在數學中，以亨利·龐加萊命名的龐加萊對偶定理，是關於流形的同調和上同調群的結構的一個基本結果。它表明，如果M是一個n維方向的閉廖(緊且無邊界)，那麼對於所有的整數k, M的第k個上同調群同構於M的(n-k)個上同調群
In mathematics, the Poincaré dualitytheorem, 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 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
Hk≌Hn-k(M).
龐加萊對偶對任何係數環都成立，只要取了這個係數環的方向; 特別地，由於每個流形都有一個唯一的方向模2，龐加萊二元性在沒有任何方向假設的情況下保持模2。
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.
歷史
History
龐加萊二象性的一種形式是由亨利·龐加萊在1893年首次提出的，沒有證據。它是用Betti數來表示的:a閉的第k個Betti數和(n-k)個Betti數。可定向n流形是等價的。上同調概念在當時距離被澄清大約40年。1895年發表的論文《西圖斯分析》(Analysis Situs)中，龐加萊試圖用他發明的拓撲交理論來證明這個定理。保羅·希加德對他的工作的批評使他意識到他的證明有嚴重的缺陷。在Situs分析的前兩個補充中，龐加萊給出了雙重三角形的新證明。
A form of Poincaré duality was first stated, without proof, by Henri Poincaré in 1893. It was stated in terms of Betti numbers: 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 Heegaardled 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.
龐加萊二元性直到20世紀30年代上同調出現後才有了現代的形式，當愛德華·哈斯勒ech和惠特尼發明了杯子和帽產品和制定龐加萊在這些新術語二元性。
Poincaré duality did not take on its modern form until the advent of cohomology in the 1930s, when Eduard ech and Hassler Whitney invented the cup and cap products and formulated Poincaré duality in these new terms.
現代公式
Modern formulation
龐加萊對偶定理的現代表述在同調和上同調方面:如果M是一個封閉的有向n流形，k是一個小於n的自然數，然後有一個規範定義的同構Hk(M,Z)→Hn-k(M,Z)。為了定義這樣的同構，選擇M的一個固定的基類[M]，如果M是有方向的，這個基類就會存在。然後通過將一個元素α∈Hk(M)映射到它的cap乘積[M]︵α來定義同構。
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 a natural number smaller than n, then there is a canonically defined isomorphism Hk(M,Z)→Hn-k(M,Z).
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 α∈Hk(M) to its cap product [M]︵α.
對於負度，上同島和上同島定義為零，特別地，龐加萊對偶性意味著可定向閉n流形的同調和上同調群在大於n度時為零。
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.
在這裡，同調和上同調是整的，但是同構在任何係數環上都是有效的。在有向流形不緊的情況下，必須用緊支上的上同調來代替上同調。
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 cohomology by cohomology with compact support.
雙細胞結構
Dual cell structures
對於三角流形，存在相應的對偶多面體分解。二元多面體分解是流形的細胞分解，使得二元多面體分解的k細胞與三角剖分的(n-k)細胞是雙射對應的，推廣對偶多面體的概念。
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.
精確地說，設T為n-流形M的三角剖分。設S是T的一個單純形。設為包含S的T的上維單純形，所以我們可以認為S是頂點的一個子集。定義與S對應的雙單元DS，使∩DS為內的凸殼，為包含S的所有頂點子集的重心。我們可以檢驗一下，如果S是i維的，那麼DS就是一個(n-i)維的單元格。此外，與T的雙胞形成M的CW分解，因此,配對CiMCn-iM→通過交集引發一個同構CiM→Cn-iM, 其中Ci是三角剖分T的細胞同源性，而Cn-iM和Cn-iM分別是流形的二重多面體/CW分解的細胞同源性和共同源性。這是鏈配合物的同構，這一事實證明了龐加萊對偶性。粗略地說，這相當於三角剖分T的邊界關係是對應S→DS下對偶多面體分解的關聯關係。
Precisely, let T be a triangulation of an n-manifold M . Let S be a simplex of T . Let be a top-dimensional simplex of T containing S, so we can think of S as a subset of the vertices of .
Define the dual cell DS corresponding to S so that ∩DS is the convex hull in of the barycentres of all subsets of the vertices of 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 CiMCn-iM→ given by taking intersections induces an isomorphism CiM→Cn-iM, where Ci is the cellular homology of the triangulation T , and Cn-iM and Cn-iM 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 S→DS.
自然性 Naturality
注意到Hk是一個逆變函子，而Hn-k是協變的。同構族同構族
Note that Hk is a contravariant functor while Hn-k is covariant. The family of isomorphisms
DM: Hk(M)→Hn-k(M)
在以下意義上是自然的:如果
is natural in the following sense: if
f :M→N
為兩個有向n流形之間的連續映射，與有向兼容，即把M的基類映射到N的基類，那麼
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，
其中f和f分別是f在同調和上同調中誘導的映射。
where f and f are the maps induced by f in homology and cohomology, respectively.
注意到一個非常重要的假設f把M的基類映射到N的基類。自然性並不適用於任意連續映射f,因為一般f 不是注射上同調。例如，如果f是一個覆蓋映射那麼它將M的基類映射到N的基類的倍數。這個倍數是映射f的次數。
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.
雙線性配對公式
Bilinear pairings formulation
假設流形M是緊的，無邊界的，可定向的，讓
Assuming the manifold M is compact, boundaryless, and orientable, let
τHiM
表示它的扭轉子群，令
denote the torsion subgroup of HiM and let
fHiM = HiM / τHiM
為自由部分 – 在本節中所有同調群取整係數。然後是雙線性映射，它是對偶的(解釋如下)。
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).
fHiMfHn-iM→
and
τHiM τ Hn-i-1M→/.
這裡/商理性的整數,作為添加劑組。注意在扭轉連接形式中，維數中有一個-1，所以配對的維度加起來是n-1而不是n。
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 n-1 rather than to n.
第一種形式通常稱為交積，第二種形式稱為扭聯形式。假設流形M是光滑的，通過使同調類變為橫向並計算它們的有向交叉數來計算它們的交叉積。對於撓性連接形式，通過將nx作為某類z的邊界來計算x和y的配對。形式為分子分數，z與y的橫交數，分母為n。
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.
對偶是對偶的說法意味著伴隨映射
The statement that the pairings are duality pairings means that the adjoint maps
fHiM→Hom( fHn-iM,)
and
τHiM→Hom( τHn-i-1M,/)
是群的同構。
are isomorphisms of groups.
這個結果是龐加萊對偶的一個應用
This result is an application of Poincaré Duality
HiMHn-iM
並結合普遍係數定理，給出了一個證明
together with the universal coefficient theorem, which gives an identification
fHn-iM ≡ Hom(Hn-iM;)
and
τHn-iM ≡ Ext(Hn-i-1M;) ≡ Hom(τHn-i-1M;/).
因此，龐加萊對偶性說，fHiM和fHn-iM是同構的，雖然沒有自然地圖給出同構，並且相似地τHiM 和τ Hn-i-1M也是同構, 雖然不是自然地。
Thus, Poincaré duality says that fHiM and fHn-iM are isomorphic, although there is no natural map giving the isomorphism, and similarly τHiM and τ Hn-i-1M are also isomorphic, though not naturally.
中間維度 Middle dimension
而對於大多數維度，龐加萊對偶性在不同的同源群之間引入了雙線性配對，在中間維度，它在一個單同島上導出了雙線性形式。得到的交形是一個非常重要的拓撲不變式。
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.
「中間維度」的含義取決於奇偶性。對於更常見的偶數維n = 2k，這實際上是中間維k，在中間同調的自由部分有一個形式:
What is meant by "middle dimension" depends on parity. For even dimension n = 2k which is more common, this is literally the middle dimension k, and there is a form on the free part of the middle homology:
相比之下，對於較少討論的奇數維n = 2k+1，它是最簡單的中下維k，在這個維上同調的扭轉部分有一個形式:
By contrast, for odd dimension n = 2k+1 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:
然而，在中下維k中同源性的自由部分與中上維k+1中也存在配對:
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:
所得到的群，雖然不是具有雙線性形式的單一群，卻是一個簡單的鏈復群，用代數L理論進行了研究。
The resulting groups, while not a single group with a bilinear form, are a simple chain complex and are studied in algebraic L-theory.
應用 Applications
Jozef Przytycki和Akira Yasuhara利用這種方法給出了三維透鏡空間的基本同倫和差分同構分類。
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.
託姆同構公式
Thom Isomorphism Formulation
龐加萊對偶性與湯姆同構定理密切相關，我們將在這裡解釋。對於這個證明，讓M是一個緊的，無邊界的有向n流形。設M×M為M與自身的乘積，設V為M×M中對角線的開管鄰域。考慮到映射:
Poincaré Duality is closely related to the Thom Isomorphism Theorem, as we will explain here.
For this exposition, let M be a compact, boundaryless oriented n-manifold. Let M×M be the product of M with itself, let V be an open tubular neighbourhood of the diagonal in M×M. Consider the maps:
HMHM→H(M×M) the Homology Cross ProductH(M×M)→H(M×M,(M×M)\ V) inclusion.H(M×M,(M×M)\ V)→H(νM,νM) excision map where νM is the normal disc bundle of the diagonal in M×M.H(νM,νM)→H-nM the Thom Isomorphism. This map is well-defined as there is a standard identification νM ≡ TM which is an oriented bundle, so the Thom Isomorphism applies.
結合起來，就得到了一個映射
Combined, this gives a map
也就是交點積，嚴格地說，它是上面的交點積的推廣，：但它也被稱為交點積。與Künneth定理相似的論證給出了撓性連接形式。
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.
這種形式的龐加萊二象性已經變得相當流行，因為它提供了一種方法來定義龐加萊二象性的任何廣義同調理論，只要你有一個湯姆同調同構的同調理論。同調理論的湯姆同構定理現在被認為是同調理論的可定向性的廣義概念。例如，在複雜拓撲k理論的意義上，流形上的一個spinc結構恰恰是可定向所需要的。
This formulation of Poincaré Duality has become quite popular as it provides a means to define Poincaré Duality for any generalized homology theory 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, aspinc-structure on a manifold turns out to be precisely what is needed to be orientable in the sense of complex topological k-theory.
歸納及相關結果
Generalizations and related results
龐加萊-萊夫謝茲對偶定理是有邊界流形的一個推廣。在不可定向的情況下，考慮到局部定向的束，可以給出一個與方向性無關的陳述:看扭曲的龐加萊二象性。
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: see Twisted Poincaré duality.
布蘭奇菲爾德對偶性是龐加萊對偶性的一個版本，它在流形的阿貝耳覆蓋空間的同調和相應的緊支上同調之間提供了一個同構。它用於獲得關於Alexander模塊的基本結構結果，並可用於定義結的特徵。
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.
隨著同調理論的發展，從1955年開始包括了k理論和其他傑出的理論，意識到同源H的可以取代了其他理論,一旦產品集合管構造;現在有教科書上的一般性治療。更具體地說，廣義同調理論有一個一般的龐加萊對偶定理，它需要一個同調理論的方向概念，並根據廣義的託姆同構定理來表示。在這方面的託姆同構定理可以被認為是廣義同調理論的龐加萊對偶的原始思想。
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 a generalized homology theorywhich 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對偶性是對(可能是奇異的)幾何對象的適當概括，例如分析空間或模式，當交叉同源性被發展成羅伯特·麥克弗森和馬克·高瑞斯基的分層空間時，例如實的或復的代數變種，正是為了將龐加萊二元性推廣到這種分層空間。
Verdier duality is the appropriate generalization to (possibly singular) geometric objects, such as analytic spaces or schemes, while intersection homology was developed 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.
代數拓撲中還有許多其他形式的幾何二象性，包括萊夫舍茨二象性、亞歷山大二象性、霍奇二象性和s二象性。
There are many other forms of geometric duality in algebraic topology, including Lefschetz duality, Alexander duality, Hodge duality, and S-duality.
在代數上，我們可以抽象出龐加萊復形的概念，龐加萊復形是一個代數對象，它的行為類似於流形的奇異鏈復形，值得注意的是，滿足龐加萊對偶在它的同島上，相對於一個可分辨的元素(對應於基類)。這些在外科學理論中用於對流形問題進行代數化。龐加萊空間的奇異鏈復形就是龐加萊復形。這些並不都是流形，但它們不是流形的失敗可以用阻礙理論來測量。
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 spaceis 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.