Poincaré duality
Encyclopedia
In mathematics
, the Poincaré duality theorem named after Henri Poincaré
, is a basic result on the structure of the homology
and cohomology
group
s of manifold
s. 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 1893. It was stated in terms of Betti number
s: 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 product
s and formulated Poincaré duality in these new terms.
of M, which will exist for oriented M.
For non-compact oriented manifolds, one has to replace cohomology by cohomology with compact support
.
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.
.
Precisely, let T be a triangulation of an n-manifold M. Let S be a simplex of T. We denote the dual cell (to be defined precisely) corresponding to S by DS. Let
be a top-dimensional simplex of T containing S. So we can think of S as a subset of the vertices of 
. Then 
is defined to be the convex hull (in 
) of the barycentres of all subsets of the vertices of 
that contain 
. One can check that if S is i-dimensional, then DS is an n-i-dimensional cell. Moreover, the dual cells to T form a CW-decomposition of M, and the only n-i-dimensional dual cell that intersects an i-cell S is DS. Thus the pairing 
given by taking intersections induces an isomorphism 
, where here 
is the cellular homology of the triangulation T, and 
and 
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 T is the incidence relation for the dual polyhedral decomposition under the correspondence 
.
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 f∗ and f∗ are the maps induced by f in homology and cohomology, respectively.
Note the very strong and crucial hypothesis that f maps the fundamental class of M to the fundamental class of N. Naturality does not hold for an arbitrary continuous map f, since in general f∗ is not an injection on cohomology. For example if f is a covering map then it maps the fundamental class of M to a multiple of the fundamental class of N. This multiple is the degree of the map f.
let
denote the torsion subgroup of 
and let 
be the free
part – all homology groups taken with integer coefficients in this section. Then there are bilinear maps
which are duality pairings (explained below).
and
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
and
are isomorphisms of groups.
This result is an application of Poincaré Duality
together with the Universal coefficient theorem
which gives an identification
and 
. Thus, Poincaré duality says that 
and 
are isomorphic, although there is no natural map giving the isomorphism, and similarly 
and 
are also isomorphic, though not naturally.
This approach to Poincaré duality was used by Przytycki and Yasuhara to give an elementary homotopy and diffeomorphism classification of 3-dimensional lens spaces.
, as we will explain here. For this exposition, let
be a compact, boundaryless oriented n-manifold. Let 
be the product of 
with itself, let 
be an open tubular neighbourhood of the diagonal in 
. Consider the maps:
Combined, this gives a map
, which is the intersection product -- strictly speaking it is a generalization of the intersection product above, but it is also called the intersection product. A similar argument with the Künneth theorem
gives the torsion linking form.
This formulation of Poincaré Duality has become quite popular as it provides a means to define Poincaré Duality for any generalized homology theories
provided one has a Thom Isomorphism for that homology theory. A Thom isomorphism theorem for a homology theory is now accepted as the generalized notion of orientability
for a homology theory. For example, a
-structure
on a manifold turns out to be precisely what is needed to be orientable in the sense of complex topological k-theory
.
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 H* 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 generalized homology theories
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 space
s 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
, Hodge duality, and S-duality.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Poincaré duality theorem named after Henri Poincaré
Henri Poincaré
Jules Henri Poincaré was a French mathematician, theoretical physicist, engineer, and a philosopher of science...
, is a basic result on the structure of the homology
Homology (mathematics)
In mathematics , homology is a certain general procedure to associate a sequence of abelian groups or modules with a given mathematical object such as a topological space or a group...
and cohomology
Cohomology
In mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a co-chain complex. That is, cohomology is defined as the abstract study of cochains, cocycles, and coboundaries...
group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
s of manifold
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
s. It states that if M is an n-dimensional oriented
Orientability
In mathematics, orientability is a property of surfaces in Euclidean space measuring whether or not it is possible to make a consistent choice of surface normal vector at every point. A choice of surface normal allows one to use the right-hand rule to define a "clockwise" direction of loops in the...
closed manifold
Closed manifold
In mathematics, a closed manifold is a type of topological space, namely a compact manifold without boundary. In contexts where no boundary is possible, any compact manifold is a closed manifold....
(compact
Compact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
and without boundary), then the kth cohomology group of M is isomorphic
Group isomorphism
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. If there exists an isomorphism between two groups, then the groups are called 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
A form of Poincaré duality was first stated, without proof, by Henri PoincaréHenri Poincaré
Jules Henri Poincaré was a French mathematician, theoretical physicist, engineer, and a philosopher of science...
in 1893. It was stated in terms of Betti number
Betti number
In algebraic topology, a mathematical discipline, the Betti numbers can be used to distinguish topological spaces. Intuitively, the first Betti number of a space counts the maximum number of cuts that can be made without dividing the space into two pieces....
s: 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
Intersection theory
In mathematics, intersection theory is a branch of algebraic geometry, where subvarieties are intersected on an algebraic variety, and of algebraic topology, where intersections are computed within the cohomology ring. The theory for varieties is older, with roots in Bézout's theorem on curves and...
, which he had invented. Criticism of his work by Poul Heegaard
Poul Heegaard
Poul Heegaard was a Danish mathematician active in the field of topology. His 1898 thesis introduced a concept now called the Heegaard splitting of a 3-manifold. Heegaard's ideas allowed him to make a careful critique of work of Henri Poincaré...
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
Eduard Cech
Eduard Čech was a Czech mathematician born in Stračov, Bohemia . His research interests included projective differential geometry and topology. In 1921–1922 he collaborated with Guido Fubini in Turin...
and Hassler Whitney
Hassler Whitney
Hassler Whitney was an American mathematician. He was one of the founders of singularity theory, and did foundational work in manifolds, embeddings, immersions, and characteristic classes.-Work:...
invented the cup
Cup product
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. This defines an associative graded commutative product operation in cohomology, turning the cohomology of a space X into a...
and cap product
Cap product
In algebraic topology the cap product is a method of adjoining a chain of degree p with a cochain of degree q, such that q ≤ p, to form a composite chain of degree p − q. It was introduced by Eduard Čech in 1936, and independently by Hassler Whitney in 1938.-Definition:Let X be a topological...
s and formulated Poincaré duality in these new terms.
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 Hk(M) to the (n − k)th cohomology group Hn − 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 Hk(M) to its cap product with a fundamental classFundamental class
In mathematics, the fundamental class is a homology class [M] associated to an oriented manifold M, which corresponds to "the whole manifold", and pairing with which corresponds to "integrating over the manifold"...
of M, which will exist for oriented M.
For non-compact oriented manifolds, one has to replace cohomology by cohomology with compact support
Cohomology with compact support
In mathematics, cohomology with compact support refers to certain cohomology theories, usually with some condition requiring that cocycles should have compact support.-de Rham cohomology with compact support for smooth manifolds:...
.
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.
Dual cell structures
Given a triangulated manifold, there is a corresponding dual polyhedral decomposition. The dual polyhedral decomposition is a cell decomposition of the manifold such that the k-cells of the dual polyhedral decomposition are in bijective correspondence with the n-k-cells of the triangulation, generalising the notion of dual polyhedraDual polyhedron
In geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other. The dual of the dual is the original polyhedron. The dual of a polyhedron with equivalent vertices is one with equivalent faces, and of one with equivalent edges is another...
.
Precisely, let T be a triangulation of an n-manifold M. Let S be a simplex of T. We denote the dual cell (to be defined precisely) corresponding to S by DS. Let
be a top-dimensional simplex of T containing S. So we can think of S as a subset of the vertices of . Then is defined to be the convex hull (in ) of the barycentres of all subsets of the vertices of that contain . One can check that if S is i-dimensional, then DS is an n-i-dimensional cell. Moreover, the dual cells to T form a CW-decomposition of M, and the only n-i-dimensional dual cell that intersects an i-cell S is DS. Thus the pairing given by taking intersections induces an isomorphism , where here is the cellular homology of the triangulation T, and and 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 T is the incidence relation for the dual polyhedral decomposition under the correspondence .Naturality
Note that Hk is a contravariant functor while Hn − k is covariant. The family of isomorphisms- DM : Hk(M) → Hn − k(M)
is natural
Natural transformation
In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure of the categories involved. Hence, a natural transformation can be considered to be a "morphism of functors". Indeed this intuition...
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
- DN = f∗ DM f∗,
where f∗ and f∗ are the maps induced by f in homology and cohomology, respectively.
Note the very strong and crucial hypothesis that f maps the fundamental class of M to the fundamental class of N. Naturality does not hold for an arbitrary continuous map f, since in general f∗ is not an injection on cohomology. For example if f is a covering map then it maps the fundamental class of M to a multiple of the fundamental class of N. This multiple is the degree of the map f.
Bilinear pairings formulation
Assuming M is compact boundaryless and orientable,let
denote the torsion subgroup of and let be the freeFree group
In mathematics, a group G is called free if there is a subset S of G such that any element of G can be written in one and only one way as a product of finitely many elements of S and their inverses...
part – all homology groups taken with integer coefficients in this section. Then there are bilinear maps
Bilinear operator
In mathematics, a bilinear operator is a function combining elements of two vector spaces to yield an element of a third vector space that is linear in each of its arguments. Matrix multiplication is an example.-Definition:...
which are duality pairings (explained below).
and
- (Here
is the quotient of the rationals by the integers, taken as an additive group.) - (Notice that in the torsion linking form, there is a −1 in the dimension, so the paired dimensions add up to
rather than to 
)
The first form is typically called the intersection product
Intersection theory
In mathematics, intersection theory is a branch of algebraic geometry, where subvarieties are intersected on an algebraic variety, and of algebraic topology, where intersections are computed within the cohomology ring. The theory for varieties is older, with roots in Bézout's theorem on curves and...
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
and
are isomorphisms of groups.
This result is an application of Poincaré Duality
together with the Universal coefficient theoremUniversal coefficient theorem
In mathematics, the universal coefficient theorem in algebraic topology establishes the relationship in homology theory between the integral homology of a topological space X, and its homology with coefficients in any abelian group A...
which gives an identification
and . Thus, Poincaré duality says that and are isomorphic, although there is no natural map giving the isomorphism, and similarly and are also isomorphic, though not naturally.This approach to Poincaré duality was used by Przytycki and Yasuhara to give an elementary homotopy and diffeomorphism classification of 3-dimensional lens spaces.
Thom Isomorphism Formulation
Poincaré Duality is closely related to the Thom Isomorphism TheoremThom space
In mathematics, the Thom space, Thom complex, or Pontryagin-Thom construction of algebraic topology and differential topology is a topological space associated to a vector bundle, over any paracompact space....
, as we will explain here. For this exposition, let
be a compact, boundaryless oriented n-manifold. Let be the product of with itself, let be an open tubular neighbourhood of the diagonal in . Consider the maps:-
the Homology Cross ProductKünneth theoremIn mathematics, especially in homological algebra and algebraic topology, a Künneth theorem is a statement relating the homology of two objects to the homology of their product. The classical statement of the Künneth theorem relates the singular homology of two topological spaces X and Y and their...
-
inclusion.
-
excision mapExcision theoremIn algebraic topology, a branch of mathematics, the excision theorem is a useful theorem about relative homology—given topological spaces X and subspaces A and U such that U is also a subspace of A, the theorem says that under certain circumstances, we can cut out U from both spaces such...
where
is the normal disc bundleNormal bundleIn differential geometry, a field of mathematics, a normal bundle is a particular kind of vector bundle, complementary to the tangent bundle, and coming from an embedding .-Riemannian manifold:...
of the diagonal in
.
-
the Thom IsomorphismThom spaceIn mathematics, the Thom space, Thom complex, or Pontryagin-Thom construction of algebraic topology and differential topology is a topological space associated to a vector bundle, over any paracompact space....
. This map is well-defined as there is a standard identification
which is an oriented bundle, so the Thom Isomorphism applies.
Combined, this gives a map
, which is the intersection product -- strictly speaking it is a generalization of the intersection product above, but it is also called the intersection product. A similar argument with the Künneth theoremKünneth theorem
In mathematics, especially in homological algebra and algebraic topology, a Künneth theorem is a statement relating the homology of two objects to the homology of their product. The classical statement of the Künneth theorem relates the singular homology of two topological spaces X and Y and their...
gives the torsion linking form.
This formulation of Poincaré Duality has become quite popular as it provides a means to define Poincaré Duality for any generalized homology theories
Homology theory
In mathematics, homology theory is the axiomatic study of the intuitive geometric idea of homology of cycles on topological spaces. It can be broadly defined as the study of homology theories on topological spaces.-The general idea:...
provided one has a Thom Isomorphism for that homology theory. A Thom isomorphism theorem for a homology theory is now accepted as the generalized notion of orientability
Orientability
In mathematics, orientability is a property of surfaces in Euclidean space measuring whether or not it is possible to make a consistent choice of surface normal vector at every point. A choice of surface normal allows one to use the right-hand rule to define a "clockwise" direction of loops in the...
for a homology theory. For example, a
-structureSpin structure
In differential geometry, a spin structure on an orientable Riemannian manifold \,allows one to define associated spinor bundles, giving rise to the notion of a spinor in differential geometry....
on a manifold turns out to be precisely what is needed to be orientable in the sense of complex topological k-theory
K-theory
In mathematics, K-theory originated as the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is an extraordinary cohomology theory known as topological K-theory. In algebra and algebraic geometry, it is referred to as algebraic K-theory. It...
.
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 sheafSheaf (mathematics)
In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space. The data can be restricted to smaller open sets, and the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of...
of local orientations, one can give a statement that is independent of orientability: see Twisted Poincaré duality
Twisted Poincaré duality
In mathematics, the twisted Poincaré duality is a theorem removing the restriction on Poincaré duality to oriented manifolds. The existence of a global orientation is replaced by carrying along local information, by means of a local coefficient system....
.
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
Alexander polynomial
In mathematics, the Alexander polynomial is a knot invariant which assigns a polynomial with integer coefficients to each knot type. James Waddell Alexander II discovered this, the first knot polynomial, in 1923...
and can be used to define the signatures of a knot
Signature of a knot
The signature of a knot is a topological invariant in knot theory. It may be computed from the Seifert surface.Given a knot K in the 3-sphere, it has a Seifert surface S whose boundary is K...
.
With the development of homology theory
Homology theory
In mathematics, homology theory is the axiomatic study of the intuitive geometric idea of homology of cycles on topological spaces. It can be broadly defined as the study of homology theories on topological spaces.-The general idea:...
to include K-theory
K-theory
In mathematics, K-theory originated as the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is an extraordinary cohomology theory known as topological K-theory. In algebra and algebraic geometry, it is referred to as algebraic K-theory. It...
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. More specifically, there is a general Poincaré duality theorem for generalized homology theories
Homology theory
In mathematics, homology theory is the axiomatic study of the intuitive geometric idea of homology of cycles on topological spaces. It can be broadly defined as the study of homology theories on topological spaces.-The general idea:...
which requires a notion of orientation with respect to a homology theory, and is formulated in terms of a generalized Thom Isomorphism Theorem
Thom space
In mathematics, the Thom space, Thom complex, or Pontryagin-Thom construction of algebraic topology and differential topology is a topological space associated to a vector bundle, over any paracompact space....
. The Thom Isomorphism Theorem
Thom space
In mathematics, the Thom space, Thom complex, or Pontryagin-Thom construction of algebraic topology and differential topology is a topological space associated to a vector bundle, over any paracompact space....
in this regard can be considered as the germinal idea for Poincaré duality for generalized homology theories.
Verdier duality
Verdier duality
In mathematics, Verdier duality is a generalization of the Poincaré duality of manifolds to locally compact spaces with singularities. Verdier duality was introduced by , as an analog for locally compact spaces of the coherent duality for schemes due to Grothendieck...
is the appropriate generalization to (possibly singular
Singularity theory
-The notion of singularity:In mathematics, singularity theory is the study of the failure of manifold structure. A loop of string can serve as an example of a one-dimensional manifold, if one neglects its width. What is meant by a singularity can be seen by dropping it on the floor...
) geometric objects, such as analytic space
Analytic space
An analytic space in the theory of functions of several complex variables is a generalization of the concept of an analytic manifold: heuristically, an analytic space is a set which is locally isomorphic to an analytic variety, while an analytic manifold is locally isomorphic to a complex euclidean...
s or schemes
Scheme (mathematics)
In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. Schemes were introduced by Alexander Grothendieck so as to broaden the notion of algebraic variety; some consider schemes to be the basic object of study of modern...
, 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
Algebraic topology
Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.Although algebraic topology...
, including Lefschetz duality
Lefschetz duality
In mathematics, Lefschetz duality is a version of Poincaré duality in geometric topology, applying to a manifold with boundary. Such a formulation was introduced by Solomon Lefschetz in the 1920s, at the same time introducing relative homology, for application to the Lefschetz fixed-point theorem...
, Alexander duality
Alexander duality
In mathematics, Alexander duality refers to a duality theory presaged by a result of 1915 by J. W. Alexander, and subsequently further developed, particularly by P. S. Alexandrov and Lev Pontryagin...
, Hodge duality, and S-duality.