Poincaré duality Guide, Meaning , Facts, Information and Description
In mathematics, the Poincaré duality theorem is a basic result on the structure of the homology and cohomology groupss of manifolds. It states that if M is an n-dimensional compact oriented manifold, then the k-th cohomology group of M is isomorphic to the (n − k)-th homology group of M. It further states that if mod 2 homology and cohomology is used, then the assumption of orientability can be dropped.
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2 Dual cell structures 3 Modern formulations 4 Generalizations |
A form of Poincaré duality was first stated, without proof, by Henri Poincaré in 1893. It was stated in terms of Betti numbers: The k-th and (n − k)-th Betti numbers of a closed n-manifold are equal. 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 1930's, when Eduard Cech and Hassler Whitney invented the cup and cap products and formulated Poincaré duality in these new terms.
Poincaré duality was classically thought of in terms of dual triangulations, which are generalizations of dual polyhedra. Given a triangulation X of an n-dimensional manifold M, one replaces each k-simplex with a (n − k)-cell to produce a new decomposition Y of M. If each (n − k)-cell is indeed a simplex then one says that Y is the dual triangulation of X. Considering the tetrahedron as a triangulation of the 2-sphere, the dual triangulation of the tetrahedron is another tetrahedron. This construction does not necessarily yield another triangulation, as the examples of the octahedron and icosahedron demonstrate. Poincaré used a (not entirely correct) method involving barycentric subdivision to show that we may always obtain a dual triangulation for compact oriented manifolds.
In more precise terms, one may describe the dual of a triangulation X as a triangulation Y such that given a k-simplex α in X, there is one (n − k)-simplex in Y whose intersection number with α is 1, and such that the intersection number of α with any other (n − k)-simplex of Y is 0.
The boundary operator in a chain complex can be viewed as a matrix. Let M be a closed n-manifold, X a triangulation of M, and Y the dual triangulation of X. Then one can show that the boundary operator
The modern statement of the Poincaré duality theorem is in terms of cohomology.
It states that if M is a closed oriented n-manifold, then there is a naturally defined isomorphism from Hk(M) to Hn − k(M). Specifically, one maps an element of Hk(M) to its cap product with a fundamental class of M, which will exist for oriented M.
There are also statements for manifolds with boundary; and taking into account the sheaf of local orientations, in the non-orientable case, one can give a statement that is independent of orientability. With the development of homology theory to include K-theory and other extraordinary theories from about 1955, it was realised that the homology H. could be replaced by other theories, once the products on manifolds were constructed; and there are now textbook treatments in generality.History
Dual cell structures
is the transpose of the boundary operator
Using the fact that the homology groups of a manifold are independent of the triangulation used to compute them, one can easily show that the k-th and (n − k)-th Betti numbers of M are equal.Modern formulations