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In [[mathematics]], the '''Poincaré duality''' theorem, named after [[Henri Poincaré]], is a basic result on the structure of the [[homology (mathematics)|homology]] and [[cohomology]] [[group (mathematics)|group]]s of [[manifold]]s. It states that if ''M'' is an ''n''-dimensional [[Orientability|oriented]] [[closed manifold]] ([[Compact space|compact]] and without boundary), then the ''k''th cohomology group of ''M'' is [[Group isomorphism|isomorphic]] to the (''n'' − ''k'')th homology group of ''M'', for all integers ''k''
:<math>H^k(M) \cong H_{n-k}(M).</math>

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 Poincaré]] in 1893. It was stated in terms of [[Betti number]]s: 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 (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 product|cup]] and [[cap product]]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 cohomology group ''H''''k''(''M'') to the (''n'' − ''k'')th homology 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''.

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''.

== 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 polyhedron|dual polyhedra]].

[[File:Dually007.png|thumb|350px|right|<math>\cup_{S \in T} \Delta \cap DS</math> -- a picture of the parts of the dual-cells in a top-dimensional simplex.]]

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 <math>\Delta</math> be a top-dimensional simplex of ''T'' containing ''S''. So we can think of ''S'' as a subset of the vertices of <math>\Delta</math>. Then <math>\Delta \cap DS</math> is defined to be the convex hull (in <math>\Delta</math>) of the barycentres of all subsets of the vertices of <math>\Delta</math> that contain <math>S</math>. 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 <math>C_i M \otimes C^{n-i} M \to \mathbb Z</math> given by taking intersections induces an isomorphism <math>C_i M \to C^{n-i} M</math>, where here <math>C_i</math> is the cellular homology of the triangulation ''T'', and <math>C_{n-i} M</math> and <math>C^{n-i} M</math> 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]]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 <math>S \longmapsto DS</math>.

=== Naturality ===
Note that ''H''''k'' is a [[contravariant functor]] while ''H''''n'' − ''k'' is [[covariant functor|covariant]]. The family of isomorphisms

:''D''''M'' : ''H''''k''(''M'') → ''H''''n'' − ''k''(''M'')

is [[natural transformation|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

:''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 manifold|orientable]],　let

:<math>\tau H_i M</math>

denote the [[torsion (algebra)|torsion]] subgroup of <math>H_i M</math> and let

:<math>fH_i M = H_i M / \tau H_i M</math>

be the [[free group|free]] part – all homology groups taken with integer coefficients in this section. Then there are [[bilinear operator|bilinear maps]] which are [[duality pairing]]s (explained below).

:<math>fH_i M \otimes fH_{n-i} M \to \Bbb Z</math>

and

:<math>\tau H_i M \otimes \tau H_{n-i-1} M \to \Bbb Q / \Bbb Z.</math>
:(Here <math>\Bbb Q / \Bbb Z</math> 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 <math>n-1,</math> rather than to <math>n.</math>)

The first form is typically called the ''[[intersection theory|intersection product]]'' and the 2nd the ''torsion linking form''.{{anchor|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
:<math>fH_i M \to \mathrm{Hom}_{\Bbb Z}(fH_{n-i} M,\Bbb Z)</math>
and
:<math>\tau H_i M \to \mathrm{Hom}_{\Bbb Z}(\tau H_{n-i-1} M, \Bbb Q/\Bbb Z)</math>
are isomorphisms of groups.

This result is an application of Poincaré Duality

:<math>H_i M \simeq H^{n-i} M</math>

together with the [[Universal coefficient theorem]] which gives an identification
:<math>fH^{n-i} M \equiv \mathrm{Hom}(H_{n-i} M; \mathbb Z)</math>
and
:<math>\tau H^{n-i} M \equiv \mathrm{Ext}(H_{n-i-1} M; \mathbb Z) \equiv \mathrm{Hom}(\tau H_{n-i-1} M; \mathbb Q/\mathbb Z)</math>.
Thus, Poincaré duality says that <math>fH_i M</math> and <math>fH_{n-i} M</math> are isomorphic, although there is no natural map giving the isomorphism, and similarly <math>\tau H_i M</math> and <math>\tau H_{n-i-1} M</math> 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|bilinear ''form'']] on a single homology group. The resulting [[Intersection theory#Topological intersection form|intersection form]] is a very important topological invariant.

What is meant by "middle dimension" depends on parity. For even dimension <math>n = 2k,</math> which is more common, this is literally the middle dimension ''k,'' and there is a form on the free part of the middle homology:
:<math>fH_k M \otimes fH_k M \to \Bbb Z</math>

By contrast, for odd dimension <math>n = 2k+1,</math> 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:
:<math>\tau H_k M \otimes \tau H_k M \to \Bbb Q / \Bbb Z.</math>
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 ''k''+1:
:<math>fH_k M \otimes fH_{k+1} M \to \Bbb Z.</math>
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 Przytycki and Yasuhara to give an elementary homotopy and diffeomorphism classification of 3-dimensional [[lens space]]s.<ref>
{{citation|last1=Przytycki|authorlink=Józef H. Przytycki|last2=Yasuhara|title=Symmetry of Links and Classification of Lens Spaces|journal=Geom. Ded.|volume=98|year=2003|issue=1}}</ref>

== Thom Isomorphism Formulation ==

Poincaré Duality is closely related to the [[Thom space|Thom Isomorphism Theorem]], as we will explain here. For this exposition, let <math>M</math> be a compact, boundaryless oriented n-manifold. Let <math>M \times M</math> be the product of <math>M</math> with itself, let <math>V</math> be an open tubular neighbourhood of the diagonal in <math>M \times M</math>. Consider the maps:

:* <math> H_* M \otimes H_* M \to H_* (M \times M)</math> the [[Künneth theorem|Homology Cross Product]]

:* <math>H_* (M \times M) \to H_* \left(M \times M, (M \times M) \setminus V\right)</math> inclusion.

:* <math>H_* \left(M \times M, (M \times M) \setminus V\right) \to H_* (\nu M, \partial \nu M) </math> [[excision theorem|excision map]] where <math>\nu M</math> is the [[normal bundle|normal disc bundle]] of the diagonal in <math>M \times M</math>.

:* <math>H_* (\nu M, \partial \nu M) \to H_{*-n} M</math> the [[Thom space|Thom Isomorphism]]. This map is well-defined as there is a standard identification <math>\nu M \equiv TM</math> which is an oriented bundle, so the Thom Isomorphism applies.

Combined, this gives a map <math>H_i M \otimes H_j M \to H_{i+j-n} M</math>, 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<ref>{{citation |last=Rudyak |first=Y. |title=On Thom Spectra, Orientablility and Cobordism |location= |publisher=Springer SMM |year=1998 |isbn=3-540-62043-5 }}</ref> as it provides a means to define Poincaré Duality for any [[homology theory|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 [[spin structure|<math>spin^c</math>-structure]] on a manifold turns out to be precisely what is needed to be orientable in the sense of [[k-theory|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 (mathematics)|sheaf]] 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 polynomial|Alexander module]] and can be used to define the [[signature of a knot|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 [[homology theory|generalized homology theories]] which requires a notion of orientation with respect to a homology theory, and is formulated in terms of a generalized [[Thom space|Thom Isomorphism Theorem]]. The [[Thom space|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 [[singularity theory|singular]]) geometric objects, such as [[analytic space]]s or [[scheme (mathematics)|schemes]], while [[intersection homology]] was developed [[Robert MacPherson (mathematician)|R. MacPherson]] and [[M. Goresky]] for [[stratified space]]s, 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 (homotopy theory)|S-duality]].

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]] 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]].

== See also ==
* [[Bruhat decomposition]]
* [[Fundamental class]]
* [[Weyl group]]

==References==
{{Reflist}}

==Further reading==
*{{citation |first=R. C. |last=Blanchfield |title=Intersection theory of manifolds with operators with applications to knot theory |journal=[[Annals of Mathematics]] |volume=65 |issue=2 |year=1957 |pages=340–356 |jstor=1969966 }}
* {{Citation | last1=Griffiths | first1=Phillip | author1-link= Phillip Griffiths | last2=Harris | first2=Joseph | author2-link= Joe Harris (mathematician) | title=Principles of algebraic geometry | publisher=Wiley | location=New York | series=Wiley Classics Library | isbn=978-0-471-05059-9 | mr=1288523 | year=1994}}
== External links ==
*[http://www.map.mpim-bonn.mpg.de/Intersection_form Intersection form] at the Manifold Atlas
*[http://www.map.mpim-bonn.mpg.de/Linking_form Linking form] at the Manifold Atlas
{{DEFAULTSORT:Poincare duality}}
[[Category:Homology theory]]
[[Category:Manifolds]]
[[Category:Duality theories]]

Classical general equilibrium model - Revision history

en>Rsrikanth05: Reverted 1 good faith edit by 110.37.93.148 using STiki

en>Duoduoduo: /* Output function */ fix incorrect wording