nLab Poincaré duality

Idea

Poincaré duality over some space is an equivalence (if it exists) relating cohomology with homology on that space.

The canonical example is the Poincaré duality in ordinary cohomology $H^\bullet(X)$/ordinary homology $H_\bullet(X)$ which exists over an orientable closed manifold $X$ of dimension $n$: any choice of volume form $\omega$ induces by the cap product an isomorphism

$$(-)\cap \omega \;\colon\; H_\bullet(X) \stackrel{\simeq}{\to} H^{n-\bullet}(X) \,.$$

More generally Poincaré duality is about dual objects in a generalized cohomology theory. For instance if $X$ above is furthermore K-oriented, hence oriented in the K-theory cohomology theory (hence if it has spin^c structure) then there is an isomorphism between its K-homology and K-theory. For more on this see at Poincaré duality algebra.

Poincaré duality is the mechanism behind Umkehr maps/push-forward in generalized cohomology: given a map of spaces $f \colon X \to Y$ which enjoy Poincaré duality with respect to some generalized cohomology theory $R$, one can pass from the canonically given pullback morphism

$$f^\ast \colon R^\bullet(Y) \to R^\bullet(X)$$

to the dual morphism

$$f_! \colon R^\bullet(X) \simeq R_\bullet(X) \stackrel{f_\ast}{\to} R_\bullet(Y) \simeq R^\bullet(Y) \,.$$

Here the duality is typically exhibited in two steps:

1. an Atiyah duality identifies the dual of $R^\bullet(X)$ with the $R$-cohomology of a Thom space of $X$;

2. a Thom isomorphism identified the $R$-cohomology of the Thom space back with that of $X$.

More generally, spaces $X$ are not self-dual in this way, but may at least be dual to themselves but equipped with a twist. This yields the twisted Umkehr maps in twisted cohomology (see for instance at Freed-Witten-Kapustin anomaly cancellation).

For more on this see at twisted Umkehr map.

Definition and statement

We first state the

• Traditional formulation

and then its

• Refinement in homotopy theory.

Traditional formulation

Around 1895 Henri Poincaré made an observation about Betti numbers of closed manifolds, which in the 1930s was then formulated by Eduard Čech and Hassler Whitney in the following modern form:

Theorem

Let $X$ be a closed manifold of dimension $n$ that is orientable. Then the cap product with any choice of orientation – in the form of a fundamental class $[X] \in H_n(X)$ – induces isomorphisms of the form

$$(-)\cap [X] \;\colon\; H^k(X) \stackrel{\simeq}{\to} H_{n-k}(X)$$

between the ordinary cohomology and the ordinary homology groups of $X$.

Later this was turned around and more general topological spaces satisfying this condition were considered:

Definition

A topological space for which there is $d \in \mathbb{N}$ and a class $[X] \in H_d(X)$ such that the cap product induces isomorphisms

$$(-) \cap [X] \;\colon\; H^\bullet(X) \stackrel{\simeq}{\to} H_{d-\bullet}(X)$$

is called a Poincaré duality space.

Refinement to homotopy theory

Traditionally Poincaré duality is stated as a duality of chain homology groups. The passage from chain complexes to their homology groups, hence the passage from full homotopy theory to just some invariants, however forgets a lot of information. But it turns out that this can always be lifted:

Theorem

Let $X$ be a Poincaré duality space of dimension $n$, def. . Then there is a quasi-isomorphism

$$C^\bullet(X) \stackrel{\simeq}{\to} \Sigma^{-d} C_\bullet(X)$$

between the chain complex of ordinary cohomology of $X$ with that $d$-fold de-suspension of the chain complex of ordinary homology of $X$.

This is such that its image in chain homology is (up to sign) the traditional Poincaré duality isomorphism

$$H^\bullet(X) \stackrel{\simeq}{\to} H_{d -\bullet}(X)$$

of theorem . When $\bullet=r$, the sign is $-1^{r(r+1)/2}$.

This is (EM 11, theorem, 2.5.2).

Remark

Since $C^\bullet(X) \simeq [C_\bullet(X,), \mathbb{I}] \simeq (C_\bullet(X))^\vee$ is the dual object to $C_\bullet(X)$ in the (∞,1)-category of unbounded chain complexes, this says that for a Poincaré duality space $X$ the chain complex $C_\bullet(X)$ is almost a self-dual objects, except for a degree-shift, hence except for a twist of (itself) degree 0.

Properties

Recognition of manifolds

From [ALGTOP-L], Oct 5, 2010.

• Jim Stasheff:

  Any one have a reference for obstructions which detect whether a space whose cohomology has Poincare duality is actually a manifold? thanks

• John Klein:

  Ranicki’s total surgery obstruction does it in the top case, in surgery dimensions.

  • The total surgery obstruction. Algebraic topology, Aarhus 1978 (Proc. Sympos., Univ. Aarhus, Aarhus, 1978), pp. 275–316, Lecture Notes in Math., 763, Springer, Berlin, 1979. (pdf)

• Nathanien Rounds:

  I believe this question is answered (simply connected case, over Q) in Dennis Sullivan’s paper Infinitesmal Computations in Topology (Theorem 13.2). The answer, as I understand it, is that outside dimension 4k any graded commutative algebra over Q wtih first betti number 0 satisfying Poincare Duality can be realized as the cohomoloyg ring of a manifold. In dimension 4k there is an obstruction related to the signature which is given in that paper. There won’t in general be a unique manifold corresponding to the ring; for example one can choose different rational Pontragin classes and change the homemorphism type but not the cohomology.

  The general answer (not simply connected, over Z) is given by Ranicki’s total surgery obstruction, as John Klein has already pointed out. One way to interpret this obstruction geometrically is that the Poincare duality map is always “local” but it need not have a “local” inverse, and the lack of a local inverse is an obstruction to having a anifold structure. See for example McCrory’s paper “A Characterization of Homology Manifolds” and also my thesis, which if you’re interested is here:

  www.math.purdue.edu/~nrounds

See also (Ranicki 96) and see at Poincaré complex.

Relation to Thom isomorphism

The Poincaré dual of a submanifold can be identified with the Thom class on its normal bundle

(…)

Relation to push-forward in cohomology

Given Poincare duality and hence (twisted-)self-dual objects, the induced dagger-structure allows torevert morphisms in cohomology. These “Umkehr maps” describe fiber integration in cohomology.

Generalizations

The six operations of Grothendieck and Grothendieck duality are designed to ensure a generalized and relative versions of Poincaré duality and related phenomena in the setups like sheaf and topos theory, algebraic geometry. See Grothendieck duality for references.

Another generalization, for singular spaces, is with help of stratifications and via intersection cohomology.

Variants and generalizations

• in topology: Poincaré duality complex

• in noncommutative topology: Poincaré duality algebra

• in derived abelian sheaf and topos theory (for abelian sheaf cohomology): Verdier duality

• in nonabelian cohomology: nonabelian Poincaré duality

Related concepts

• Poincaré complex

• Poincaré duality algebra

• KO-dimension

• Serre duality

References

General

This historical work of Henri Poincaré is reviewed around page 28. pf

• Jean Dieudonné, A History of Algebraic and Differential Topology, 1900 - 1960

Summaries of the traditional modern statement of Poincaré duality are for instance in

• eom, Poincaré duality

• Manifold Atlas, Poincaré Duality Spaces

Discussion of generalizations to “chain duality” (see also at Poincare complex) is at

• Andrew Ranicki, 45 slides on chain duality, September 1996 (pdf)

Discussion in etale cohomology is in

• James Milne, sections 14 and 24 of Lectures on Étale Cohomology

With an eye towards generalization in spectral geometry (spectral triples):

• Alain Connes, page 10 of Noncommutative geometry and reality, J. Math. Phys. 36 (11), 1995 (pdf)

In the generality of supermanifolds:

• Konstantin Eder, John Huerta, Simone Noja, Poincaré Duality for Supermanifolds, Higher Cartan Structures and Geometric Supergravity [arXiv:2312.05224]

For equivariant cohomology

In equivariant cohomology:

For finite groups

• Steven Costenoble, Stefan Waner, Equivariant Poincaré duality, Michigan Math. J. 39 (1992), no. 2, 325–351

For compact Lie groups

• Michel Brion, Poincaré duality and equivariant (co)homology, Michigan mathematical journal, Volume 48, Issue 1 (2000), 1-624 (EUCLID)

• Steven Costenoble, Stefan Waner, Equivariant ordinary homology and cohomology (arXiv:math/0310237)

• Christopher Allday, Matthias Franz, Volker Puppe, Equivariant Poincaré-Alexander-Lefschetz duality and the Cohen-Macaulay property (arXiv:1303.1146)

In equivariant K-theory:

• Jean-Louis Tu, Twisted K-theory and Poincare duality (arXiv:0609556)

• Heath Emerson, Ralf Meyer, Dualities in equivariant Kasparov theory, New York J. Math. 16 (2010), 245-313 (arXiv:0711.0025)

• Heath Emerson, Duality, correspondences and the Lefschetz map in equivariant KK-theory: a survey (arXiv:0904.4744)

For Hochschild cohomology

Poincaré duality on Hochschild (co)homology

• M. Van den Bergh, A relation between Hochschild homology and cohomology for Gorenstein rings . Proc. Amer. Math. Soc. 126 (1998), 1345–1348; (JSTOR)

  Correction: Proc. Amer. Math. Soc. 130 (2002), 2809–2810.

with more on that in

• U. Krähmer, Poincaré duality in Hochschild cohomology (pdf)

That this Poincare duality takes the Connes coboundary operator to the BV operator is shown in

• Victor Ginzburg, Calabi-Yau Algebras, arXiv/math.AG/0612139

For spaces in higher geometry

Poincaré duality on Deligne-Mumford stacks/orbifolds is discussed in

• Dan Abramovich, Tom Graber, Angelo Vistoli, Gromov-Witten theory of Deligne-Mumford stacks (arXiv:math/0603151)

• Weimin Chen, Yongbin Ruan, A New Cohomology Theory for Orbifold, Commun. Math. Phys. 248 (2004) 1-31 (arXiv:math/0004129)

For disucssion in noncommutative topology/KK-theory see at Poincaré duality algebra.

On the level of chains

Poincaré duality is traditional considered on the level of cohomology groups. One may asks if it lifts to a duality on the underlying chain complexes.

Such “derived” Poincaré duality on the level of chain complexes is claimed in theorem 2.5.2 of

• Eric J. Malm, String topology and the based loop space (arXiv:1103.6198)

based on – or at least inspired by – the discussion in (section 10) of

• William Dwyer, John Greenlees, S. Iyengar, Duality in algebra and topology (arXiv:math/0510247)

The article

• Thomas Tradler, Mahmoud Zeinalian, Dennis Sullivan, Infinity Structure of Poincare Duality Spaces (arXiv:math/0309455)

claims a chain duality (even of $A_\infty$-comodule chains but then possibly only exhibited by a bimodule) for the singular chain complex of any Poincaré duality topological space.

A review is in

• Thomas Tradler, Infinity inner products on A-infinity algebras (arXiv:0806.0065)