In mathematics, specifically in algebraic topology, the Euler class is a characteristic class of oriented, real vector bundles. Like other characteristic classes, it measures how "twisted" the vector bundle is. In the case of the tangent bundle of a smooth manifold, it generalizes the classical notion of Euler characteristic. It is named after Leonhard Euler because of this.

Throughout this article is an oriented, real vector bundle of rank over a base space .

Formal definitionEdit

The Euler class   is an element of the integral cohomology group

 

constructed as follows. An orientation of   amounts to a continuous choice of generator of the cohomology

 

of each fiber   relative to the complement   of zero. From the Thom isomorphism, this induces an orientation class

 

in the cohomology of   relative to the complement   of the zero section  . The inclusions

 

where   includes into   as the zero section, induce maps

 

The Euler class e(E) is the image of u under the composition of these maps.

PropertiesEdit

The Euler class satisfies these properties, which are axioms of a characteristic class:

  • Functoriality: If   is another oriented, real vector bundle and   is continuous and covered by an orientation-preserving map  , then  . In particular,  .
  • Whitney sum formula: If   is another oriented, real vector bundle, then the Euler class of their direct sum is given by  
  • Normalization: If   possesses a nowhere-zero section, then  .
  • Orientation: If   is   with the opposite orientation, then  .

Note that "Normalization" is a distinguishing feature of the Euler class. The Euler class obstructs the existence of a non-vanishing section in the sense that if   then   has no non-vanishing section.

Also unlike other characteristic classes, it is concentrated in a degree which depends on the rank of the bundle:  . By contrast, the Stiefel Whitney classes   live in   independent of the rank of  . This reflects the fact that the Euler class is unstable, as discussed below.

Vanishing Locus of Generic SectionEdit

The Euler class corresponds to the vanishing locus of a section of   in the following way. Suppose that   is an oriented smooth manifold of dimension  . Let   be a smooth section that transversely intersects the zero section. Let   be the zero locus of  . Then   is a codimension   submanifold of   which represents a homology class   and   is the Poincaré dual of  .

Self-intersectionEdit

For example, if   is a compact submanifold, then the Euler class of the normal bundle of   in   is naturally identified with the self-intersection of   in  .

Relations to other invariantsEdit

In the special case when the bundle E in question is the tangent bundle of a compact, oriented, r-dimensional manifold, the Euler class is an element of the top cohomology of the manifold, which is naturally identified with the integers by evaluating cohomology classes on the fundamental homology class. Under this identification, the Euler class of the tangent bundle equals the Euler characteristic of the manifold. In the language of characteristic numbers, the Euler characteristic is the characteristic number corresponding to the Euler class.

Thus the Euler class is a generalization of the Euler characteristic to vector bundles other than tangent bundles. In turn, the Euler class is the archetype for other characteristic classes of vector bundles, in that each "top" characteristic class equals the Euler class, as follows.

Modding out by 2 induces a map

 

The image of the Euler class under this map is the top Stiefel-Whitney class wr(E). One can view this Stiefel-Whitney class as "the Euler class, ignoring orientation".

Any complex vector bundle E of complex rank d can be regarded as an oriented, real vector bundle E of real rank 2d. The Euler class of E is given by the highest dimensional Chern class  

Squares to top Pontryagin classEdit

The Pontryagin class   is defined as the Chern class of the complexifiaction of E  .

The complexification   is isomorphic as an oriented bundle to  . Comparing Euler classes, we see that

 

If the rank r of E is even then   where   is the top dimensional Pontryagin class of  .

UnstabiliyEdit

A characteristic class  is stable if   where   is a rank one trivial bundle. Unlike most other characteristic classes, the Euler class is unstable. In fact,  .

The Euler class is represented by a cohomology class in the classifying space BSO(k)  . The unstability of the Euler class shows that it is not the pull-back of a class in   under the inclusion  .

This can be seen intuitively in that the Euler class is a class whose degree depends on the dimension of the bundle (or manifold, if the tangent bundle): the Euler class is an element of   where   is the dimension of the bundle, while the other classes have a fixed dimension (e.g., the first Stiefel-Whitney class is an element of  ).

The fact that the Euler class is unstable should not be seen as a "defect": rather, it means that the Euler class "detects unstable phenomena". For instance, the tangent bundle of an even dimensional sphere is stably trivial but not trivial (the usual inclusion of the sphere   has trivial normal bundle, thus the tangent bundle of the sphere plus a trivial line bundle is the tangent bundle of Euclidean space, restricted to  , which is trivial), thus other characteristic classes all vanish for the sphere, but the Euler class does not vanish for even spheres, providing a non-trivial invariant.

ExamplesEdit

SpheresEdit

The Euler characteristic of the n-sphere Sn is:

 

Thus, there is no non-vanishing section of the tangent bundle of even spheres (this is known as the Hairy ball theorem). In particular, the tangent bundle of an even sphere is nontrivial—i.e.,   is not a parallelizable manifold, and cannot admit a Lie group structure.

For odd spheres, S2n−1R2n, a nowhere vanishing section is given by

 

which shows that the Euler class vanishes; this is just n copies of the usual section over the circle.

As the Euler class for an even sphere corresponds to  , we can use the fact that the Euler class of a Whitney sum of two bundles is just the cup product of the Euler class of the two bundles to see that there are no non-trivial subbundles of the tangent bundle of an even sphere.

Since the tangent bundle of the sphere is stably trivial but not trivial, all other characteristic classes vanish on it, and the Euler class is the only ordinary cohomology class that detects non-triviality of the tangent bundle of spheres: to prove further results, one must use secondary cohomology operations or K-theory.

CircleEdit

The cylinder is a line bundle over the circle, by the natural projection  . It is a trivial line bundle, so it possesses a nowhere-zero section, and so its Euler class is 0. It is also isomorphic to the tangent bundle of the circle; the fact that its Euler class is 0 corresponds to the fact that the Euler characteristic of the circle is 0.

See alsoEdit

Other classesEdit

ReferencesEdit

  • Bott, Raoul and Tu, Loring W. (1982). Differential Forms in Algebraic Topology. Springer-Verlag. ISBN 0-387-90613-4.CS1 maint: multiple names: authors list (link)
  • Bredon, Glen E. (1993). Topology and Geometry. Springer-Verlag. ISBN 0-387-97926-3.
  • Milnor, John W.; Stasheff, James D. (1974). Characteristic Classes. Princeton University Press. ISBN 0-691-08122-0.