The Poincare Duality Theorem states that for a closed, orientable manifold, the homology groups in one dimension are isomorphic to the cohomology groups in the complementary dimension. Named for Henri Poincare, it is a foundational result of algebraic topology.
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Proof YearPoincare first stated a form of the result without proof in 1893, in terms of Betti numbers. His 1895 paper Analysis Situs attempted the first proof, using topological intersection theory; criticism from Poul Heegaard showed that proof was flawed, and Poincare gave a corrected proof, in terms of dual triangulations, in the first two complements to Analysis Situs. The modern cohomological form was not reached until the 1930s. StatementFor a closed, oriented n-dimensional manifold, the kth cohomology group is isomorphic to the (n minus k)th homology group, for every integer k. 1 Classification
Statement FormCharacterization Theorem 1 Connections
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1. Poincare Duality Theorem (Wikipedia)
Wikimedia FoundationLead paragraph, statement sentence
It states that if M is an n-dimensional oriented closed manifold (compact and without boundary), then the kth cohomology group of M is isomorphic to the (n - k)th homology group of M, for all integers k
History
In his 1895 paper Analysis Situs, Poincaré tried to prove the theorem using topological intersection theory, which he had invented.
In Group: Algebraic Topology, Lead paragraph
In mathematics, the Poincaré duality theorem, named after Henri Poincaré, is a basic result on the structure of the homology and cohomology groups of manifolds.
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