Lefschetz duality

E623376

duality principle mathematical theorem

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.

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All labels observed (3)

Label	Occurrences
Lefschetz duality canonical	2
Poincaré–Alexander–Lefschetz duality	1
Poincaré–Lefschetz duality for local coefficients	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]

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
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 ⓘ
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 ⓘ linked to: Lefschetz duality
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 ⓘ linked to: "Algebraic Topology" Spanier's Algebraic Topology ⓘ linked to: "Algebraic Topology" textbooks on algebraic topology ⓘ
isRelatedTo	Alexander duality ⓘ Poincaré–Lefschetz duality ⓘ linked to: Poincaré duality
isSpecialCaseOf	Verdier duality ⓘ
namedAfter	Solomon Lefschetz ⓘ
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 ⓘ

How these facts were elicited

Referenced by (4)

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

Poincaré duality → relatedConcept → Lefschetz duality ⓘ

Alexander duality → relatedConcept → Lefschetz duality ⓘ

Alexander duality → relatedConcept → Poincaré–Alexander–Lefschetz duality ⓘ

linked to: Lefschetz duality

Lefschetz duality → hasVariant → Poincaré–Lefschetz duality for local coefficients ⓘ