Poincaré duality
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.
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.
The modern statement of the Poincaré duality theorem is in terms of homology and cohomology: if M is a closed oriented n-manifold, then there is a canonically defined isomorphism
for any integer k. To define such an isomorphism, one chooses a fixed fundamental class [M] of M, which will exist if
[1] 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. Here, homology and cohomology are integral, but the isomorphism remains valid over any coefficient ring.
In the case where an oriented manifold is not compact, one has to replace homology by Borel–Moore homology or replace cohomology by cohomology with compact support Given a triangulated manifold, there is a corresponding dual polyhedral decomposition.
are the cellular homologies and cohomologies of the dual polyhedral/CW decomposition the manifold respectively.
The fact that this is an isomorphism of chain complexes is a proof of Poincaré duality.
Roughly speaking, this amounts to the fact that the boundary relation for the triangulation
The family of isomorphisms is natural in the following sense: if 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 where
Assuming the manifold M is compact, boundaryless, and orientable, let denote the torsion subgroup of
and let be the free part – all homology groups taken with integer coefficients in this section.
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 then takes the value equal to the fraction whose numerator is the transverse intersection number of z with y, and whose denominator is n. The statement that the pairings are duality pairings means that the adjoint maps and are isomorphisms of groups.
While for most dimensions, Poincaré duality induces a bilinear pairing between different homology groups, in the middle dimension it induces a bilinear form on a single homology group.
This approach to Poincaré duality was used by Józef Przytycki and Akira Yasuhara to give an elementary homotopy and diffeomorphism classification of 3-dimensional lens spaces.
[2] 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.
Poincaré duality is closely related to the Thom isomorphism theorem.
A similar argument with the Künneth theorem gives the torsion linking form.
This formulation of Poincaré duality has become popular[3] as it defines Poincaré duality for any generalized homology theory, given a Künneth theorem and a Thom isomorphism for that homology theory.
For example, a spinC-structure on a manifold is a precise analog of an orientation within complex topological k-theory.
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.
could be replaced by other theories, once the products on manifolds were constructed; and there are now textbook treatments in generality.
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.
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).
