In mathematics, the Poincaré duality theorem, 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 th homology group of M, for all integers k
Hk(M)\congHn-k(M).
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.
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 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 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
Hk(M,\Z)\toHn-k(M,\Z)
M
\alpha\inHk(M)
[M]\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
Hi(X)\stackrel{\cong}{\to}
BM | |
H | |
n-i |
(X),
i | |
H | |
c(X) |
\stackrel{\cong}{\to}Hn-i(X).
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
Precisely, let T be a triangulation of an n-manifold M. Let S be a simplex of T. Let
\Delta
\Delta
\Delta\capDS
\Delta
\Delta
S
n-i
CiM ⊗ Cn-iM\to\Z
CiM\toCn-iM
Ci
Cn-iM
Cn-iM
S\longmapstoDS
Note that
Hk
Hn-k
DM\colonHk(M)\toHn-k(M)
f\colonM\toN
DN=f*\circDM\circf*,
f*
f*
f
Note the very strong and crucial hypothesis that
f
f
f*
f
f
Assuming the manifold M is compact, boundaryless, and orientable, let
\tauHiM
HiM
fHiM=HiM/\tauHiM
fHiM ⊗ fHn-iM\to\Z
\tauHiM ⊗ \tauHn-i-1M\to\Q/\Z
Here
\Q/\Z
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 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
fHiM\toHom\Z(fHn-iM,\Z)
\tauHiM\toHom\Z(\tauHn-i-1M,\Q/\Z)
This result is an application of Poincaré duality
HiM\simeqHn-iM
fHn-iM\equivHom(Hn-iM;\Z)
\tauHn-iM\equivExt(Hn-i-1M;\Z)\equivHom(\tauHn-i-1M;\Q/\Z)
fHiM
fHn-iM
\tauHiM
\tauHn-i-1M
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:
fHkM ⊗ fHkM\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:
\tauHkM ⊗ \tauHkM\to\Q/\Z.
fHkM ⊗ fHk+1M\to\Z.
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. Let
M
H*M ⊗ H*M\toH*(M x M)
H*(M x M)\toH*\left(M x M,(M x M)\setminusV\right)
H*\left(M x M,(M x M)\setminusV\right)\toH*(\nuM,\partial\nuM)
\nuM
M x M
H*(\nuM,\partial\nuM)\toH*-nM
\nuM\equivTM
Combined, this gives a map
HiM ⊗ HjM\toHi+j-nM
This formulation of Poincaré duality has become popular[2] 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 for that theory. For example, a spinC-structure on a manifold is a precise analog of an orientation within complex topological k-theory.
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.
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'*
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.
There are many other forms of geometric duality in algebraic topology, including Lefschetz duality, Alexander duality, Hodge duality, and 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.
. Allen Hatcher. Algebraic Topology. 2002. Cambridge University Press. 9780521795401. 1st. Cambridge. English. 1867354.