Relates the homology of a compact subset of a sphere to the cohomology of its complement. Named for James Waddell Alexander, it is a foundational duality result of algebraic topology, extending simpler facts like the Jordan curve theorem to general dimensions.
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StatementAlexander duality relates the reduced homology of the complement of a compact, locally contractible subspace of a sphere to the reduced cohomology of the subspace itself, in complementary degrees; it is used, for example, to compute the cohomology of knot and link complements. 1 Classification
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1. Alexander duality (Wikipedia)
Wikimedia FoundationAlexander duality, lead section, sentence 2
It applies to the homology theory properties of the complement of a subspace X in Euclidean space, a sphere, or another manifold.
Alexander duality, lead section, sentence 1
In mathematics, Alexander duality refers to a duality theory initiated by a result of J. W. Alexander in 1915, and subsequently further developed, particularly by Pavel Alexandrov and Lev Pontryagin.
In Group: Algebraic Topology, Lead or opening section
It applies to the homology theory properties of the complement of a subspace X in Euclidean space, a sphere, or another manifold.
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