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.
is an oriented, real vector bundle of rank
is an element of the integral cohomology group constructed as follows.
amounts to a continuous choice of generator of the cohomology of each fiber
From the Thom isomorphism, this induces an orientation class in the cohomology of
The Euler class obstructs the existence of a non-vanishing section in the sense that if
Also unlike other characteristic classes, it is concentrated in a degree which depends on the rank of the bundle:
By contrast, the Stiefel Whitney classes
This reflects the fact that the Euler class is unstable, as discussed below.
The Euler class corresponds to the vanishing locus of a section of
is an oriented smooth manifold of dimension
is a compact submanifold, then the Euler class of the normal bundle of
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.
Modding out by 2 induces a map The image of the Euler class under this map is the top Stiefel-Whitney class wr(E).
Any complex vector bundle E of complex rank d can be regarded as an oriented, real vector bundle E of real rank 2d.
is defined as the Chern class of the complexification of E:
Comparing Euler classes, we see that If the rank r of E is even then
is the top dimensional Pontryagin class of
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
, 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 the n-sphere Sn is: Thus, there is no non-vanishing section of the tangent bundle of even spheres (this is known as the Hairy ball theorem).
is not a parallelizable manifold, and cannot admit a Lie group structure.
For odd spheres, S2n−1 ⊂ R2n, 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.
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.
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 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.