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\section*{Poincaré duality}

[[!redirects Poincare duality]]

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{duality}{}\paragraph*{{Duality}}\label{duality}

[[!include duality - contents]]

\hypertarget{integration_theory}{}\paragraph*{{Integration theory}}\label{integration_theory}

[[!include integration theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{Idea}{Idea}\dotfill \pageref*{Idea} \linebreak
\noindent\hyperlink{DefinitionAndStatement}{Definition and statement}\dotfill \pageref*{DefinitionAndStatement} \linebreak
\noindent\hyperlink{TraditionalFormulation}{Traditional formulation}\dotfill \pageref*{TraditionalFormulation} \linebreak
\noindent\hyperlink{RefinementInHomotopyTheory}{Refinement to homotopy theory}\dotfill \pageref*{RefinementInHomotopyTheory} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{recognition_of_manifolds}{Recognition of manifolds}\dotfill \pageref*{recognition_of_manifolds} \linebreak
\noindent\hyperlink{relation_to_thom_isomorphism}{Relation to Thom isomorphism}\dotfill \pageref*{relation_to_thom_isomorphism} \linebreak
\noindent\hyperlink{relation_to_pushforward_in_cohomology}{Relation to push-forward in cohomology}\dotfill \pageref*{relation_to_pushforward_in_cohomology} \linebreak
\noindent\hyperlink{generalizations}{Generalizations}\dotfill \pageref*{generalizations} \linebreak
\noindent\hyperlink{variants_and_generalizations}{Variants and generalizations}\dotfill \pageref*{variants_and_generalizations} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{for_equivariant_cohomology}{For equivariant cohomology}\dotfill \pageref*{for_equivariant_cohomology} \linebreak
\noindent\hyperlink{for_hochschild_cohomology}{For Hochschild cohomology}\dotfill \pageref*{for_hochschild_cohomology} \linebreak
\noindent\hyperlink{for_spaces_in_higher_geometry}{For spaces in higher geometry}\dotfill \pageref*{for_spaces_in_higher_geometry} \linebreak
\noindent\hyperlink{on_the_level_of_chains}{On the level of chains}\dotfill \pageref*{on_the_level_of_chains} \linebreak


\hypertarget{Idea}{}\subsection*{{Idea}}\label{Idea}

Poincar\'e{} duality over some [[space]] is an [[equivalence]] (if it exists) relating [[cohomology]] with [[homology]] on that space.

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

\begin{displaymath}
(-)\cap \omega \;\colon\; H_\bullet(X) \stackrel{\simeq}{\to} H^{n-\bullet}(X)
  \,.
\end{displaymath}
More generally Poincar\'e{} duality is about [[dual objects]] in a [[generalized cohomology theory]]. For instance if $X$ above is furthermore [[K-orientation|K-oriented]], hence [[orientation in generalized cohomology |oriented]] in the [[K-theory]] [[cohomology theory]] (hence if it has [[spin{\tt \symbol{94}}c structure]]) then there is an isomorphism between its [[K-homology]] and [[K-theory]]. For more on this see at \emph{[[Poincaré duality algebra]]}.

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

\begin{displaymath}
f^\ast \colon R^\bullet(Y) \to R^\bullet(X)
\end{displaymath}
to the [[dual morphism]]

\begin{displaymath}
f_! \colon R^\bullet(X) \simeq R_\bullet(X) \stackrel{f_\ast}{\to} R_\bullet(Y) \simeq R^\bullet(Y)
  \,.
\end{displaymath}
Here the duality is typically exhibited in two steps:

\begin{enumerate}%
\item an [[Atiyah duality]] identifies the dual of $R^\bullet(X)$ with the $R$-cohomology of a [[Thom space]] of $X$;


\item a [[Thom isomorphism]] identified the $R$-cohomology of the Thom space back with that of $X$.



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

For more on this see at \emph{[[twisted Umkehr map]]}.

\hypertarget{DefinitionAndStatement}{}\subsection*{{Definition and statement}}\label{DefinitionAndStatement}

We first state the

\begin{itemize}%
\item \emph{\hyperlink{TraditionalFormulation}{Traditional formulation}}

\end{itemize}
and then its

\begin{itemize}%
\item \emph{\hyperlink{RefinementInHomotopyTheory}{Refinement in homotopy theory}}.

\end{itemize}
\hypertarget{TraditionalFormulation}{}\subsubsection*{{Traditional formulation}}\label{TraditionalFormulation}

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

\begin{theorem}
\label{TraditionalPoincareDuality}\hypertarget{TraditionalPoincareDuality}{}
Let $X$ be a [[closed manifold]] of [[dimension]] $n$ that is [[orientation|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

\begin{displaymath}
(-)\cap [X] \;\colon\; H^k(X) \stackrel{\simeq}{\to} H_{n-k}(X)
\end{displaymath}
between the [[ordinary cohomology]] and the [[ordinary homology]] groups of $X$.

\end{theorem}
Later this was turned around and more general [[topological spaces]] satisfying this condition were considered

\begin{defn}
\label{PoincareDualitySpace}\hypertarget{PoincareDualitySpace}{}
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]]

\begin{displaymath}
(-) \cap [X] \;\colon\; H^\bullet(X) \stackrel{\simeq}{\to} H_{d-\bullet}(X)
\end{displaymath}
is called a \textbf{[[Poincaré duality space]]}.

\end{defn}
\hypertarget{RefinementInHomotopyTheory}{}\subsubsection*{{Refinement to homotopy theory}}\label{RefinementInHomotopyTheory}

Traditionally Poincar\'e{} duality is stated as a [[dual object|duality]] of [[chain homology|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:

\begin{theorem}
\label{}\hypertarget{}{}
Let $X$ be a [[Poincaré duality space]] of [[dimension]] $n$, def. \ref{PoincareDualitySpace}. Then there is a [[quasi-isomorphism]]

\begin{displaymath}
C^\bullet(X) \stackrel{\simeq}{\to} \Sigma^{-d} C_\bullet(X)
\end{displaymath}
between the [[chain complex]] of [[ordinary cohomology]] of $X$ with that $d$-fold [[suspension|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 Poincar\'e{} duality isomorphism

\begin{displaymath}
H^\bullet(X) \stackrel{\simeq}{\to} H_{d -\bullet}(X)
\end{displaymath}
of theorem \ref{TraditionalPoincareDuality}. When $\bullet=r$, the sign is $-1^{r(r+1)/2}$.

\end{theorem}
This is (\hyperlink{EM11}{EM 11, theorem, 2.5.2}).

\begin{remark}
\label{}\hypertarget{}{}
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 [[twisted cohomology|twist]] of (itself) degree 0.

\end{remark}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{recognition_of_manifolds}{}\subsubsection*{{Recognition of manifolds}}\label{recognition_of_manifolds}

From ALGTOP-L, Oct 5, 2010.

\begin{itemize}%
\item [[Jim Stasheff]]:

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


\item John Klein:

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

\begin{itemize}%
\item 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. (\href{http://www.maths.ed.ac.uk/~aar/papers/total.pdf}{pdf})

\end{itemize}

\item 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/{\tt \symbol{126}}nrounds



\end{itemize}
See also (\hyperlink{Ranicki96}{Ranicki 96}) and see at \emph{[[Poincaré complex]]}.

\hypertarget{relation_to_thom_isomorphism}{}\subsubsection*{{Relation to Thom isomorphism}}\label{relation_to_thom_isomorphism}

The Poincar\'e{} dual of a [[submanifold]] can be identified with the [[Thom class]] on its [[normal bundle]]

(\ldots{})

\hypertarget{relation_to_pushforward_in_cohomology}{}\subsubsection*{{Relation to push-forward in cohomology}}\label{relation_to_pushforward_in_cohomology}

Given Poincare duality and hence (twisted-)[[self-dual objects]], the \href{self-dual+object#RelationToDaggerCompactStructure}{induced dagger-structure} allows torevert morphisms in [[cohomology]]. These ``[[Umkehr maps]]'' describe \emph{[[fiber integration in cohomology]]}.

\hypertarget{generalizations}{}\subsubsection*{{Generalizations}}\label{generalizations}

The [[six operations]] of [[Grothendieck]] and [[Grothendieck duality]] are designed to ensure a generalized and relative versions of Poincar\'e{} duality and related phenomena in the setups like sheaf and [[topos theory]], algebraic geometry. See \emph{[[Grothendieck duality]]} for references.

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

\hypertarget{variants_and_generalizations}{}\subsection*{{Variants and generalizations}}\label{variants_and_generalizations}

\begin{itemize}%
\item in [[topology]]: [[Poincaré duality complex]]


\item in [[noncommutative topology]]: [[Poincaré duality algebra]]


\item in [[derived category|derived]] [[abelian sheaf|abelian]] [[sheaf and topos theory]] (for [[abelian sheaf cohomology]]): [[Verdier duality]]


\item in [[nonabelian cohomology]]: [[nonabelian Poincaré duality]]



\end{itemize}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[Poincaré complex]]


\item [[Poincaré duality algebra]]


\item [[KO-dimension]]


\item [[Serre duality]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

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

\begin{itemize}%
\item [[Jean Dieudonné]], \emph{A History of Algebraic and Differential Topology, 1900 - 1960}

\end{itemize}
Summaries of the traditional modern statement of Poincar\'e{} duality are for instance in

\begin{itemize}%
\item [[eom]], \emph{\href{http://www.encyclopediaofmath.org/index.php/Poincar%C3%A9_duality}{Poincar\'e{} duality}}


\item Manifold Atlas, \emph{\href{http://www.map.mpim-bonn.mpg.de/Poincar%C3%A9_Duality_Spaces}{Poincar\'e{} Duality Spaces}}



\end{itemize}
Discussion of generalizations to ``chain duality'' (see also at \emph{[[Poincare complex]]}) is at

\begin{itemize}%
\item [[Andrew Ranicki]], \emph{45 slides on chain duality}, September 1996 (\href{http://www.math.uiuc.edu/K-theory/0154/45slides.pdf}{pdf})

\end{itemize}
Discussion in [[etale cohomology]] is in

\begin{itemize}%
\item [[James Milne]], sections 14 and 24 of \emph{[[Lectures on Étale Cohomology]]}

\end{itemize}
With an eye towards generalization in [[spectral geometry]] ([[spectral triples]]):

\begin{itemize}%
\item [[Alain Connes]], page 10 of \emph{Noncommutative geometry and reality}, J. Math. Phys. 36 (11), 1995 (\href{http://www.alainconnes.org/docs/reality.pdf}{pdf})

\end{itemize}
\hypertarget{for_equivariant_cohomology}{}\subsubsection*{{For equivariant cohomology}}\label{for_equivariant_cohomology}

In [[equivariant cohomology]]:

For [[finite groups]]

\begin{itemize}%
\item [[Steven Costenoble]], [[Stefan Waner]], \emph{Equivariant Poincar\'e{} duality}, Michigan Math. J. 39 (1992), no. 2, 325--351

\end{itemize}
For [[compact Lie groups]]

\begin{itemize}%
\item [[Michel Brion]], \emph{Poincar\'e{} duality and equivariant (co)homology}, Michigan mathematical journal, Volume 48, Issue 1 (2000), 1-624 (\href{http://projecteuclid.org/euclid.mmj/1030132709}{EUCLID})


\item [[Steven Costenoble]], [[Stefan Waner]], \emph{Equivariant ordinary homology and cohomology} (\href{http://arxiv.org/abs/math/0310237}{arXiv:math/0310237})


\item Christopher Allday, Matthias Franz, [[Volker Puppe]], \emph{Equivariant Poincar\'e{}-Alexander-Lefschetz duality and the Cohen-Macaulay property} (\href{http://arxiv.org/abs/1303.1146}{arXiv:1303.1146})



\end{itemize}
In [[equivariant K-theory]]:

\begin{itemize}%
\item [[Jean-Louis Tu]], \emph{Twisted K-theory and Poincare duality} (\href{http://arxiv.org/abs/math/0609556}{arXiv:0609556})


\item [[Heath Emerson]], [[Ralf Meyer]], \emph{Dualities in equivariant Kasparov theory}, New York J. Math. 16 (2010), 245-313 (\href{http://arxiv.org/abs/0711.0025}{arXiv:0711.0025})


\item [[Heath Emerson]], \emph{Duality, correspondences and the Lefschetz map in equivariant KK-theory: a survey} (\href{http://arxiv-web3.library.cornell.edu/abs/0904.4744}{arXiv:0904.4744})



\end{itemize}
\hypertarget{for_hochschild_cohomology}{}\subsubsection*{{For Hochschild cohomology}}\label{for_hochschild_cohomology}

[[nLab:Poincaré duality]] on [[nLab:Hochschild cohomology|Hochschild (co)homology]]

\begin{itemize}%
\item M. Van den Bergh, \emph{A relation between Hochschild homology and cohomology for Gorenstein rings} . Proc. Amer. Math. Soc. 126 (1998), 1345--1348; (\href{http://www.jstor.org/stable/118786}{JSTOR})

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



\end{itemize}
with more on that in

\begin{itemize}%
\item U. Kr\"a{}hmer, \emph{Poincar\'e{} duality in Hochschild cohomology} (\href{http://www.maths.gla.ac.uk/~ukraehmer/brussels.pdf}{pdf})

\end{itemize}
That this Poincare duality takes the [[Connes coboundary operator]] to the [[BV operator]] is shown in

\begin{itemize}%
\item [[nLab:Victor Ginzburg]], \emph{Calabi-Yau Algebras}, \href{http://arxiv.org/abs/math.AG/0612139}{arXiv/math.AG/0612139}

\end{itemize}
\hypertarget{for_spaces_in_higher_geometry}{}\subsubsection*{{For spaces in higher geometry}}\label{for_spaces_in_higher_geometry}

Poincar\'e{} duality on [[Deligne-Mumford stacks]]/[[orbifolds]] is discussed in

\begin{itemize}%
\item Dan Abramovich, Tom Graber, [[Angelo Vistoli]], \emph{Gromov-Witten theory of Deligne-Mumford stacks} (\href{http://arxiv.org/abs/math/0603151}{arXiv:math/0603151})


\item Weimin Chen, [[Yongbin Ruan]], \emph{A New Cohomology Theory for Orbifold}, Commun. Math. Phys. 248 (2004) 1-31 (\href{http://arxiv.org/abs/math/0004129}{arXiv:math/0004129})



\end{itemize}
For disucssion in [[noncommutative topology]]/[[KK-theory]] see at \emph{[[Poincaré duality algebra]]}.

\hypertarget{on_the_level_of_chains}{}\subsubsection*{{On the level of chains}}\label{on_the_level_of_chains}

Poincar\'e{} 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'' Poincar\'e{} duality on the level of chain complexes is claimed in theorem 2.5.2 of

\begin{itemize}%
\item Eric J. Malm, \emph{String topology and the based loop space} (\href{http://arxiv.org/abs/1103.6198}{arXiv:1103.6198})

\end{itemize}
based on -- or at least inspired by -- the discussion in (section 10) of

\begin{itemize}%
\item [[William Dwyer]], [[John Greenlees]], S. Iyengar, \emph{Duality in algebra and topology} (\href{http://arxiv.org/abs/math/0510247}{arXiv:math/0510247})

\end{itemize}
The article

\begin{itemize}%
\item [[Thomas Tradler]], [[Mahmoud Zeinalian]], [[Dennis Sullivan]], \emph{Infinity Structure of Poincare Duality Spaces} (\href{http://arxiv.org/abs/math/0309455}{arXiv:math/0309455})

\end{itemize}
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 Poincar\'e{} duality topological space.

A review is in

\begin{itemize}%
\item [[Thomas Tradler]], \emph{Infinity inner products on A-infinity algebras} (\href{http://arxiv.org/abs/0806.0065}{arXiv:0806.0065})

\end{itemize}
[[!redirects Poincare duality]] [[!redirects Poincaré duality]] [[!redirects Poincar\%C3\%A9 duality]]



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