Mayer–Vietoris sequence in de Rham cohomology

E620669

construction in differential geometry mathematical concept tool in algebraic topology

The Mayer–Vietoris sequence in de Rham cohomology is a long exact sequence that computes the de Rham cohomology of a manifold by relating it to the cohomology of an open cover and their intersection.

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Observed surface forms (1)

Surface form	Occurrences
Mayer–Vietoris sequence	4

Statements (46)

Predicate	Object
instanceOf	construction in differential geometry ⓘ mathematical concept ⓘ tool in algebraic topology ⓘ
analogOf	Mayer–Vietoris sequence in singular cohomology NERFINISHED ⓘ
appearsIn	advanced textbooks on differential geometry ⓘ textbooks on algebraic topology ⓘ
appliesTo	differentiable manifolds ⓘ smooth manifolds ⓘ
assumes	U and V form an open cover of M ⓘ
basedOn	de Rham complex NERFINISHED ⓘ short exact sequence of complexes ⓘ
category	cochain-level construction ⓘ
clarifies	relationship between local and global properties of manifolds ⓘ
compatibleWith	homotopy invariance of de Rham cohomology ⓘ sheaf-theoretic viewpoint on differential forms ⓘ
constructs	connecting homomorphism in cohomology ⓘ
field	algebraic topology ⓘ de Rham cohomology NERFINISHED ⓘ differential geometry ⓘ
hasComponent	difference map on intersections ⓘ restriction maps of differential forms ⓘ
hasForm	⋯ → H^{k-1}(U∩V) → H^{k}(M) → H^{k}(U)⊕H^{k}(V) → H^{k}(U∩V) → ⋯ ⓘ
hasPrerequisite	knowledge of cochain complexes ⓘ knowledge of differential forms ⓘ knowledge of exact sequences ⓘ
hasProperty	long exact sequence ⓘ
isSpecialCaseOf	Mayer–Vietoris sequence for sheaf cohomology NERFINISHED ⓘ
isToolFor	computing cohomology of manifolds built by gluing ⓘ computing cohomology of projective spaces ⓘ computing cohomology of spheres ⓘ computing cohomology of tori ⓘ
namedAfter	Leopold Vietoris NERFINISHED ⓘ Waldo R. Mayer NERFINISHED ⓘ
purpose	compute de Rham cohomology of a manifold ⓘ
relates	cohomology of a manifold ⓘ cohomology of intersections of open subsets ⓘ cohomology of open subsets ⓘ
requires	good open cover ⓘ
usedFor	gluing local differential form data into global cohomology classes ⓘ
usedIn	excision-type arguments in differential topology ⓘ inductive computations on cell decompositions ⓘ proofs of Poincaré duality ⓘ
uses	de Rham cohomology groups ⓘ exactness of de Rham complex ⓘ intersection of open sets ⓘ open cover of a manifold ⓘ

Referenced by (5)

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

Poincaré lemma → relatedTo → Mayer–Vietoris sequence in de Rham cohomology ⓘ

Alexandrov–Čech cohomology → hasFeature → Mayer–Vietoris sequence in de Rham cohomology ⓘ

this entity surface form: Mayer–Vietoris sequence

Weil cohomology → hasAxiom → Mayer–Vietoris sequence in de Rham cohomology ⓘ

this entity surface form: Mayer–Vietoris sequence

étale cohomology → satisfies → Mayer–Vietoris sequence in de Rham cohomology ⓘ

this entity surface form: Mayer–Vietoris sequence

"Algebraic Topology" → developsConcept → Mayer–Vietoris sequence in de Rham cohomology ⓘ

subject surface form: Algebraic Topology

this entity surface form: Mayer–Vietoris sequence