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.
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 ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.