Poincaré duality
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. 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
Poincaré duality holds for any coefficient ring, so long as one has taken an orientation with respect to that coefficient ring; in particular, since every manifold has a unique orientation mod 2, Poincaré duality holds mod 2 without any assumption of orientation.
History[edit]
A form of Poincaré duality was first stated, without proof, by Henri Poincaré in 1893. It was stated in terms of Betti numbers: The kth and (n − k)th Betti numbers of a closed (i.e., compact and without boundary) orientable 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 1930s, when Eduard Čech and Hassler Whitney invented the cup and cap products and formulated Poincaré duality in these new terms.
Application to Euler Characteristics[edit]
An immediate result from Poincaré duality is that any closed odd-dimensional manifold M has Euler characteristic zero, which in turn gives that any manifold that bounds has even Euler characteristic.
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The Poincaré–Lefschetz duality theorem is a generalisation for manifolds with boundary. In the non-orientable case, taking into account the sheaf of local orientations, one can give a statement that is independent of orientability: see twisted Poincaré duality.
Blanchfield duality is a version of Poincaré duality which provides an isomorphism between the homology of an abelian covering space of a manifold and the corresponding cohomology with compact supports. It is used to get basic structural results about the Alexander module and can be used to define the signatures of a knot.
With the development of homology theory to include K-theory and other extraordinary theories from about 1955, it was realised that the homology could be replaced by other theories, once the products on manifolds were constructed; and there are now textbook treatments in generality. More specifically, there is a general Poincaré duality theorem for a generalized homology theory which requires a notion of orientation with respect to a homology theory, and is formulated in terms of a generalized Thom isomorphism theorem. The Thom isomorphism theorem in this regard can be considered as the germinal idea for Poincaré duality for generalized homology theories.
Verdier duality is the appropriate generalization to (possibly singular) geometric objects, such as analytic spaces or schemes, while intersection homology was developed by Robert MacPherson and Mark Goresky for stratified spaces, such as real or complex algebraic varieties, precisely so as to generalise Poincaré duality to such stratified spaces.
There are many other forms of geometric duality in algebraic topology, including Lefschetz duality, Alexander duality, Hodge duality, and S-duality.
More algebraically, one can abstract the notion of a Poincaré complex, which is an algebraic object that behaves like the singular chain complex of a manifold, notably satisfying Poincaré duality on its homology groups, with respect to a distinguished element (corresponding to the fundamental class). These are used in surgery theory to algebraicize questions about manifolds. A Poincaré space is one whose singular chain complex is a Poincaré complex. These are not all manifolds, but their failure to be manifolds can be measured by obstruction theory.
