Inmathematics, specifically inalgebraic topology, theEuler class is acharacteristic class oforiented, realvector bundles. Like other characteristic classes, it measures how "twisted" the vector bundle is. In the case of thetangent bundle of a smoothmanifold, it generalizes the classical notion ofEuler characteristic. It is named afterLeonhard Euler because of this.

Throughout this article${\displaystyle E}$ is an oriented, real vector bundle ofrank${\displaystyle r}$ over a base space${\displaystyle X}$.

The Euler class${\displaystyle e(E)}$ is an element of the integralcohomology group

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

constructed as follows. Anorientation of${\displaystyle E}$ amounts to a continuous choice of generator of the cohomology

${\displaystyle H^r({\mathbf{R}}^r,{\mathbf{R}}^r\setminus \{0\};\mathbf{Z})\cong {\tilde{H}}^{r-1}(S^{r-1};\mathbf{Z})\cong \mathbf{Z}}$

of each fiber${\displaystyle {\mathbf{R}}^r}$relative to the complement${\displaystyle {\mathbf{R}}^r\setminus \{0\}}$ of zero. From theThom isomorphism, this induces anorientation class

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

in the cohomology of${\displaystyle E}$ relative to the complement${\displaystyle E\setminus E_0}$ of thezero section${\displaystyle E_0}$. The inclusions

${\displaystyle (X,\mathrm{\varnothing })\hookrightarrow (E,\mathrm{\varnothing })\hookrightarrow (E,E\setminus E_0),}$

where${\displaystyle X}$ includes into${\displaystyle E}$ as the zero section, induce maps

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

TheEuler classe(E) is the image ofu under the composition of these maps.

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

• Functoriality: If${\displaystyle F\to Y}$ is another oriented, real vector bundle and${\displaystyle f:Y\to X}$ is continuous and covered by an orientation-preserving map${\displaystyle F\to E}$, then${\displaystyle e(F)=f^{\ast }(e(E))}$. In particular,${\displaystyle e(f^{\ast }(E))=f^{\ast }(e(E))}$.
• Whitneysum formula: If${\displaystyle F\to X}$ is another oriented, real vector bundle, then the Euler class of theirdirect sum is given by${\displaystyle e(E\oplus F)=e(E)⌣e(F).}$
• Normalization: If${\displaystyle E}$ possesses a nowhere-zero section, then${\displaystyle e(E)=0}$.
• Orientation: If${\displaystyle \overline{E}}$ is${\displaystyle E}$ with the opposite orientation, then${\displaystyle e(\overline{E})=-e(E)}$.

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${\displaystyle e(E)\ne 0}$ then${\displaystyle E}$ has no non-vanishing section.

Alsounlike other characteristic classes, it is concentrated in a degree which depends on the rank of the bundle:${\displaystyle e(E)\in H^r(X)}$. By contrast, the Stiefel Whitney classes${\displaystyle w_i(E)}$ live in${\displaystyle H^i(X;\mathbb{Z}/2)}$ independent of the rank of${\displaystyle E}$. This reflects the fact that the Euler class isunstable, as discussed below.

The Euler class corresponds to the vanishing locus of a section of${\displaystyle E}$ in the following way. Suppose that${\displaystyle X}$ is an oriented smooth manifold of dimension${\displaystyle d}$. Let${\displaystyle \sigma :X\to E}$ be a smooth section thattransversely intersects the zero section. Let${\displaystyle Z\subseteq X}$ be the zero locus of${\displaystyle \sigma }$. Then${\displaystyle Z}$ is acodimension${\displaystyle r}$ submanifold of${\displaystyle X}$ which represents ahomology class${\displaystyle [Z]\in H_{d-r}(X;\mathbf{Z})}$ and${\displaystyle e(E)}$ is thePoincaré dual of${\displaystyle [Z]}$.

For example, if${\displaystyle Y}$ is a compact submanifold, then the Euler class of thenormal bundle of${\displaystyle Y}$ in${\displaystyle X}$ is naturally identified with theself-intersection of${\displaystyle Y}$ in${\displaystyle X}$.

In the special case when the bundleE 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 thefundamental homology class. Under this identification, the Euler class of the tangent bundle equals the Euler characteristic of the manifold. In the language ofcharacteristic 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

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

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

Any complex vector bundleE of complex rankd can be regarded as an oriented, real vector bundleE of real rank 2d. The Euler class ofE is given by the highest dimensional Chern class${\displaystyle e(E)=c_d(E)\in H^{2d}(X)}$

The Pontryagin class${\displaystyle p_r(E)}$ is defined as the Chern class of the complexification ofE:${\displaystyle p_r(E)=c_{2r}(\mathbf{C}\otimes E)}$.

The complexification${\displaystyle \mathbf{C}\otimes E}$ is isomorphic as an oriented bundle to${\displaystyle E\oplus E}$. Comparing Euler classes, we see that

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

If the rankr ofE is even then${\displaystyle e(E)⌣e(E)=c_r(E)=p_{r/2}(E)}$ where${\displaystyle p_{r/2}(E)}$ is the top dimensionalPontryagin class of${\displaystyle E}$.

A characteristic class${\displaystyle c}$ isstable if${\displaystyle c(E\oplus {\underset{\_}{R}}^1)=c(E)}$ where${\displaystyle {\underset{\_}{R}}^1}$ is a rank one trivial bundle.Unlike most other characteristic classes, the Euler class isunstable. In fact,${\displaystyle e(E\oplus {\underset{\_}{R}}^1)=e(E)⌣e({\underset{\_}{R}}^1)=0}$.

The Euler class is represented by a cohomology class in theclassifying spaceBSO(k)${\displaystyle e\in H^k(\mathrm{B}\mathrm{S}\mathrm{O}(k))}$. The unstability of the Euler class shows that it is not the pull-back of a class in${\displaystyle H^k(\mathrm{B}\mathrm{S}\mathrm{O}(k+1))}$ under the inclusion${\displaystyle \mathrm{B}\mathrm{S}\mathrm{O}(k)\to \mathrm{B}\mathrm{S}\mathrm{O}(k+1)}$.

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${\displaystyle H^d(X)}$ where${\displaystyle d}$ 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${\displaystyle H^1(X)}$).

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${\displaystyle S^n\subseteq {\mathrm{R}}^{n+1}}$ 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${\displaystyle S^n}$, 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.

The Euler characteristic of then-sphereSⁿ is:

Thus, there is no non-vanishing section of the tangent bundle of even spheres (this is known as theHairy ball theorem). In particular, the tangent bundle of an even sphere is nontrivial—i.e.,${\displaystyle S^{2n}}$ is not aparallelizable manifold, and cannot admit aLie group structure.

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

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

which shows that the Euler class vanishes; this is justn copies of the usual section over the circle.

As the Euler class for an even sphere corresponds to${\displaystyle 2[S^{2n}]\in H^{2n}(S^{2n},\mathbf{Z})}$, 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 usesecondary cohomology operations orK-theory.

The cylinder is a line bundle over the circle, by the natural projection${\displaystyle {\mathrm{R}}^1\times S^1\to S^1}$. 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.

• Vandermonde polynomial
• Thom isomorphism
• Generalized Gauss–Bonnet theorem

• Chern class
• Pontryagin class
• Stiefel-Whitney class

• Bott, Raoul and Tu, Loring W. (1982).Differential Forms in Algebraic Topology. Springer-Verlag.ISBN 0-387-90613-4.{{cite book}}: CS1 maint: multiple names: authors list (link)
• 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.

• Characteristic classes
• Vector bundles

• Articles with short description
• Short description is different from Wikidata
• CS1 maint: multiple names: authors list