Lefschetz duality
E623376
Lefschetz duality is a generalization of Poincaré duality that relates the homology of a compact manifold with boundary to the cohomology of the manifold relative to its boundary.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Lefschetz duality canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T6801417 — 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: Lefschetz duality Context triple: [Poincaré duality, relatedConcept, Lefschetz duality]
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A.
Poincaré duality
Poincaré duality is a fundamental theorem in algebraic topology that relates the homology and cohomology groups of an oriented closed manifold in complementary dimensions.
-
B.
Mayer–Vietoris sequence in de Rham cohomology
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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C.
Serre duality
Serre duality is a fundamental theorem in algebraic geometry that generalizes classical duality for Riemann surfaces to higher-dimensional projective varieties, relating cohomology groups of coherent sheaves via a dualizing sheaf.
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D.
Lefschetz operator
The Lefschetz operator is a linear operator in Kähler geometry that acts on differential forms by wedging with the Kähler form, playing a central role in the Hard Lefschetz theorem and Hodge theory.
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E.
Lefschetz fixed-point theorem
The Lefschetz fixed-point theorem is a fundamental result in algebraic topology that relates the number of fixed points of a continuous map on a topological space to traces of the induced maps on its homology groups.
- 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: Lefschetz duality Target entity description: Lefschetz duality is a generalization of Poincaré duality that relates the homology of a compact manifold with boundary to the cohomology of the manifold relative to its boundary.
-
A.
Poincaré duality
Poincaré duality is a fundamental theorem in algebraic topology that relates the homology and cohomology groups of an oriented closed manifold in complementary dimensions.
-
B.
Mayer–Vietoris sequence in de Rham cohomology
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.
-
C.
Serre duality
Serre duality is a fundamental theorem in algebraic geometry that generalizes classical duality for Riemann surfaces to higher-dimensional projective varieties, relating cohomology groups of coherent sheaves via a dualizing sheaf.
-
D.
Lefschetz operator
The Lefschetz operator is a linear operator in Kähler geometry that acts on differential forms by wedging with the Kähler form, playing a central role in the Hard Lefschetz theorem and Hodge theory.
-
E.
Lefschetz fixed-point theorem
The Lefschetz fixed-point theorem is a fundamental result in algebraic topology that relates the number of fixed points of a continuous map on a topological space to traces of the induced maps on its homology groups.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
duality principle
ⓘ
mathematical theorem ⓘ |
| appliesTo | compact manifold with boundary ⓘ |
| assumes |
compactness
ⓘ
manifold with boundary ⓘ oriented manifold ⓘ |
| context |
manifolds with boundary
ⓘ
relative (co)homology ⓘ |
| dimensionCondition | n-dimensional manifold ⓘ |
| field |
algebraic topology
ⓘ
differential topology ⓘ homological algebra ⓘ |
| formalism | expressed via cap product with fundamental class ⓘ |
| generalizes | Poincaré duality NERFINISHED ⓘ |
| givesIsomorphismBetween |
H_i(M) and H^{n-i}(M,\partial M)
ⓘ
H_i(M,\partial M) and H^{n-i}(M) ⓘ |
| hasConsequence |
duality between Betti numbers of M and (M,\partial M)
ⓘ
nondegenerate pairing between homology and relative cohomology ⓘ |
| hasVariant | Poincaré–Lefschetz duality for local coefficients NERFINISHED ⓘ |
| holdsFor |
coefficients in a field
ⓘ
coefficients in a principal ideal domain ⓘ |
| involves |
boundary inclusion map
ⓘ
long exact sequence of a pair ⓘ orientation class in H_n(M,\partial M) ⓘ |
| isAnalogOf | Poincaré duality for manifolds without boundary ⓘ |
| isDiscussedIn |
Hatcher's Algebraic Topology
NERFINISHED
ⓘ
Spanier's Algebraic Topology NERFINISHED ⓘ textbooks on algebraic topology ⓘ |
| isRelatedTo |
Alexander duality
NERFINISHED
ⓘ
Poincaré–Lefschetz duality NERFINISHED ⓘ |
| isSpecialCaseOf | Verdier duality NERFINISHED ⓘ |
| namedAfter | Solomon Lefschetz NERFINISHED ⓘ |
| relates |
homology of a compact manifold with boundary
ⓘ
relative cohomology of the manifold with respect to its boundary ⓘ |
| requires |
existence of a fundamental class in H_n(M,\partial M)
ⓘ
local orientability ⓘ |
| timePeriod | 20th century mathematics ⓘ |
| typeOf | homology–cohomology duality ⓘ |
| usedIn |
Morse theory on manifolds with boundary
ⓘ
algebraic geometry via comparison theorems ⓘ intersection theory ⓘ topological invariants of manifolds with boundary ⓘ |
| usedToProve | properties of manifolds with nonempty boundary ⓘ |
| uses |
cap product
ⓘ
fundamental class ⓘ singular cohomology ⓘ singular homology ⓘ |
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Input
Subject: Lefschetz duality Description of subject: Lefschetz duality is a generalization of Poincaré duality that relates the homology of a compact manifold with boundary to the cohomology of the manifold relative to its boundary.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.