Alexander duality

E620671

theorem in algebraic topology topological duality theorem

Alexander duality is a theorem in algebraic topology that relates the homology (or cohomology) of a subspace of a sphere to the reduced cohomology of its complement.

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Statements (46)

Predicate	Object
instanceOf	theorem in algebraic topology ⓘ topological duality theorem ⓘ
appearsIn	standard graduate textbooks on algebraic topology ⓘ
appliesTo	locally contractible subsets of spheres ⓘ subspaces of spheres ⓘ
assumes	A is a nonempty closed subset of S^n ⓘ n \ge 1 ⓘ
codomain	cohomology groups ⓘ homology groups ⓘ
domain	spheres ⓘ topological spaces ⓘ
expresses	isomorphism between homology of a subspace and reduced cohomology of its complement ⓘ
field	algebraic topology ⓘ cohomology theory ⓘ homology theory ⓘ
generalizationOf	Jordan curve theorem (via homological methods) NERFINISHED ⓘ
hasVariant	Alexander–Spanier cohomology version ⓘ Borel–Moore homology version of Alexander duality ⓘ cohomological Alexander duality NERFINISHED ⓘ
historicalPeriod	early 20th century mathematics ⓘ
holdsFor	finite CW-complexes embedded in spheres ⓘ polyhedra embedded in spheres ⓘ
involves	complements in spheres ⓘ reduced cohomology ⓘ reduced homology ⓘ singular cohomology ⓘ singular homology ⓘ
isPartOf	classical results of algebraic topology ⓘ
namedAfter	James Waddell Alexander II NERFINISHED ⓘ
relatedConcept	Lefschetz duality NERFINISHED ⓘ Poincaré–Alexander–Lefschetz duality NERFINISHED ⓘ knot complement ⓘ link complement ⓘ
relates	homology of a subspace of a sphere ⓘ reduced cohomology of the complement of a subspace in a sphere ⓘ
relatesTo	Poincaré duality NERFINISHED ⓘ
requires	Mayer–Vietoris sequence NERFINISHED ⓘ excision in homology ⓘ long exact sequence of a pair ⓘ
typicalAssumption	A is locally contractible ⓘ coefficients in a principal ideal domain ⓘ
typicalForm	\tilde H_i(S^n \setminus A) \cong \tilde H^{n-i-1}(A) ⓘ
usedFor	computing homology of complements of subsets in spheres ⓘ knot theory ⓘ linking phenomena in topology ⓘ studying embeddings of complexes in spheres ⓘ

Referenced by (1)

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

Poincaré duality → relatedConcept → Alexander duality ⓘ