Euler class

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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 \to X is an oriented, real vector bundle of rank r.

Contents

Formal definition

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

H^r(X; \mathbb{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; \mathbb{Z})

of each fiber F relative to the complement F \setminus F_0 to its zero element F0. This induces an orientation class

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

in the cohomology of E relative to the complement E \setminus E_0 to 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; \mathbb{Z}) \to H^r(E; \mathbb{Z}) \to H^r(X; \mathbb{Z}).

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

Properties

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

Functoriality
If F \to Y is another oriented, real vector bundle and f : Y \to X is continuous and covered by an orientation-preserving map F \to E, then e(F) = f * e(E). In particular, e(f * E) = f * e(E).
Whitney sum formula
If F \to X is another oriented, real vector bundle, then the Euler class of the direct sum E \oplus F is given by
e(E \oplus F) = e(E) \cup e(F).

Its distinguishing feature is that it detects the existence of a non-vanishing section:

Normalization
If E possesses a nowhere-zero section, then e(E) = 0.

It also satisfies:

Orientation
If \bar E is E with the opposite orientation, then e(\bar E) = -e(E).

Note that unlike other characteristic classes, it is concentrated in a single dimension, which depends on the rank of the bundle: e(E) \in H^r — there are no e_0, e_1, \dots. In particular, c_0(E) = p_0(E) = 1 \in H^0(X;\mathbf{Z}) and w_0(E) = 1 \in H^0(X;\mathbf{Z}/2\mathbf{Z}), 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

\sigma \colon X \to E

be a generic smooth section and Z \subseteq 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, \mathbb{Z}) \to H^r(X, \mathbb{Z}/2).

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

The Whitney sum E \oplus E is isomorphic to the complexification E \otimes \mathbb{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 \mathbb{C}) = c_r(E \otimes \mathbb{C}) \in H^{2r}(X, \mathbb{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 pr / 2(E).

Under the splitting principle, this corresponds to the square of the Vandermonde polynomial equaling the discriminant: the Euler class corresponds to the Vandermonde polynomial, the basic alternating polynomial, while the top Pontryagin class corresponds to the discriminant, a symmetric 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. Concretely, this means that if 1 is a trivial bundle, then e(V\oplus 1) \neq 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 \oplus 1) = 0.

More abstractly, the cohomology class in the classifying space BO(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 BO(k + 1) under the inclusion BO(k) \to BO(k+1). Intuitively, it is not "consistently defined independently of dimension".

This can be seen intuitively in that the Euler class is only 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 "detects unstable phenomena". For instance, the tangent bundle of spheres is stable trivial but not trivial (the usual inclusion of the sphere S^n \subset \mathbf{R}^{n+1} 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:

\chi(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, and they do not admit a Lie group structure.

For odd spheres, S^{2n-1} \subset \mathbf{R}^n, 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.

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 \mathbb{R} \times \mathbb{S}^1 \to \mathbb{S}^1. 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 also

Other classes

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. and Stasheff, James D. (1974). Characteristic Classes. Princeton University Press. ISBN 0-691-08122-0. 

Wikipedia content modification information:

  • This page was last modified on 20 December 2008, at 01:01.

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