In the topology, a part of mathematics, the Poincaré duality statement, named after Henri Poincaré, is a fundamental result of the structure of the homology and cohomology groups of varieties. She states that if M is a n-dimensionally oriented closed-chain (compact and non-bound), then the k-th cohomology group of M for all integers K is isomorph to the (n-k) -de homology group of M, < / p> H k ( M ) & # x2245; H n & # x2212; k ( M ) . {\displaystyle H^{k}(M)\cong H_{n-k}(M).}
Poincaré duality applies to each coefficient ring, as long as one has adopted an orientation regarding that coefficient ring; since each variety has a unique orientation mod 2, it is particularly important that Poincare duality mod 2 applies without any further assumption regarding orientation. History
In 1893, a form of Poincaré duality was still unsuccessfully drafted by Henri Poincaré. The duality was put in terms of Betti numbers: The k-e and (n-k) -eh Betti numbers of a closed (ie compact and unlimited) oriented n-variety are the same. A clarification of the cohomology concept was still about 40 years in the future. In his work of 1895, Analysis Situs, Poincaré tried to prove the statement by using the topological cross-sectional theory that he himself formulated. However, criticism of his work by Poul Heegaard led him to realize that his evidence showed serious flaws. In the first two complements on his Analysis Situs, Poincaré gave a new proof in the form of dual triangulations.
Only after the arrival of cohomology in the 1930s, the Poincaré duality became its modern form. Then Eduard Čech and Hassler Whitney came with the cup and cap products and formulated the Poincaré duality in these new terms.
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