Mayer–Vietoris sequence in de Rham cohomology
E620669
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.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Mayer–Vietoris sequence | 4 |
| Mayer–Vietoris sequence in de Rham cohomology canonical | 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 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 ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Mayer–Vietoris sequence in de Rham cohomology Description of subject: 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.
Referenced by (5)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Mayer–Vietoris sequence
this entity surface form:
Mayer–Vietoris sequence
this entity surface form:
Mayer–Vietoris sequence
subject surface form:
Algebraic Topology
this entity surface form:
Mayer–Vietoris sequence