Euler class

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 formula_1 is an oriented, real vector bundle of rank formula_2 over a base space formula_3.
Formal definition.
The Euler class formula_4 is an element of the integral cohomology group
constructed as follows. An orientation of formula_1 amounts to a continuous choice of generator of the cohomology
of each fiber formula_8 relative to the complement formula_9 of zero. From the Thom isomorphism, this induces an orientation class
in the cohomology of formula_1 relative to the complement formula_12 of the zero section formula_13. The inclusions
where formula_3 includes into formula_1 as the zero section, induce maps
The Euler class "e"("E") is the image of "u" under the composition of these maps.
Properties.
The Euler class satisfies these properties, which are axioms of a characteristic class:
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 formula_30 then formula_1 has no non-vanishing section. 
Also "unlike" other characteristic classes, it is concentrated in a degree which depends on the rank of the bundle: formula_32. By contrast, the Stiefel Whitney classes formula_33 live in formula_34 independent of the rank of formula_1. This reflects the fact that the Euler class is unstable, as discussed below.
Vanishing Locus of Generic Section.
The Euler class corresponds to the vanishing locus of a section of formula_1 in the following way. Suppose that formula_3 is an oriented smooth manifold of dimension formula_38. Let formula_39 be a smooth section that transversely intersects the zero section. Let formula_40 be the zero locus of formula_41. Then formula_42 is a codimension formula_2 submanifold of formula_3 which represents a homology class formula_45 and formula_4 is the Poincaré dual of formula_47.
Self-intersection.
For example, if formula_48 is a compact submanifold, then the Euler class of the normal bundle of formula_48 in formula_3 is naturally identified with the self-intersection of formula_48 in formula_3.
Relations to other invariants.
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 2"d". The Euler class of "E" is given by the highest dimensional Chern class formula_54
Squares to top Pontryagin class.
The Pontryagin class formula_55 is defined as the Chern class of the complexification of "E": formula_56.
The complexification formula_57 is isomorphic as an oriented bundle to formula_58. Comparing Euler classes, we see that
If the rank "r" of "E" is even then formula_60 where formula_61 is the top dimensional Pontryagin class of formula_1.
Instability.
A characteristic class formula_63 is "stable" if formula_64 where formula_65 is a rank one trivial bundle.
Unlike most other characteristic classes, the Euler class is "unstable". In fact, formula_66.
The Euler class is represented by a cohomology class in the classifying space BSO("k") formula_67. The unstability of the Euler class shows that it is not the pull-back of a class in formula_68 under the inclusion formula_69.
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 formula_70 where formula_38 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 formula_72).
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 formula_73 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 formula_74, 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.
Examples.
Spheres.
The Euler characteristic of the "n"-sphere S"n" 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., formula_76 is not a parallelizable manifold, and cannot admit a Lie group structure.
For odd spheres, S2"n"−1 ⊂ R2"n", 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 formula_78, 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.
Circle.
The cylinder is a line bundle over the circle, by the natural projection formula_79. 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.