundles. This is especially true for shallow bundle: one used on the effect of broken - linear transformations.