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 E is an oriented, real vector bundle of rank r over a base space X .

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 E is an oriented, real vector bundle of rank r over a base space X . Functoriality: If F \to Y is another oriented, real vector bundle and f\colon 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 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 \overline{E} is E with the opposite orientation, then e(\overline{E}) = -e(E) . 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 e(E)\neq 0 then E has no non-vanishing section. Also unlike other characteristic classes, it is concentrated in a degree which depends on the rank of the bundle: e(E)\in H^r(X) . By contrast, the Stiefel Whitney classes w_i(E) live in H^i(X;\mathbb{Z}/2) independent of the rank of E . This reflects the fact that the Euler class is unstable, as discussed below. 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 is given by the highest dimensional Chern class e(E)=c_d(E)\in H^{2d}(X) ===Squares to top Pontryagin class=== The Pontryagin class p_r(E) is defined as the Chern class of the complexification of E : p_r(E)=c_{r}(\mathbf{C}\otimes E) . The complexification \mathbf{C}\otimes E is isomorphic as an oriented bundle to E\oplus E . Comparing Euler classes, we see that e(E) \smile e(E) = e(E \oplus E) = e(E \otimes \mathbf{C}) = c_r(E \otimes \mathbf{C}) \in H^{2r}(X, \mathbf{Z}). If the rank of E is odd, then e(E) \smile e(E) = 0 . If the rank r of E is even then e(E) \smile e(E) = c_{r}(E)=p_{r/2}(E) where p_{r/2}(E) is the top dimensional Pontryagin class of E . ===Instability=== A characteristic class c is stable if c(E\oplus \underline{R}^1)=c(E) where \underline{R}^1 is a rank one trivial bundle. Unlike most other characteristic classes, the Euler class is unstable. In fact, e(E\oplus\underline{R}^1)=e(E)\smile e(\underline{R}^1)=0 . The Euler class is represented by a cohomology class in the classifying space BSO(k) e\in H^k(\mathrm{BSO}(k)) . The unstability of the Euler class shows that it is not the pull-back of a class in H^k(\mathrm{BSO}(k+1)) under the inclusion \mathrm{BSO}(k) \to \mathrm{BSO}(k+1) . 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 H^d(X) where d 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 H^1(X) ). 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 S^n\subseteq \mathrm{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, restricted to S^n , 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. \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 (this is known as the hairy ball theorem). In particular, the tangent bundle of an even sphere is nontrivial—i.e., S^{2n} is not a parallelizable manifold, and cannot admit a Lie group structure. For odd spheres, S 2n−1 ⊂ R 2n , 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[S^{2n}] \in H^{2n}(S^{2n}, \mathbf{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 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.

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