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\begin{document}

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\section*{Poincaré duality algebra}

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

\hypertarget{noncommutative_geometry}{}\paragraph*{{Noncommutative geometry}}\label{noncommutative_geometry}

[[!include noncommutative geometry - contents]]

\hypertarget{operator_algebra}{}\paragraph*{{Operator algebra}}\label{operator_algebra}

[[!include AQFT and operator algebra 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{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{ForGradedCommutativeAlgebras}{For graded-commutative algebras}\dotfill \pageref*{ForGradedCommutativeAlgebras} \linebreak
\noindent\hyperlink{ForCStarAlgebras}{For $C^\ast$-algebras}\dotfill \pageref*{ForCStarAlgebras} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{for_dgalgebras}{For dg-Algebras}\dotfill \pageref*{for_dgalgebras} \linebreak
\noindent\hyperlink{PropertiesForCStarAlgebras}{For $C^\ast$-algebras}\dotfill \pageref*{PropertiesForCStarAlgebras} \linebreak
\noindent\hyperlink{PropertiesCAstDualsAndTwists}{Duals and twists}\dotfill \pageref*{PropertiesCAstDualsAndTwists} \linebreak
\noindent\hyperlink{PropertiesKOrientationAndUmkehrMaps}{K-Orientation and Umkehr maps}\dotfill \pageref*{PropertiesKOrientationAndUmkehrMaps} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{for_graded_associative_algebras}{For graded associative algebras}\dotfill \pageref*{for_graded_associative_algebras} \linebreak
\noindent\hyperlink{for_algebras_3}{For $C^\ast$-algebras}\dotfill \pageref*{for_algebras_3} \linebreak


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

Generally, a \emph{Poincar\'e{} duality dg-algebra} is a [[dg-algebra]] with structure mimicking [[Poincaré duality]] in [[ordinary cohomology]].

On the other hand a \emph{Poincar\'e{} duality $C^\ast$-algebra} is a [[C\emph{-algebra]] which represents a [[space]] in [[noncommutative topology]] for which there is a sensible notion of [[Poincaré duality]] in [[K-theory]] ([[operator K-theory]]/[[K-homology]]).}

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\hypertarget{ForGradedCommutativeAlgebras}{}\subsubsection*{{For graded-commutative algebras}}\label{ForGradedCommutativeAlgebras}

\begin{defn}
\label{}\hypertarget{}{}
The structure of a Poincar\'e{} duality algebra in [[dimension]] $n$ on a graded-commutative [[graded vector space|graded]] [[associative algebra]] $A$ is a [[linear function]] $\epsilon \colon A_n \to k$ to the ground field such that all the induced [[bilinear forms]]

\begin{displaymath}
A_k \otimes A_{n-k} \stackrel{\otimes}{\to} A^n \stackrel{\epsilon}{\to} k
\end{displaymath}
are non-degenerate.

\end{defn}
e.g. (\hyperlink{LambrechstStanley07}{Lambrechst-Stanley 07})

\hypertarget{ForCStarAlgebras}{}\subsubsection*{{For $C^\ast$-algebras}}\label{ForCStarAlgebras}

For [[C\emph{-algebras]] hence in [[noncommutative topology]] there is the following notion of Poincar\'e{} duality, which is really Poincar\'e{} with respect not to [[ordinary cohomology]] but [[K-theory]] ([[operator K-theory]]).}

We start with the definition of Poincar\'e{} \emph{self}-duality and then generalize to Poincar\'e{} dual pairs.

\begin{defn}
\label{PDAlgebra}\hypertarget{PDAlgebra}{}
A [[separable C\emph{-algebra]] $A \in$ [[C}Alg]] is a \textbf{Poincar\'e{} duality algebra} (or \emph{PD algebra}, for short ) if it is [[dualizable object]] when regarded as an object of the [[KK-theory]]-category, with [[dual object]] its [[opposite algebra]].

The element $\Delta$ in def. \ref{PDAlgebra} is called a \textbf{[[fundamental class]]} of $A$.

\end{defn}
This appears as (\hyperlink{BMRS07}{BMRS 07, def. 2.1}, following \hyperlink{Connes}{Connes, p. 601}) following (\hyperlink{Connes}{Connes}).

\begin{remark}
\label{}\hypertarget{}{}
Explicitly def. \ref{PDAlgebra} says that $A$ is a PD algebra if there exists $\Delta \in KK(A \otimes A^{op}, \mathbb{C})$ and $\Delta^\vee \in KK(\mathbb{C}, A \otimes A^{op})$ such that

\begin{displaymath}
\Delta^\vee \otimes_{A^{op}} \Delta = id_A \in KK(A,A)
\end{displaymath}
and

\begin{displaymath}
\Delta^\vee \otimes_A \Delta = id_{A^{op}} \in KK(A^{op}, A^{op})
  \,.
\end{displaymath}
\end{remark}
\begin{prop}
\label{}\hypertarget{}{}
For $A$ $B$ two Poincar\'e{} duality algebras, def. $\backslash$ref\{PDAlgebra\}, and for $f \colon A \to B$ a [[homomorphism]] between them, regarded as a morphism $f^\ast \colon B \to A$ in [[KK-theory]], the correspondung [[dual morphism]] $f! \colon A \to B$ is the one such that postcomposition in $KK$ with this corresponds to the [[Umkehr map]]/[[push forward in generalized cohomology]] in [[KK-theory]].

\end{prop}
For more on this see below at \emph{\hyperlink{PropertiesKOrientationAndUmkehrMaps}{Properties -- K-Orientation and Umkehr mpas}}.

\begin{remark}
\label{OppositeConvolutionAlgebraAndInverseTwist}\hypertarget{OppositeConvolutionAlgebraAndInverseTwist}{}
For $C^\ast$-algebras which are [[groupoid convolution algebras]] $C^\ast(\mathcal{G})$ the [[opposite algebra]] is [[Morita equivalence|Morita equivlant]] (since a [[groupoid]] $\mathcal{G}$ is [[equivalence of categories|equivalent]] to its [[opposite category|opposite groupoid]] $\mathcal{G}^{op}$, the equivalence being induced by the [[functor]] which sends a morphism to its [[inverse]]). But given a [[circle 2-bundle]] $\chi \colon \mathcal{G} \to \mathbf{B}^2 U(1)$ the corresponding [[twisted groupoid convolution algebra]] is such that passing to the opposite corresponds to passing to the inverse twist $-\chi$.

\end{remark}
Therefore it makes sense to consider more generally

\begin{defn}
\label{}\hypertarget{}{}
For $A$ a [[C\emph{-algebra]] a \emph{Poincar\'e{} dual} for $A$ is a [[dual object]] $B \in C^\ast Alg \to KK$ in [[KK-theory]].}

\end{defn}
Below in the \hyperlink{PropertiesForCStarAlgebras}{Proposition-Section} is discussed how under Poincar\'e{}-duality the [[twisted K-theory|twist]] changes.

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

\hypertarget{for_dgalgebras}{}\subsubsection*{{For dg-Algebras}}\label{for_dgalgebras}

\hypertarget{PropertiesForCStarAlgebras}{}\subsubsection*{{For $C^\ast$-algebras}}\label{PropertiesForCStarAlgebras}

\hypertarget{PropertiesCAstDualsAndTwists}{}\paragraph*{{Duals and twists}}\label{PropertiesCAstDualsAndTwists}

\begin{prop}
\label{}\hypertarget{}{}
Let $X$ be a [[closed manifold]] with [[spin{\tt \symbol{94}}c-structure]]. Then there is a Poincar\'e{} duality [[isomorphism]]

\begin{displaymath}
K^\bullet(X) \simeq K_\bullet(X)
  \,.
\end{displaymath}
\end{prop}
For instance (\hyperlink{Connes}{Connes, chapter 2.7, prop. 5}).

(\ldots{}) The relaton between Poincar\'e{} duality on algebras of functions and [[spin{\tt \symbol{94}}c-structure]] is discussed in (\hyperlink{Connes}{Connes, around p. 603}). (\ldots{})

Notice that the [[obstruction]] to [[spin{\tt \symbol{94}}c structure]] is the third [[integral Stiefel-Whitney class]] $W_3 \colon B SO \to B^2 U(1)$. If this does not vanish on a [[manifold]], then a Poincar\'e{} dual/[[dual object]] in [[KK-theory]] still exists, but is the same manifold equipped with a [[twisted K-theory|twist]] \emph{shifted} by $W_3(\tau_X)$, where $\tau_X$ denotes the ([[cotangent bundle|co]])[[tangent bundle]] of $X$.

\begin{defn}
\label{CStarAlgebraOf2BundleOnManifold}\hypertarget{CStarAlgebraOf2BundleOnManifold}{}
For $X$ a ([[compact topological space|compact]]) [[manifold]] and $c \in H^3(X,\mathbb{Z})$ the class of a [[circle 2-bundle]]/[[bundle gerbe]] $\mathcal{G}$ on $X$, write

\begin{displaymath}
C_c(X) \in C^\ast Alg \to KK
\end{displaymath}
for the corresponding [[twisted groupoid convolution algebra]], the one whose [[operator K-theory]] is the $c$-[[twisted K-theory]] of $X$:

\begin{displaymath}
KK_\bullet(\mathbb{C}, C_c(X)) \simeq K_{\bullet + c}(X)
  \,.
\end{displaymath}
\end{defn}
\begin{prop}
\label{DualOfCompactManifoldWithTwist}\hypertarget{DualOfCompactManifoldWithTwist}{}
Let $X$ be a [[compact topological space|compact]] [[manifold]] with [[tangent bundle]] $\tau_X$ and let $c \in H^3(X,\mathbb{Z})$ be a [[twisted K-theory|twist]]. Then the [[C\emph{-algebra]] $C_{c}(X)$ of def. \ref{CStarAlgebraOf2BundleOnManifold} has a [[dual object]] in the [[full subcategory]] of [[KK-theory]] on [[separable C}-algebras]], given by

\begin{displaymath}
(C_c(X))^\vee
  \simeq
  C_{\frac{1}{c\otimes W_3(\tau_X)}}(X)
  \,,
\end{displaymath}
hence by the same manifold but with twist the inverse of the third [[integral Stiefel-Whitney class]] and the original twist.

The same remains true in $G$-equivariant KK-theory, for $G$ a [[locally compact topological space|locally compact]] [[topological group]].

\end{prop}
The non-equivariant case is in (\hyperlink{BrodzkiMathaiRosenbergSzabo06}{Brodzki-Mathai-Rosenberg-Szabo 06, section 7.3}) and the generalization to the equivariant case in (\hyperlink{Tu06}{Tu 06, theorem 3.1}) (where we use remark \ref{OppositeConvolutionAlgebraAndInverseTwist} in order to interpret the opposite twisted convolution algebra up to equivalence as inducing the inverse twist).

\hypertarget{PropertiesKOrientationAndUmkehrMaps}{}\paragraph*{{K-Orientation and Umkehr maps}}\label{PropertiesKOrientationAndUmkehrMaps}

We discuss [[Umkehr maps]]/[[fiber integration in generalized cohomology]] in [[K-theory]] using Poincar\'e{} duality algebras / [[dual objects]] in [[KK-theory]].

\begin{prop}
\label{}\hypertarget{}{}
Every homomorphism $f \colon A \to B$ between PD $C^\ast$-algebras is [[K-orientation|K-orientable in KK-theory]]. The [[K-orientation]] is given by the corresponding [[dual morphism]], hence the element $f! \colon B \to A$ given as the composite

\begin{displaymath}
f!
  \coloneqq
  \Delta^\vee_A \otimes_{A^{op}} f^{op} \otimes_{B^{op}} \Delta_B
  \,.
\end{displaymath}
\end{prop}
(\hyperlink{BMRS07}{BMRS 07, 3.3})

More generally we have the following.

\begin{example}
\label{PushAlongInclusionOfManifoldsAndDBraneCharge}\hypertarget{PushAlongInclusionOfManifoldsAndDBraneCharge}{}
Let $i \colon Q \to X$ be a map of [[compact topological space|compact]] [[manifolds]] and let $\chi \colon X \to B^2 U(1)$ modulate a [[circle 2-bundle]] regarded as a [[twisted K-theory|twist for K-theory]]. Then forming [[twisted groupoid convolution algebras]] yields a [[KK-theory]] morphism of the form

\begin{displaymath}
C_{i^\ast \chi}(Q)
  \stackrel{i^\ast}{\longleftarrow}
  C_{\chi}(X)
  \,,
\end{displaymath}
with notation as in def. \ref{CStarAlgebraOf2BundleOnManifold}. By prop. \ref{DualOfCompactManifoldWithTwist} the [[dual morphism]] is of the form

\begin{displaymath}
C_{\frac{1}{i^\ast \chi \otimes W_3(T Q)}}(Q)
  \stackrel{i_!}{\longrightarrow}
  C_{\frac{1}{\chi \otimes W_3(T X)}}(X)
  \,.
\end{displaymath}
If we redefine the twist on $X$ to absorb this ``quantum correction'' as $\chi \mapsto \frac{1}{\chi \otimes W_3(T X)}$ then this is

\begin{displaymath}
C_{i^\ast \chi\frac{W_3(i^\ast T X)}{W_3(T Q)}}(Q)
  \stackrel{i_!}{\longrightarrow}
  C_{\chi}(X)
  \,,
\end{displaymath}
Postcomposition with this map in [[KK-theory]] now yields a map from the $i^\ast \chi \otimes W_3(N Q)$-[[twisted K-theory]] of $Q$ to the $\chi$-[[twisted K-theory]] of $X$:

\begin{displaymath}
i_! \colon K_{\bullet + W_3(N Q) + i^\ast \chi}(Q) \to K_{\bullet +\chi}
  \,.
\end{displaymath}
This is the \emph{[[twisted Umkehr map]]} in this context.

If we here think of $i \colon Q \hookrightarrow X$ as being the inclusion of a [[D-brane]] [[worldvolume]], then $\chi$ would be the class of the [[background gauge field|background]] [[B-field]] and an element

\begin{displaymath}
[\xi] \in K_{\bullet + W_3(N Q) + i^\ast \chi}(Q)
\end{displaymath}
is called (the K-class of) a \emph{[[Chan-Paton gauge field]]} on the D-brane satisfying the [[Freed-Witten-Kapustin anomaly cancellation]] mechanism. (The orginal [[Freed-Witten anomaly cancellation]] assumes $\xi$ given by a [[twisted unitary bundle|twisted line bundle]] in which case it exhibits a [[twisted spin{\tt \symbol{94}}c structure]] on $Q$.) Finally its [[fiber integration|push-forward]]

\begin{displaymath}
[i_! \xi] \in K_{\bullet- \chi}(X)
\end{displaymath}
is called the corresponding \emph{[[D-brane charge]]}.

\end{example}
See (\hyperlink{Nuiten13}{Nuiten 13}).

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{example}
\label{}\hypertarget{}{}
For $A = C_0(X)$ the [[algebra of functions]] on a [[compact topological space|compact]] [[complex manifold]] $X$, then $A$ is a PD algebra with fundamental class $\Delta$ in [[K-homology]] given by the [[Dolbeault operator]] on $X \times X$.

\end{example}
(\hyperlink{BMRS07}{BMRS 07, example 3.2})

\begin{example}
\label{}\hypertarget{}{}
For $A = C_0(X)$ the [[algebra of functions]] [[vanishing at infinity]] of a [[manifold]] $X$ with [[spin{\tt \symbol{94}}c structure]]. Take $B = C_0(T^\ast X) \simeq_{KK} A^{op} \simeq A$. Then $\Delta$ constructed from the [[Dirac operator]] on the [[Clifford algebra bundle]] over $T^\ast X$ is a fundamental class.

\end{example}
(\hyperlink{BMRS07}{BMRS 07, proof of theorem 2.9})

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

\begin{itemize}%
\item [[fundamental class]]


\item [[virtual fundamental class]]


\item [[Poincaré duality complex]]



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

\hypertarget{for_graded_associative_algebras}{}\subsubsection*{{For graded associative algebras}}\label{for_graded_associative_algebras}

\begin{itemize}%
\item Pascal Lambrechts, Don Stanley, \emph{Poincar\'e{} duality and commutative differential graded algebras} (\href{http://arxiv.org/abs/math/0701309}{arXiv:math/0701309})

\end{itemize}
\hypertarget{for_algebras_3}{}\subsubsection*{{For $C^\ast$-algebras}}\label{for_algebras_3}

For [[C\emph{-algebras]]/in [[noncommutative topology]]:}

\begin{itemize}%
\item [[Henri Moscovici]], \emph{Eigenvalue inequalities and Poincar\'e{} duality in noncommutative geometry}, Commun. Math. Phys. 184 , 3 (1997) 619

\end{itemize}
Chapter 6.4 $\beta$ (starting p. 601) in

\begin{itemize}%
\item [[Alain Connes]], \emph{[[Noncommutative Geometry]]}

\end{itemize}
Def. 2.1 in

\begin{itemize}%
\item Jacek Brodzki, [[Varghese Mathai]], [[Jonathan Rosenberg]], [[Richard Szabo]], \emph{Noncommutative correspondences, duality and D-branes in bivariant K-theory}, Adv. Theor. Math. Phys.13:497-552,2009 (\href{http://arxiv.org/abs/0708.2648}{arXiv:0708.2648})

\end{itemize}
Duality including the [[twisted K-theory]] induced by [[twisted spin{\tt \symbol{94}}c structure]] over [[manifolds]] is discussed in section 7 of

\begin{itemize}%
\item Jacek Brodzki, [[Varghese Mathai]], [[Jonathan Rosenberg]], [[Richard Szabo]], \emph{D-Branes, RR-Fields and Duality on Noncommutative Manifolds}, Commun. Math. Phys. 277:643-706,2008 (\href{http://arxiv.org/abs/hep-th/0607020}{arXiv:hep-th/0607020})

\end{itemize}
and generalized to equivariant [[KK-theory]] in

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

\end{itemize}
More on dual objects in KK is in

\begin{itemize}%
\item [[Heath Emerson]], [[Ralf Meyer]], \emph{Bivariant K-theory via correspondences}, Adv. Math. 225 (2010), 2883-2919 (\href{http://arxiv.org/abs/0812.4949}{arXiv:0812.4949})


\item [[Heath Emerson]], [[Ralf Meyer]], \emph{Dualities in equivariant Kasparov theory} (\href{http://arxiv.org/abs/0711.0025}{arXiv:0711.0025})



\end{itemize}
Discussion of the [[twisted Umkehr map]] and the [[Freed-Witten-Kapustin anomaly]] in this context is in

\begin{itemize}%
\item [[Joost Nuiten]], \emph{[[schreiber:master thesis Nuiten|Cohomological quantization of local prequantum boundary field theory]]}, master thesis, August 2013

\end{itemize}
[[!redirects Poincaré duality algebras]]

[[!redirects Poincare duality algebra]] [[!redirects Poincare duality algebras]]

[[!redirects Poincaré duality C\emph{-algebra]] [[!redirects Poincaré duality C}-algebras]]

[[!redirects Poincare duality C\emph{-algebra]] [[!redirects Poincare duality C}-algebras]]



\end{document}