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 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.

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.

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.

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", and the chain groups are related by isomorphisms

:$C\_k(Y)\; o\; Hom\_Z(C\_\{n-k\}(X),Z)\; .$

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

:$C\_k(X)\; o\; C\_\{k-1\}(X)$

is the transpose of the boundary operator

:$C\_\{n-k+1\}(Y)=Hom\_Z(C\_\{k-1\}(X),Z)\; o\; C\_\{n-k\}(Y)=Hom\_Z(C\_k(X),Z).,$

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".

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

:"D"_"M" : "H"^"k"("M") → "H"_{"n" − "k"}("M")

is natural in the following sense: if

:"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

:"D_N" = "f"_∗ "D_M" "f"^∗,

where "f"_∗ and "f"^∗ are the maps induced by "f" in homology and cohomology, respectively.

Bilinear pairings formulation

Assuming "M" is compact boundaryless and orientable,let $au\; H\_i\; M$ denote the torsion subgroup of $H\_i\; M$ and let $fH\_i\; M\; =\; H\_i\; M\; /\; au\; H\_i\; M$ be the free part. Then there are bilinear maps which are duality pairings

: $fH\_i\; M\; otimes\; fH\_\{n-i\}\; M\; o\; Bbb\; Z$

and

: $au\; H\_i\; M\; otimes\; au\; H\_\{n-i-1\}\; M\; o\; Bbb\; Q\; /\; Bbb\; Z.$

$Bbb\; Q\; /\; Bbb\; Z$ is the quotient of the rationals by the integers (taken as an additive quotient group).

The first form is typically called the "intersection product" and the 2nd the "torsion linking form". Assuming the manifold "M" is smooth, the intersection product is computed by perturbing the homology classes to be transverse and computing their oriented intersection number. For the torsion linking form, one computes the pairing of "x" and "y" by realizing "nx" as the boundary of some class "z". The form is the fraction with numerator the transverse intersection number of "z" with "y" and denominator "n".

The statement that the pairings are duality pairings means that the adjoint maps :$fH\_i\; M\; o\; Hom\_\{Bbb\; Z\}(fH\_\{n-i\}\; M,Bbb\; Z)$ and :$au\; H\_i\; M\; o\; Hom\_\{Bbb\; Z\}(\; au\; H\_\{n-i-1\}\; M,\; Bbb\; Q/Bbb\; Z)$are isomorphisms of groups.

Thus, Poincaré duality says that $fH\_i\; M$ and $fH\_\{n-i\}\; M$ are isomorphic, although there is no natural map giving the isomorphism, and similarly $au\; H\_i\; M$ and $au\; H\_\{n-i-1\}\; M$ are also isomorphic.

This approach to Poincaré duality was used by Przytycki and Yasuhara to give an elementary homotopy and diffeomorphism classification of 3-dimensional lens spaces. [Przytycki, Yasuhara. Symmetry of Links and Classification of Lens Spaces. Geom. Ded. Vol 98. No. 1. (2003)]

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.

"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 "H"_* could be replaced by other theories, once the products on manifolds were constructed; and there are now textbook treatments in generality.

Verdier duality is the appropriate generalization to (possibly singular) geometric objects, such as analytic spaces or schemes, while intersection homology was developed R. MacPherson and M. 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 and S-duality (homotopy theory).

References

Bibliography

* R.C. Blanchfield, Intersection theory of manifolds with operators with applications to knot theory, Annals of Math, 65 (1957), 340--356.

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