- Euler class
mathematics, specifically in algebraic topology, the Euler class, named after Leonhard Euler, is a characteristic classof oriented, real vector bundles. Like other characteristic classes, it measures how "twisted" the vector bundle is. In the case of the tangent bundleof a smooth manifold, it generalizes the classical notion of Euler characteristic.
Throughout this article is an oriented, real vector bundle of rank .
The Euler class is an element of the integral
constructed as follows. An
orientationof amounts to a continuous choice of generator of the cohomology
of each fiber relative to the complement to its zero element . This induces an orientation class
in the cohomology of relative to the complement to the
zero section. The inclusions
where includes into as the zero section, induce maps
The Euler class is the image of under the composite of these maps.
The Euler class satisfies these useful properties:
* Functoriality: If is another oriented, real vector bundle and is continuous and covered by an orientation-preserving map , then . In particular, .
* Orientation: If is with the opposite orientation, then .
* Whitney sum formula: If is another oriented, real vector bundle, then the Euler class of the
direct sumis given by
* Normalization: If possesses a nowhere-zero section, then .
Under mild conditions (such as a smooth, closed, oriented manifold), the Euler class corresponds to the vanishing of a section of in the following way. Let
be a generic smooth section and its zero locus. Then represents a homology class of
codimensionin , and is the Poincaré dualof .
For example, if is a compact submanifold, then the Euler class of the
normal bundleof in is naturally identified with the self-intersectionof in .
Relations to other invariants
In the special case when the bundle in question is the tangent bundle of a compact, oriented, -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 induces a map
The image of the Euler class under this map is the top
Stiefel-Whitney class. One can view this Stiefel-Whitney class as "the Euler class, ignoring orientation".
Any complex vector bundle of complex rank can be regarded as an oriented, real vector bundle of real rank . The top
Chern classof the complex bundle equals the Euler class of the real bundle.
The Whitney sum is isomorphic to the complexification , which is a complex bundle of rank . Comparing Euler classes, we see that
If the rank is even, then this cohomology class equals the top
Example: Line bundle over the circle
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 . It is also isomorphic to the tangent bundle of the circle; the fact that its Euler class is corresponds to the fact that the Euler characteristic of the circle is .
*cite book | author=Bott, Raoul and Tu, Loring W. | title=Differential Forms in Algebraic Topology
publisher=Springer-Verlag | year=1982 | id=ISBN 0-387-90613-4
*cite book | author=Bredon, Glen E. | title=Topology and Geometry | publisher=Springer-Verlag | year=1993 | id= ISBN 0-387-97926-3
*cite book | author=Milnor, John W. and Stasheff, James D. | title=Characteristic Classes
publisher=Princeton University Press | year=1974 | id=ISBN 0-691-08122-0
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