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.
A form of Poincaré duality was first stated, without proof, by Henri Poincaré in 1893. It was stated in terms of Betti numbers. The cohomology concept was at that time about 40 years from being clarified. In his 1895 paper Analysis Situs, Poincaré tried to prove the theorem using topological intersection theory, which he had invented. Criticism of his work by Poul Heegaard led him to realize that his proof was seriously flawed. In the first two complements to Analysis Situs, Poincaré gave a new proof in terms of dual triangulations.
Poincaré duality did not take on its modern form until the advent of cohomology in the 1930s, when Eduard Čech and Hassler Whitney invented the cup and cap products and formulated Poincaré duality in these new terms.
Structured and high-profile, with a flat visor and a subtle grey under visor.
• 85% acrylic, 15% wool
• Structured, 6-panel, high-profile
• Plastic snap closure
• Grey under visor
• Head circumference: 22″–24″ (55–60 cm)
|A (inches)||20 ½-24 ¼|
|B (inches)||4 ½|
|C (inches)||2 ⅝|
|D (inches)||7 ⅛|