Poincaré Duality
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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the Poincaré duality theorem, named after
Henri Poincaré Jules Henri Poincaré (, ; ; 29 April 185417 July 1912) was a French mathematician, Theoretical physics, theoretical physicist, engineer, and philosophy of science, philosopher of science. He is often described as a polymath, and in mathemati ...
, is a basic result on the structure of the homology and
cohomology In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed ...
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic iden ...
s of
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...
s. It states that if ''M'' is an ''n''-dimensional oriented
closed manifold In mathematics, a closed manifold is a manifold Manifold with boundary, without boundary that is Compact space, compact. In comparison, an open manifold is a manifold without boundary that has only ''non-compact'' components. Examples The onl ...
(
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact, a type of agreement used by U.S. states * Blood compact, an ancient ritual of the Philippines * Compact government, a t ...
and without boundary), then the ''k''th cohomology group of ''M'' is
isomorphic In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
to the th homology group of ''M'', for all integers ''k'' : H^k(M) \cong H_(M). Poincaré duality holds for any coefficient
ring (The) Ring(s) may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell Arts, entertainment, and media Film and TV * ''The Ring'' (franchise), a ...
, 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 Poincaré Jules Henri Poincaré (, ; ; 29 April 185417 July 1912) was a French mathematician, Theoretical physics, theoretical physicist, engineer, and philosophy of science, philosopher of science. He is often described as a polymath, and in mathemati ...
in 1893. It was stated in terms of
Betti number In algebraic topology, the Betti numbers are used to distinguish topological spaces based on the connectivity of ''n''-dimensional simplicial complexes. For the most reasonable finite-dimensional spaces (such as compact manifolds, finite simplicia ...
s: The ''k''th and 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 In mathematics, intersection theory is one of the main branches of algebraic geometry, where it gives information about the intersection of two subvarieties of a given variety. The theory for varieties is older, with roots in Bézout's theorem o ...
, which he had invented. Criticism of his work by
Poul Heegaard Poul Heegaard (; November 2, 1871, Copenhagen - February 7, 1948, Oslo) 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 ...
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 Čech (; 29 June 1893 – 15 March 1960) was a Czech mathematician. His research interests included projective differential geometry and topology. He is especially known for the technique known as Stone–Čech compactification (in topo ...
and
Hassler Whitney Hassler Whitney (March 23, 1907 – May 10, 1989) was an American mathematician. He was one of the founders of singularity theory, and did foundational work in manifolds, embeddings, immersion (mathematics), immersions, characteristic classes and, ...
invented the
cup A cup is an open-top vessel (container) used to hold liquids for drinking, typically with a flattened hemispherical shape, and often with a capacity of about . Cups may be made of pottery (including porcelain), glass, metal, wood, stone, pol ...
and
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\leq p, to form a composite chain of degree p-q. It was introduced by Eduard Čech in 1936, and independently by Hassl ...
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, then there is a canonically defined isomorphism H^k(M, \Z) \to H_(M, \Z) for any integer ''k''. To define such an isomorphism, one chooses a fixed fundamental class 'M''of ''M'', which will exist if M is oriented. Then the isomorphism is defined by mapping an element \alpha \in H^k(M) to the
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\leq p, to form a composite chain of degree p-q. It was introduced by Eduard Čech in 1936, and independently by Hassl ...
frown \alpha. 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 : H^i(X) \stackrel H_^(X), or replace cohomology by cohomology with compact support : H^i_c(X) \stackrel H_(X).


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, generalizing the notion of dual polyhedra. Precisely, let T be a triangulation of an n-manifold M. Let S be a simplex of T. Let \Delta be a top-dimensional simplex of T containing S, so we can think of S as a subset of the vertices of \Delta. Define the dual cell DS corresponding to S so that \Delta \cap DS is the
convex hull In geometry, the convex hull, convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space, ...
in \Delta of the barycentres of all subsets of the vertices of \Delta that contain S. 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 C_i M \otimes C_ M \to \Z given by taking intersections induces an isomorphism C_i M \to C^ M, where C_i is the
cellular homology In mathematics, cellular homology in algebraic topology is a homology theory for the category of CW-complexes. It agrees with singular homology, and can provide an effective means of computing homology modules. Definition If X is a CW-complex ...
of the triangulation T, and C_ M and C^ M are the cellular homologies and cohomologies of the dual polyhedral/CW decomposition the manifold respectively. The fact that this is an isomorphism of
chain complex In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is contained in the kernel o ...
es 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 S \longmapsto DS.


Naturality

Note that H^k is a
contravariant functor In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and ...
while H_ is covariant. The family of isomorphisms : D_M\colon H^k(M) \to H_(M) is
natural Nature is an inherent character or constitution, particularly of the ecosphere or the universe as a whole. In this general sense nature refers to the laws, elements and phenomena of the physical world, including life. Although humans are part ...
in the following sense: if : f\colon M\to N is a
continuous map In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as '' discontinuities''. More preci ...
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_ \circ D_M \circ 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 the manifold ''M'' is compact, boundaryless, and orientable, let : \tau H_i M denote the torsion subgroup of H_i M and let : fH_i M = H_i M / \tau H_i M be the free part – all homology groups taken with integer coefficients in this section. Then there are
bilinear maps Bilinear may refer to: * Bilinear sampling (also called "bilinear filtering"), a method in computer graphics for choosing the color of a texture * Bilinear form, a type of mathematical function from a vector space to the underlying field * Bilinea ...
which are duality pairings (explained below). : fH_i M \otimes fH_ M \to \Z and : \tau H_i M \otimes \tau H_ M \to \Q/\Z. Here \Q/\Z 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 ''n''. The first form is typically called the ''
intersection product In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two Line (geometry), lines in a Plane (geometr ...
'' 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 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 : fH_i M \to \mathrm_(fH_ M,\Z) and : \tau H_i M \to \mathrm_(\tau H_ M, \Q/\Z) are isomorphisms of groups. This result is an application of Poincaré duality : H_i M \simeq H^ M, together with the
universal coefficient theorem In algebraic topology, universal coefficient theorems establish relationships between homology groups (or cohomology groups) with different coefficients. For instance, for every topological space , its ''integral homology groups'': :H_i(X,\Z) ...
, which gives an identification : fH^ M \equiv \mathrm(H_ M; \Z) and : \tau H^ M \equiv \mathrm(H_ M; \Z) \equiv \mathrm(\tau H_ M; \Q/\Z). Thus, Poincaré duality says that fH_i M and fH_ M are isomorphic, although there is no natural map giving the isomorphism, and similarly \tau H_i M and \tau H_ M are also isomorphic, though not naturally. ;Middle dimension 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. The resulting intersection form is a very important topological invariant. What is meant by "middle dimension" depends on parity. For even dimension , which is more common, this is literally the middle dimension ''k'', and there is a form on the free part of the middle homology: : fH_k M \otimes fH_k M \to \Z By contrast, for odd dimension , which is less commonly discussed, it is most simply the lower middle dimension ''k'', and there is a form on the torsion part of the homology in that dimension: : \tau H_k M \otimes \tau H_k M \to \Q/\Z. However, there is also a pairing between the free part of the homology in the lower middle dimension ''k'' and in the upper middle dimension : : fH_k M \otimes fH_ M \to \Z. The resulting groups, while not a single group with a bilinear form, are a simple chain complex and are studied in algebraic L-theory. ;Applications This approach to Poincaré duality was used by
Józef Przytycki Józef Henryk Przytycki (, ; born 14 October 1953 in Warsaw, Poland), is a Polish mathematician specializing in the fields of knot theory and topology. Academic background Przytycki received a Master of Science degree in mathematics from Unive ...
and Akira Yasuhara to give an elementary homotopy and diffeomorphism classification of 3-dimensional
lens space A lens space is an example of a topological space, considered in mathematics. The term often refers to a specific class of 3-manifolds, but in general can be defined for higher dimensions. In the 3-manifold case, a lens space can be visualized ...
s.


Application to Euler characteristics

An immediate result from Poincaré duality is that any closed odd-dimensional manifold ''M'' has
Euler characteristic In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space's ...
zero, which in turn gives that any manifold that bounds has even Euler characteristic.


Thom isomorphism formulation

Poincaré duality is closely related to the Thom isomorphism theorem. Let M be a compact, boundaryless oriented ''n''-manifold, and the product of ''M'' with itself. Let ''V'' be an open tubular neighbourhood of the diagonal in . Consider the maps: :* H_* M \otimes H_* M \to H_* (M \times M) the Homology cross product :* H_* (M \times M) \to H_* \left(M \times M, (M \times M) \setminus V\right) inclusion. :* H_* \left(M \times M, (M \times M) \setminus V\right) \to H_* (\nu M, \partial \nu M) excision map where \nu M is the normal disc bundle of the diagonal in M \times M. :* H_* (\nu M, \partial \nu M) \to H_ M the Thom isomorphism. This map is well-defined as there is a standard identification \nu M \equiv TM which is an oriented bundle, so the Thom isomorphism applies. Combined, this gives a map H_i M \otimes H_j M \to H_ M, which is the ''intersection product'', generalizing the intersection product discussed above. A similar argument with the Künneth theorem gives the ''torsion linking form''. This formulation of Poincaré duality has become popular as it defines Poincaré duality for any generalized homology theory, given a Künneth theorem and a Thom isomorphism for that homology theory. A Thom isomorphism theorem for a homology theory is now viewed as the generalized notion of
orientability In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "anticlockwise". A space is o ...
for that theory. For example, a spinC-structure on a manifold is a precise analog of an orientation within complex topological k-theory.


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 Sheaf may refer to: * Sheaf (agriculture), a bundle of harvested cereal stems * Sheaf (mathematics) In mathematics, a sheaf (: sheaves) is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open s ...
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 In topology, a covering or covering projection is a continuous function, map between topological spaces that, intuitively, Local property, locally acts like a Projection (mathematics), projection of multiple copies of a space onto itself. In par ...
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 In mathematics, the term homology, originally introduced in algebraic topology, has three primary, closely-related usages. The most direct usage of the term is to take the ''homology of a chain complex'', resulting in a sequence of abelian grou ...
to include
K-theory In mathematics, K-theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is a cohomology theory known as topological K-theory. In algebra and algebraic geometr ...
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 a generalized homology theory 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 Singular may refer to: * Singular, the grammatical number that denotes a unit quantity, as opposed to the plural and other forms * Singular or sounder, a group of boar, see List of animal names * Singular (band), a Thai jazz pop duo *'' Singula ...
) 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. There are many other forms of geometric duality in
algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...
, including Lefschetz duality, Alexander duality, Hodge duality, and
S-duality In theoretical physics, S-duality (short for strong–weak duality, or Sen duality) is an equivalence of two physical theories, which may be either quantum field theories or string theories. S-duality is useful for doing calculations in theore ...
. 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). These are used in
surgery theory In mathematics, specifically in geometric topology, surgery theory is a collection of techniques used to produce one finite-dimensional manifold from another in a 'controlled' way, introduced by . Milnor called this technique ''surgery'', while An ...
to algebraicize questions about manifolds. A Poincaré space is one whose singular chain complex is a Poincaré complex. These are not all manifolds, but their failure to be manifolds can be measured by
obstruction theory Obstruction may refer to: Places * Obstruction Island, in Washington state * Obstruction Islands, east of New Guinea Medicine * Obstructive jaundice * Obstructive sleep apnea * Airway obstruction, a respiratory problem ** Recurrent airway obstr ...
.


See also

* Bruhat decomposition * Fundamental class *
Weyl group In mathematics, in particular the theory of Lie algebras, the Weyl group (named after Hermann Weyl) of a root system Φ is a subgroup of the isometry group of that root system. Specifically, it is the subgroup which is generated by reflections t ...


References


Further reading

* *


External links


Intersection form
at the Manifold Atlas
Linking form
at the Manifold Atlas {{DEFAULTSORT:Poincare duality Homology theory Manifolds Duality theories Theorems in algebraic geometry