Euler class

In mathematics, specifically in algebraic topology, the Euler class, named after Leonhard Euler, 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.

Throughout this article E → X is an oriented, real vector bundle of rank r.

Formal definition

The Euler class e(E) is an element of the integral cohomology group

H^r(X; \mathbf{Z}),

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

H^r(F, F \setminus F_0; \mathbf{Z})

of each fiber F relative to the complement F\F₀ of its zero element F₀. From Thom isomorphism, this induces an orientation class

u \in H^r(E, E \setminus E_0; \mathbf{Z})

in the cohomology of E relative to the complement E\E₀ of the zero section E₀. The inclusions

(X, \emptyset) \hookrightarrow (E, \emptyset) \hookrightarrow (E, E \setminus E_0),

where X includes into E as the zero section, induce maps

H^r(E, E \setminus E_0; \mathbf{Z}) \to H^r(E; \mathbf{Z}) \to H^r(X; \mathbf{Z}).

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: If F → Y is another oriented, real vector bundle and f : Y → X is continuous and covered by an orientation-preserving map F → E, then e(F) = f*e(E). In particular, e(f*E) = f*e(E).
Whitney sum formula: If F → X is another oriented, real vector bundle, then the Euler class of their direct sum is given by $e(E \oplus F) = e(E) \smile e(F).$
Normalization: If E possesses a nowhere-zero section, then e(E) = 0.
Orientation: If E is E with the opposite orientation, then e(E) = −e(E).

Note that "Normalization" is a distinguishing feature of the Euler class, so that it detects the existence of a non-vanishing section

Also unlike other characteristic classes, it is concentrated in a single dimension, which depends on the rank of the bundle: e(E) ∈ H^r — there are no e₀, e₁, .... In particular, c₀(E) = p₀(E) = 1 ∈ H⁰(X; Z) and w₀(E) = 1 ∈ H⁰(X; Z/2Z), but there is no e₀. This reflects the fact that the Euler class is unstable, as discussed below.

Vanishing of section

Under mild conditions (such as X a smooth, closed, oriented manifold), the Euler class corresponds to the vanishing of a section of E in the following way. Let σ : X → E be a generic smooth section and Z ⊆ X its zero locus. Then Z represents a homology class [Z] of codimension r in X, and e(E) is the Poincaré dual of [Z].

Self-intersection

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

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

H^r(X, \mathbf{Z}) \to H^r(X, \mathbf{Z}/2\mathbf{Z}).

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

Any complex vector bundle V of complex rank d can be regarded as an oriented, real vector bundle E of real rank 2d. The top Chern class c_d(V) of the complex bundle equals the Euler class e(E) of the real bundle.

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

e(E) \cup e(E) = e(E \oplus E) = e(E \otimes \mathbf{C}) = c_r(E \otimes \mathbf{C}) \in H^{2r}(X, \mathbf{Z}).

Squares to top Pontryagin class

If the rank r is even, then this cohomology class $e(E) \cup e(E)$ equals the top Pontryagin class p_r/2(E).

Unstable

Unlike the other characteristic classes, the Euler class is unstable, in the sense of stable homotopy theory. Concretely, this means that if 1 is a trivial bundle, then e(V ⊕ 1) ≠ e(V); 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 e(V ⊕ 1) = 0.

More abstractly, the cohomology class in the classifying space BSO(k) that represents the Euler class of a k-dimensional bundle is an unstable class: it is not the pull-back of a class in BSO(k+1) under the inclusion BSO(k) → BSO(k+1). 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 H¹, 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 Sⁿ ⊂ Rⁿ⁺¹ 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 Sⁿ, 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:

\chi(\mathbf{S}^n) = 1 + (-1)^n = \begin{cases} 2 & n\text{ even}\\ 0 & n\text{ odd}. \end{cases}

Thus, there is no non-vanishing section of the tangent bundle of even spheres, so the tangent bundle is not trivial—i.e., S²ⁿ is not a parallelizable manifold, and in particular does not admit a Lie group structure.

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

(x_2,-x_1,x_4,-x_3,\dots,x_{2n},-x_{2n-1})

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 2[S²ⁿ] ∈ H²ⁿ(S²ⁿ, Z), 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 R × S¹ → S¹. 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.

References

Bott, Raoul and Tu, Loring W. (1982). Differential Forms in Algebraic Topology. Springer-Verlag. ISBN 0-387-90613-4.
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.

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