Euler class

In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, specifically in algebraic topology

Algebraic topology

Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.Although algebraic topology...

, the Euler class, named after Leonhard Euler

Leonhard Euler

Leonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...

, is a characteristic class

Characteristic class

In mathematics, a characteristic class is a way of associating to each principal bundle on a topological space X a cohomology class of X. The cohomology class measures the extent to which the bundle is "twisted" — particularly, whether it possesses sections or not...

 of oriented, real vector bundle

Vector bundle

In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X : to every point x of the space X we associate a vector space V in such a way that these vector spaces fit together...

s. Like other characteristic classes, it measures how "twisted" the vector bundle is. In the case of the tangent bundle

Tangent bundle

In differential geometry, the tangent bundle of a differentiable manifold M is the disjoint unionThe disjoint union assures that for any two points x1 and x2 of manifold M the tangent spaces T1 and T2 have no common vector...

 of a smooth manifold

Manifold

In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....

, it generalizes the classical notion of Euler characteristic

Euler characteristic

In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent...

.

Throughout this article is an oriented, real vector bundle of rank .

Formal definition

The Euler class is an element of the integral cohomology

Cohomology

In mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a co-chain complex. That is, cohomology is defined as the abstract study of cochains, cocycles, and coboundaries...

 group


constructed as follows. An orientation

Orientability

In mathematics, orientability is a property of surfaces in Euclidean space measuring whether or not it is possible to make a consistent choice of surface normal vector at every point. A choice of surface normal allows one to use the right-hand rule to define a "clockwise" direction of loops in the...

 of amounts to a continuous choice of generator of the cohomology


of each fiber relative to the complement of its zero element . From 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 is the image of under the composition of these maps.

Properties

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

Hassler Whitney

Hassler Whitney was an American mathematician. He was one of the founders of singularity theory, and did foundational work in manifolds, embeddings, immersions, and characteristic classes.-Work:...

 sum formula: If is another oriented, real vector bundle, then the Euler class of the direct sum  is given by


Its distinguishing feature is that it detects the existence of a non-vanishing section:
Normalization: If possesses a nowhere-zero section, then .

It also satisfies:
Orientation: If is with the opposite orientation, then .

Note that unlike other characteristic classes, it is concentrated in a single dimension, which depends on the rank of the bundle: — there are no . In particular, and , but there is no . This reflects the fact that the Euler class is unstable, as discussed below.

Vanishing of section

Under mild conditions (such as a smooth, closed, oriented manifold), the Euler class corresponds to the vanishing of a section of in the following way. Let


be a generic

Generic property

In mathematics, properties that hold for "typical" examples are called generic properties. For instance, a generic property of a class of functions is one that is true of "almost all" of those functions, as in the statements, "A generic polynomial does not have a root at zero," or "A generic...

 smooth section and its zero locus. Then represents a homology

Homology (mathematics)

In mathematics , homology is a certain general procedure to associate a sequence of abelian groups or modules with a given mathematical object such as a topological space or a group...

 class of codimension

Codimension

In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, and also to submanifolds in manifolds, and suitable subsets of algebraic varieties.The dual concept is relative dimension.-Definition:...

  in , and is the Poincaré dual of .

Self-intersection

For example, if is a compact submanifold, then the Euler class of the normal bundle

Normal bundle

In differential geometry, a field of mathematics, a normal bundle is a particular kind of vector bundle, complementary to the tangent bundle, and coming from an embedding .-Riemannian manifold:...

 of in is naturally identified with the self-intersection of in .

Relations to other invariants

In the special case when the bundle in question is the tangent bundle of a compact, oriented, -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 induces a map


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

Any complex vector bundle of complex rank can be regarded as an oriented, real vector bundle of real rank . The top Chern class

Chern class

In mathematics, in particular in algebraic topology and differential geometry, the Chern classes are characteristic classes associated to complex vector bundles.Chern classes were introduced by .-Basic idea and motivation:...

  of the complex bundle equals the Euler class of the real bundle.

The Whitney sum is isomorphic to the complexification , which is a complex bundle of rank . Comparing Euler classes, we see that

Squares to top Pontryagin class

If the rank is even, then this cohomology class equals the top Pontryagin class

Pontryagin class

In mathematics, the Pontryagin classes are certain characteristic classes. The Pontryagin class lies in cohomology groups with degree a multiple of four...

 .

Under the splitting principle

Splitting principle

In mathematics, the splitting principle is a technique used to reduce questions about vector bundles to the case of line bundles.In the theory of vector bundles, one often wishes to simplify computations, say of Chern classes. Often computations are well understood for line bundles and for direct...

, this corresponds to the square of the Vandermonde polynomial equaling the discriminant

Discriminant

In algebra, the discriminant of a polynomial is an expression which gives information about the nature of the polynomial's roots. For example, the discriminant of the quadratic polynomialax^2+bx+c\,is\Delta = \,b^2-4ac....

: the Euler class corresponds to the Vandermonde polynomial, the basic alternating polynomial

Alternating polynomial

In algebra, an alternating polynomial is a polynomial f such that if one switches any two of the variables, the polynomial changes sign:f = -f....

, while the top Pontryagin class corresponds to the discriminant, a symmetric polynomial

Symmetric polynomial

In mathematics, a symmetric polynomial is a polynomial P in n variables, such that if any of the variables are interchanged, one obtains the same polynomial...

.

More formally, the Euler class of a direct sum of line bundles is the Vandermonde polynomial (orientation determines the order of the line bundles up to sign), while top Pontryagin class is the discriminant.

Unstable

Unlike the other characteristic classes, the Euler class is unstable, in the sense of stable homotopy theory

Stable homotopy theory

In mathematics, stable homotopy theory is that part of homotopy theory concerned with all structure and phenomena that remain after sufficiently many applications of the suspension functor...

. Concretely, this means that if 1 is a trivial bundle, then ; stable would mean that these are equal. In fact, adding a trivial bundle gives an obvious section, namely a constant on the trivial component, and 0 on the other, thus .

More abstractly, the cohomology class in the classifying space

Classifying space

In mathematics, specifically in homotopy theory, a classifying space BG of a topological group G is the quotient of a weakly contractible space EG by a free action of G...

  that represents the Euler class of a k-dimensional bundle is an unstable class: it is not the pull-back of a class in under the inclusion . Intuitively, it is not "consistently defined independently of dimension".

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): it is always of top dimension, while the other classes have a fixed dimension (the first Stiefel-Whitney class is in , etc.).

The fact that the Euler class is unstable should not be seen as a "defect": rather, from the point of view of stable homotopy, it means that the Euler class "detects unstable phenomena". For instance, the tangent bundle of spheres 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, rectricted 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.

Spheres

The Euler characteristic of the n-sphere is:
Thus, there is no non-vanishing section of the tangent bundle of even spheres, so the tangent bundle is not trivial, and they do not admit a Lie group structure.

For odd spheres, , 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

K-theory

In mathematics, K-theory originated as the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is an extraordinary cohomology theory known as topological K-theory. In algebra and algebraic geometry, it is referred to as algebraic K-theory. It...

.

Circle

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 . It is also isomorphic to the tangent bundle of the circle; the fact that its Euler class is corresponds to the fact that the Euler characteristic of the circle is .