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.

Try in SPARQL Jump to: Surface forms Disambiguation Statements Elicitation Referenced by

All labels observed (4)

Label	Occurrences
Mayer–Vietoris sequence	8
Mayer–Vietoris sequence for sheaf cohomology	1
Mayer–Vietoris sequence in de Rham cohomology canonical	1
Mayer–Vietoris sequence in singular cohomology	1

How this entity was disambiguated

This entity first appeared as the object of triple T6801388 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.

NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07

Target entity: Mayer–Vietoris sequence in de Rham cohomology
Context triple: [Poincaré lemma, relatedTo, Mayer–Vietoris sequence in de Rham cohomology]

A. de Rham cohomology
de Rham cohomology is a cohomology theory for smooth manifolds that uses differential forms to capture their global topological and geometric properties.
B. Dolbeault cohomology classes
Dolbeault cohomology classes are equivalence classes of differential forms on a complex manifold defined using the ∂̄-operator, encoding the manifold’s complex-analytic and geometric structure.
C. L’Analysis Situs et la Géométrie Algébrique
L’Analysis Situs et la Géométrie Algébrique is a foundational mathematical treatise that helped establish modern algebraic topology and its connections with algebraic geometry.
D. Grothendieck spectral sequence
The Grothendieck spectral sequence is a fundamental tool in homological algebra that relates the derived functors of a composite functor to the derived functors of its components, enabling efficient computation of cohomology.
E. Hodge decomposition
Hodge decomposition is a fundamental result in differential geometry and Hodge theory that expresses differential forms on a Riemannian manifold uniquely as sums of exact, co-exact, and harmonic components.
F. None of above. chosen
G. Unsure - the case is ambiguous/there is not enough information to decide.

NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07

Target entity: Mayer–Vietoris sequence in de Rham cohomology
Target entity description: 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.

A. de Rham cohomology
de Rham cohomology is a cohomology theory for smooth manifolds that uses differential forms to capture their global topological and geometric properties.
B. Dolbeault cohomology classes
Dolbeault cohomology classes are equivalence classes of differential forms on a complex manifold defined using the ∂̄-operator, encoding the manifold’s complex-analytic and geometric structure.
C. L’Analysis Situs et la Géométrie Algébrique
L’Analysis Situs et la Géométrie Algébrique is a foundational mathematical treatise that helped establish modern algebraic topology and its connections with algebraic geometry.
D. Grothendieck spectral sequence
The Grothendieck spectral sequence is a fundamental tool in homological algebra that relates the derived functors of a composite functor to the derived functors of its components, enabling efficient computation of cohomology.
E. Hodge decomposition
Hodge decomposition is a fundamental result in differential geometry and Hodge theory that expresses differential forms on a Riemannian manifold uniquely as sums of exact, co-exact, and harmonic components.
F. None of above. chosen

Statements (46)

Predicate	Object
instanceOf	construction in differential geometry ⓘ mathematical concept ⓘ tool in algebraic topology ⓘ
analogOf	Mayer–Vietoris sequence in singular cohomology ⓘ linked to: Mayer–Vietoris sequence in de Rham cohomology
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 ⓘ linked to: Hodge–de Rham complex 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 ⓘ 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 ⓘ linked to: Mayer–Vietoris sequence in de Rham cohomology
isToolFor	computing cohomology of manifolds built by gluing ⓘ computing cohomology of projective spaces ⓘ computing cohomology of spheres ⓘ computing cohomology of tori ⓘ
namedAfter	Leopold Vietoris ⓘ Waldo R. Mayer ⓘ
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 ⓘ

How these facts were elicited