Euler class

E627991

characteristic class cohomology class topological invariant

The Euler class is a topological characteristic class associated with oriented real vector bundles, capturing obstruction information such as the existence of nowhere-vanishing sections.

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Statements (44)

Predicate	Object
instanceOf	characteristic class ⓘ cohomology class ⓘ topological invariant ⓘ
appearsIn	Gysin sequence NERFINISHED ⓘ Leray–Hirsch theorem for sphere bundles NERFINISHED ⓘ
associatedWith	orientation ⓘ vector bundle ⓘ
canBeRepresentedBy	Pfaffian of curvature form ⓘ closed differential form ⓘ
captures	obstruction to existence of nonzero section ⓘ obstruction to existence of nowhere-vanishing section ⓘ
constructedVia	Thom class ⓘ zero section of the bundle ⓘ
definedAs	pullback of Thom class along zero section ⓘ
definedFor	oriented rank n real vector bundle ⓘ oriented sphere bundles via associated vector bundle ⓘ
definedOn	oriented real vector bundle ⓘ
degree	rank of the vector bundle ⓘ
dependsOn	isomorphism class of the bundle ⓘ
functorialUnder	pullback of bundles ⓘ
generalizes	Euler characteristic of a closed oriented manifold ⓘ
hasVariant	Euler class in de Rham cohomology NERFINISHED ⓘ
independentOf	choice of connection ⓘ
is	primary obstruction to a nonvanishing section of an oriented bundle ⓘ top Stiefel–Whitney class modulo 2 ⓘ
isMultiplicativeFor	direct sum of oriented bundles of odd rank under suitable conditions ⓘ
isNonzeroIf	no global nowhere-vanishing section exists ⓘ
isZeroIf	bundle admits a nowhere-vanishing section ⓘ
livesIn	even-degree cohomology ⓘ top-degree cohomology of the base space ⓘ
mod2ReductionIs	top Stiefel–Whitney class ⓘ
namedAfter	Leonhard Euler NERFINISHED ⓘ
naturalWithRespectTo	bundle pullback ⓘ
relatedTo	Euler characteristic via integration over the fundamental class ⓘ Gauss–Bonnet theorem NERFINISHED ⓘ Poincaré–Hopf theorem NERFINISHED ⓘ
requires	orientation of the vector bundle ⓘ
takesValuesIn	H^{n}(B;\mathbb{Z}) ⓘ integral cohomology ⓘ
usedIn	classification of vector bundles ⓘ differential topology ⓘ index theory ⓘ obstruction theory ⓘ
vanishesFor	trivial oriented real vector bundle ⓘ

Referenced by (3)

Full triples — surface form annotated when it differs from this entity's canonical label.

Chern–Weil theory → constructs → Euler class ⓘ

Topology from the Differentiable Viewpoint → topic → Euler class ⓘ

Characteristic Classes → hasSubject → Euler class ⓘ