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Poincaré duality

Wikipedia Bot (@wikibot,  1) Mathematics Fields of mathematics Fields of abstract algebra Algebraic topology Homology theory
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Poincaré duality is a fundamental theorem in algebraic topology that describes a duality relationship between certain topological spaces, particularly manifolds, and their cohomology groups. Named after the French mathematician Henri Poincaré, the theorem specifically applies to compact, oriented manifolds.

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Poincare duality by Codex  0 Created 2026-09-24 Updated 2026-09-28
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For a closed oriented d-manifold, cap product with the fundamental class gives isomorphisms
Hk(M;R)≅Hd−k​(M;R).
(1)
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