In mathematics, the Poincaré duality theorem is a basic result on the structure of the homology and cohomology groups 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, for all integers k. It further states that if mod 2 homology and cohomology is used, then the assumption of orientability can be dropped. Mathematics, often abbreviated maths in Commonwealth English and math in American English, is the study of abstraction. ... In mathematics (especially algebraic topology and abstract algebra), homology (in Greek homeos = identical and logos = word) is a certain general procedure to associate a sequence of abelian groups or modules to a given mathematical object (such as a topological space or a group). ... In mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a cochain complex. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... In mathematics, a manifold M is a type of space, characterized in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. ... In mathematics, a compact space is a space that resembles a closed and bounded subset of Euclidean space Rn in that it is small in a certain sense and contains all its limit points. The modern general definition calls a topological space compact if every open cover of it has... This article or section should be merged with Orientable manifold. ... In abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations. ...

History

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 (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. Henri Poincaré, photograph from the frontispiece of the 1913 edition of Last Thoughts Jules Henri Poincaré ( April 29, 1854 – July 17, 1912) was one of Frances greatest mathematicians, theoretical scientists and a philosopher of science. ... 1893 was a common year starting on Sunday (see link for calendar). ... In algebraic topology, the Betti numbers of a topological space X are a sequence b0, b1, ... of topological invariants. ... 1895 was a common year starting on Tuesday (see link for calendar). ... Poul Heegaard (November 2, 1871 — February 7, 1948) was a mathematician active in the field of topology. ...

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. Eduard Čech (June 29, 1893 - March 15, 1960) was a mathematician born in Stracov, Bohemia (then Austria-Hungary now Czech Republic). ... Hassler Whitney (23 March 1907 – 10 May 1989) was an American mathematician, who was one of the founders of singularity theory. ... In mathematics, specifically in algebraic topology, the cup product is a method of adjoining two cocycles of degree p and q to form a composite cocycle of degree p+q. ...

Dual cell structures

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. This article is about measurement by the use of triangles: for other usages of the term triangulation, see triangulation (disambiguation). ... In geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the others. ... simplex refers to a one-way communications channel. ... A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ... For other uses, see sphere (disambiguation). ... An octahedron (plural: octahedra) is a polyhedron with eight faces. ... An icosahedron [ˌaıkəsəhiːdrən] noun (plural: -drons, -dra [-drə]) is a polyhedron having 20 faces. ... In geometry, the barycentric subdivision is a standard way of dividing an arbitrary convex polygon into triangles, a convex polyhedron into tetrahedra, or, in general, a convex polytope into simplices with the same dimension, by connecting the barycenters of their faces in a specific way. ...

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 In homological algebra, a chain complex is a sequence of abelian groups or modules A0, A1, A2. ...

is the transpose of the boundary operator See transposition for meanings of this term in telecommunication and music. ...

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 formulation

The modern statement of the Poincaré duality theorem is in terms of homology and cohomology: if M is a closed oriented n-manifold, and k is an integer, then there is a canonically defined isomorphism from the k-th homology group H_k(M) to the (n−k)-th cohomology group H^{n − k}(M). (Here, homology and cohomology is taken with coefficients in the ring of integers, but the isomorphism holds for any coefficient ring.) Specifically, one maps an element of H^k(M) to its cap product with a fundamental class of M, which will exist for oriented M. In mathematics, the fundamental class is a homology class [M] associated to a manifold M. It is defined (firstly) in cases when M is a closed manifold of dimension n, and oriented. ...

Homology and cohomology groups are defined to be zero for negative degrees, so Poincaré duality in particular implies that the homology and cohomology groups of orientable closed n-manifolds are zero for degrees bigger than n.

Naturality

Note that H^k is a contravariant functor while H_{n − k} is covariant. The family of isomorphisms For functors in computer science, see the function object article. ... For functors in computer science, see the function object article. ...

D_M : H^k(M) → H_{n − k}(M)

is natural in the following sense: if In category theory, an abstract branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i. ...

f : M → N

is a continuous map between two oriented n-manifolds which is compatible with orientation, i.e. which maps the fundamental class of M to the fundamental class of N, then In mathematics, a continuous function is one in which arbitrarily small changes in the input produce arbitrarily small changes in the output. ...

D_N = f_* D_M f^*,

where f_* and f^* are the maps induced by f in homology and cohomology, respectively.

Generalizations and related results

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. In mathematics, a sheaf F on a given topological space X gives a set or richer structure F(U) for each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and gluing smaller open sets to obtain...

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. In mathematics, homology theory is the axiomatic study of the intuitive geometric idea of homology of cycles on topological spaces. ... The topic of K-theory spans the subjects of algebraic topology, abstract algebra and some areas of application like operator algebras and algebraic geometry. ...

There are many other forms of geometric duality in algebraic topology, including Lefschetz duality, Alexander duality and S-duality (homotopy theory). Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. ...