Euler class on arvau

Wikipedia / 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.

is an oriented, real vector bundle of rank.

Formal definition

is an element of the integral cohomology group

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

relative of zero. From the Thom isomorphism, this induces an orientation class

of the zero section. The inclusions

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:

Functoriality:.
Whitney sum formula: is another oriented, real vector bundle, then the Euler class of their direct sum
Normalization:.
Orientation:.

has no non-vanishing section.

Also unlike. This reflects the fact that the Euler class is unstable, as discussed below.

Vanishing locus of generic section

be a smooth section that transversely intersects is a codimension submanifold which represents a homology is the Poincaré dual.

Self-intersection

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

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 2d. The Euler class of E

Squares to top Pontryagin class

is defined as the Chern class of the complexification of .

. Comparing Euler classes, we see that

If the rank of . If the rank r of is the top dimensional Pontryagin class.

Instability

is stable is a rank one trivial bundle. Unlike most other characteristic classes, the Euler class is unstable.

The Euler class is represented by a cohomology class in the classifying space BSO(k).

has trivial normal bundle, thus the tangent bundle of the sphere plus a trivial line bundle is the tangent bundle of Euclidean space, 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 Sn is:

Thus, there is no non-vanishing section of the tangent bundle of even spheres (this is known as the hairy ball theorem is not a parallelizable manifold, and cannot admit a Lie group structure.

For odd spheres, S2n−1 ⊂ R2n, 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.

, we can use the fact that the Euler class of a Whitney sum of two bundles is just the cup product of the Euler classes of the two bundles to see that there are no other subbundles of the tangent bundle than the tangent bundle itself and the null bundle, for any even-dimensional 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

. 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.