Poincare duality

In mathematics, the Poincaré duality theorem is a basic result in the structure of homology groups and cohomology of manifolds. Asserts that if M is an n-dimensional compact oriented manifold, then the kth cohomology group of M > is isomorphic to (n-k)-th homology group of M, for all integers k. It further states that if mod 2 homology and cohomology are used, then the assumption of orientability can be omitted.

History

A form of Poincaré duality was first stated, without proof, by Henri Poincaré in 1893. It was stated in terms of the Betti numbers: The kth and (n -k)-th Betti numbers of a closed orientable variety (ie compact and borderless) are equal. The concept of cohomology was at that time more than 40 years away from being clarified. In his 'document' of 1895, Situs Analysis, Poincaré tried to prove the theorem using the topological theory of intersection, which he had invented. Poul Heegaard's criticism of his work led him to realize that his proof was seriously incomplete. In the first two supplements to the Análisis Situs, Poincaré gave a new proof in terms of dual triangulations.

The Poincaré duality did not acquire its modern form until the advent of cohomology in the 1930s, when Eduard Cech and Hassler Whitney invented the products cup & cap (cape and cup) and formulated the duality of Poincaré in these new terms.

Dual Cell Structures

The classical Poincaré duality was thought of in terms of dual triangulations, which are generalizations of dual polyhedra. Given a triangulation X 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 Y is said to be the dual triangulation of X. Considering a tetrahedron as a triangulation of the 2-sphere, the dual triangulation of the tetrahedron is another tetrahedron. This construction does not necessarily give another triangulation, as the examples of the octahedron and the icosahedron show. Poincaré used a (not entirely correct) method involving barycentric subdivision to show that we can always obtain dual triangulation for compact oriented manifolds.

In more exact terms, one can describe the dual of a triangulation X as a triangulation Y such that given a k-simplex α in X, there is a (n-k)-simplex in Y whose intersection number with α is 1, and such that the number of intersection of α with any other (n -k)-simplex of Y is 0.

The edge operator on a complex string can be viewed as an array. Let M be a closed n-manifold, X a triangulation of M, and Y the dual triangulation of X. Then it can be shown that the edge operator

Cp(X)→ → Cp− − 1(X){displaystyle C_{p}(X)to C_{p-1}(X)}

is the transpose of the border operator

Cn− − p+1(And)→ → Cn− − p(And){displaystyle C_{n-p+1}(Y)to C_{n-p}(Y)}

Using the fact that the homology groups of a manifold are independent of the triangulation that computes them, it can easily be shown that the kth y (n -k)-th Betti numbers of M are equal.

Modern formulation

The modern presentation of the Poincaré duality theorem is in terms of homology and cohomology: if M is an oriented closed n-manifold, and >k is an integer, then there is a definite canonical isomorphism of the kth homology group H_k (M) to the (n-k)-th cohomology group H ^n-k(M). (Here, homology and cohomology are taken with coefficients in the ring of integers, but isomorphism holds for any ring of coefficients.) Specifically, an element of H ^{is mapped k}(M) to its product cap with a fundamental class of M, which will exist for M > 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 closed orientable n-manifolds - are zero for degrees greater than n.

Naturality

Note that H^k is a contravariant functor while the H_{n -k} is covariant. The family of isomorphisms

D_M: H^k (M) → H_n-k(M)

is natural in the following sense: if

f: M → N

is a continuous function between two n-oriented manifolds that is orientation-compatible, that is, it maps the fundamental class of M to the fundamental class of N, then

D_N = f_♪ D_M f^♪,

where f _* and f^* are the functions induced by f in homology and cohomology, respectively.

Generalizations and Related Results

The Poincaré-Lefschetz duality theorem is a generalization for manifolds with edge. In the non-steerable case, considering the beam of local orientations, a presentation can be given that is independent of steerability.

With the development of homology theory to include K-theory and other extraordinary theories from 1955, it was observed that H_{* could be replaced by other theories, once the array products were built; and now there are textbook treatments with all generality.}

There are many other forms of geometric duality in algebraic topology, including Lefschetz duality, Alexander duality, and S-duality (homotope theory).