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\begin{document}

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\section*{KK-theory}

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

\hypertarget{index_theory}{}\paragraph*{{Index theory}}\label{index_theory}

[[!include index theory - contents]]

\hypertarget{cohomology}{}\paragraph*{{Cohomology}}\label{cohomology}

[[!include cohomology - contents]]

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

[[!include AQFT and operator algebra contents]]

\hypertarget{functional_analysis}{}\paragraph*{{Functional analysis}}\label{functional_analysis}

[[!include functional analysis - contents]]

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

[[!include noncommutative geometry - contents]]

\hypertarget{motivic_cohomology}{}\paragraph*{{Motivic cohomology}}\label{motivic_cohomology}

[[!include motivic cohomology - 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{InTermsOfFredholmHilbertBimodules}{In terms of Fredholm-Hilbert $C^\ast$-bimodules}\dotfill \pageref*{InTermsOfFredholmHilbertBimodules} \linebreak
\noindent\hyperlink{UniversalCharacterization}{Universal category-theoretic characterization}\dotfill \pageref*{UniversalCharacterization} \linebreak
\noindent\hyperlink{RelationToHomotopyClassesOfStarHomomorphisms}{In terms of homotopy-classes of $\ast$-homomorphisms}\dotfill \pageref*{RelationToHomotopyClassesOfStarHomomorphisms} \linebreak
\noindent\hyperlink{InTermsOfCorrespondences}{In terms of correspondences/spans of groupoids}\dotfill \pageref*{InTermsOfCorrespondences} \linebreak
\noindent\hyperlink{AsAnAnalogOfMotives}{As an analog of motives in noncommutative topology}\dotfill \pageref*{AsAnAnalogOfMotives} \linebreak
\noindent\hyperlink{EquivariantKKTheory}{Equivariant KK-theory}\dotfill \pageref*{EquivariantKKTheory} \linebreak
\noindent\hyperlink{Examples}{Examples}\dotfill \pageref*{Examples} \linebreak
\noindent\hyperlink{basic_examples}{Basic examples}\dotfill \pageref*{basic_examples} \linebreak
\noindent\hyperlink{the_archetypical_examples}{The archetypical examples}\dotfill \pageref*{the_archetypical_examples} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{RelationToKCohomologyAndTwistedKTheory}{Relation to operator K-cohomology, K-homology, twisted K-theory}\dotfill \pageref*{RelationToKCohomologyAndTwistedKTheory} \linebreak
\noindent\hyperlink{relation_to_extensions}{Relation to extensions}\dotfill \pageref*{relation_to_extensions} \linebreak
\noindent\hyperlink{TriangulatedAndSpectrumEnrichedStructure}{Triangulated structure and $KU$-module structure}\dotfill \pageref*{TriangulatedAndSpectrumEnrichedStructure} \linebreak
\noindent\hyperlink{knneth_theorem}{K\"u{}nneth theorem}\dotfill \pageref*{knneth_theorem} \linebreak
\noindent\hyperlink{excision_and_relation_to_etheory}{Excision and relation to E-theory}\dotfill \pageref*{excision_and_relation_to_etheory} \linebreak
\noindent\hyperlink{PoincareDualityAndThomIsomorphism}{Poincar\'e{} duality and Thom isomorphism}\dotfill \pageref*{PoincareDualityAndThomIsomorphism} \linebreak
\noindent\hyperlink{UmkehrMap}{Push-forward in KK-theory}\dotfill \pageref*{UmkehrMap} \linebreak
\noindent\hyperlink{further_theorems}{Further Theorems}\dotfill \pageref*{further_theorems} \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{excision}{Excision}\dotfill \pageref*{excision} \linebreak
\noindent\hyperlink{in_category_theory_and_homotopy_theory}{In Category theory and Homotopy theory}\dotfill \pageref*{in_category_theory_and_homotopy_theory} \linebreak
\noindent\hyperlink{in_the_context_of_the_novikov_conjecture}{In the context of the Novikov conjecture}\dotfill \pageref*{in_the_context_of_the_novikov_conjecture} \linebreak
\noindent\hyperlink{in_the_context_of_the_atiyahsinger_index_theorem}{In the context of the Atiyah-Singer index theorem}\dotfill \pageref*{in_the_context_of_the_atiyahsinger_index_theorem} \linebreak
\noindent\hyperlink{for_convolution_algebras_and_in_geometric_quantization}{For convolution algebras and In geometric quantization}\dotfill \pageref*{for_convolution_algebras_and_in_geometric_quantization} \linebreak
\noindent\hyperlink{ReferencesInTermsOfCorrespondences}{In terms of correspondences/spans}\dotfill \pageref*{ReferencesInTermsOfCorrespondences} \linebreak
\noindent\hyperlink{for_plain_kktheory}{For plain KK-theory}\dotfill \pageref*{for_plain_kktheory} \linebreak
\noindent\hyperlink{ReferencesCorrespondencesEquivariant}{For equivariant KK-theory}\dotfill \pageref*{ReferencesCorrespondencesEquivariant} \linebreak
\noindent\hyperlink{RelationToMotives}{Relation to motives and algebraic KK-theory}\dotfill \pageref*{RelationToMotives} \linebreak
\noindent\hyperlink{in_dbrane_theory}{In D-brane theory}\dotfill \pageref*{in_dbrane_theory} \linebreak
\noindent\hyperlink{smooth_refinement_and_spectral_triples}{Smooth refinement and spectral triples}\dotfill \pageref*{smooth_refinement_and_spectral_triples} \linebreak


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

\emph{KK-theory} is a ``[[bivariant cohomology theory|bivariant]]'' joint generalization of [[operator K-theory]] and [[K-homology]]: for $A, B$ two [[C\emph{-algebras]], the \emph{KK-group} $KK(A,B)$ is a natural [[homotopy]] [[equivalence class]] of $(A,B)$-[[Hilbert bimodules]] equipped with an additional left weak [[Fredholm module]] structure. These KK-groups $KK(A,B)$ behave in the first argument as [[K-homology]] of $A$ and in the second as [[K-cohomology]]/[[operator K-theory]] of $B$.}

Abstractly, KK-theory is an [[additive category]] of [[C\emph{-algebras]] which is the split-[[exact functor|exact]] and [[homotopy]]-invariant [[localization]] of [[C}Alg]] at the [[compact operators]]. Hence, abstractly KK-theory is a fundamental notion in [[noncommutative topology]], but its standard presentation by [[Fredholm module|Fredolm]]-[[Hilbert bimodules]] as above is rooted in [[functional analysis|functional]] [[analysis]]. A slight variant of this localization process is called \emph{[[E-theory]]}.

Due to this joint root in [[functional analysis]] and ([[noncommutative topology|noncommutative]]) [[cohomology]]/[[homotopy theory]] (``[[noncommutative stable homotopy theory]]''), KK-theory is a natural home of [[index theory]], for [[elliptic operators]] on [[smooth manifolds]] as well as for their generalization to [[equivariant cohomology|equivariant]] situations, to [[foliations]] and generally to [[Lie groupoid]]-theory (via their [[groupoid convolution C\emph{-algebras]]) and [[noncommutative geometry]].}

As a special case of this, [[quantization]] in its incarnation as [[geometric quantization by push-forward]] has been argued to naturally proceed by [[index theory]] in KK-theory (\hyperlink{Landsman03}{Landsman 03}, \hyperlink{Bos07}{Bos 07}). Also the coupling of [[D-branes]] and their [[Chan-Paton bundles]] in [[twisted K-theory]] with [[RR-charge]] in [[string theory]] is naturally captured by the coupling between [[K-homology]] and [[K-cohomology]] in KK-theory (e.g. \hyperlink{Szabo}{Szabo 08}).

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

We state first the original and standard definition of $KK$-groups in terms of [[equivalence classes]] of [[Fredholm module|Fredholm]]-[[Hilbert C\emph{-module|Hilbert C}]]-[[Hilbert bimodules|bimodules]] in

\begin{itemize}%
\item \hyperlink{InTermsOfFredholmHilbertBimodules}{In terms of Fredholm Hilbert C-star-bimodules}

\end{itemize}
Then we state the abstract [[category theory|category-theoretic]] characterization by [[localization]] in

\begin{itemize}%
\item \emph{\hyperlink{UniversalCharacterization}{Universal category-theoretic characterization}.}

\end{itemize}
An equivalent and explicity [[homotopy theory|homotopy theoretic]] characterization akin to that of the standard [[homotopy category]] [[Ho(Top)]] is in

\begin{itemize}%
\item \hyperlink{RelationToHomotopyClassesOfStarHomomorphisms}{In terms of homotopy classes of star-homomorphisms}.

\end{itemize}
\hypertarget{InTermsOfFredholmHilbertBimodules}{}\subsubsection*{{In terms of Fredholm-Hilbert $C^\ast$-bimodules}}\label{InTermsOfFredholmHilbertBimodules}

\begin{defn}
\label{}\hypertarget{}{}
In all of the following, ``$C^\ast$-algebra'' means \emph{[[separable topological space|separable]]} [[C\emph{-algebra]]. We write [[C}Alg]] for for the [[category]] whose [[objects]] are separable $C^\ast$-algebras and whose [[morphisms]] are $\ast$-[[homomorphisms]] between these.

\end{defn}
\begin{example}
\label{}\hypertarget{}{}
We write

\begin{itemize}%
\item $\mathcal{B} \coloneqq \mathcal{B}(H)$ for the $C^\ast$-algebra of [[bounded operators]] on a complex, infinite-dimensional separable [[Hilbert space]];


\item $\mathcal{K} \coloneqq \mathcal{K}(H) \hookrightarrow \mathcal{B}(H)$ for the [[compact operators]].



\end{itemize}
\end{example}
\begin{defn}
\label{HilbertCStarModule}\hypertarget{HilbertCStarModule}{}
For $B \in$ [[C\emph{Alg]], a [[Hilbert C}-module]] over $B$ is

\begin{enumerate}%
\item a [[complex numbers|complex]] [[vector space]] $\mathcal{H}$;


\item equipped with a [[C\emph{-representation]] of $B$ from the right;}


\item equipped with a [[sesquilinear map]] (linear in the second argument)

\begin{displaymath}
\langle -,-\rangle \colon \mathcal{H} \times \mathcal{H} \to B
\end{displaymath}
(the $B$-valued [[inner product]])



\end{enumerate}
such that

\begin{enumerate}%
\item $\langle -,-\rangle$ behaves indeed like a positive definitine inner product over $B$:

\begin{enumerate}%
\item $\langle x,y\rangle^\ast = \langle y,x\rangle$


\item $\langle x,x\rangle \geq 0$ (in the sense of [[positive elements]] in $B$)


\item $\langle x,x\rangle = 0$ precisely if $x = 0$;


\item $\langle x,y \cdot b\rangle = \langle x,y \rangle \cdot b$



\end{enumerate}

\item $H$ is [[complete space|complete]] with respect to the [[norm]]:

${\Vert x \Vert_H} \coloneqq {\Vert \langle x,x\rangle\Vert_B}$.



\end{enumerate}
\end{defn}
\begin{defn}
\label{HilbertBimodule}\hypertarget{HilbertBimodule}{}
For $A,B \in C^\ast Alg$ an $(A,B)$-[[Hilbert C\emph{-bimodule]] is an $B$-[[Hilbert C}-module]], def. \ref{HilbertCStarModule} $(\mathcal{H}, \langle \rangle)$ equipped with a [[C-star representation]] of $A$ from the left such that all $a \in A$ are ``adjointable'' in the $B$-valued inner product, meaning that

\begin{displaymath}
\langle a^\ast \cdot x,y\rangle = \langle x, a y\rangle
  \,.
\end{displaymath}
\end{defn}
\begin{defn}
\label{}\hypertarget{}{}
For $A, B \in$ [[C\emph{Alg]], \textbf{Kasparov $(A,B)$-bimodule} is a $\mathbb{Z}_2$-[[graded vector space|graded]] $(A,B)$-[[Hilbert bimodules]] $\mathcal{H}, \langle -,-\rangle$, def. \ref{HilbertBimodule}, equipped with an adjointable odd-graded [[bounded operator]] $F \in \mathcal{B}_B(\mathcal{H})$ such that}

\begin{enumerate}%
\item $(F^2 - 1)\pi(a) \in \mathcal{K}_B(\mathcal{H})$


\item $[F, \pi(a)] \in \mathcal{K}_B(\mathcal{H})$


\item $(F - F^\ast) \pi(a)\in \mathcal{K}_B(\mathcal{H})$



\end{enumerate}
for all $a \in A$,

hence such that $F$ squares to the identity, commutes with multiplication operators and is self-adjoint \emph{up to} [[compact operators]].

\end{defn}
For instance (\hyperlink{Blackadar99}{Blackadar 99, p. 144}).

\begin{example}
\label{}\hypertarget{}{}
For $B = \mathbb{C}$ a Kasparov $(A,B)$-bimodule is equivalently an $A$-[[Fredholm module]] for an essentially self-adjoint [[Fredholm operator]]

\end{example}
\begin{defn}
\label{HomotopyOfKasparovBimodules}\hypertarget{HomotopyOfKasparovBimodules}{}
A [[homotopy]] between two Kasparov $(A,B)$-bimodules is an $(A, C([0,1],B))$-bimodule which interpolates between the two.

(\ldots{})

\end{defn}
\begin{defn}
\label{}\hypertarget{}{}
Writes $KK(A,B)$ for the set of [[equivalence classes]] of Kasparov $(A,B)$-bimodules under homotopy, def. \ref{HomotopyOfKasparovBimodules}.

\end{defn}
\begin{prop}
\label{}\hypertarget{}{}
$KK(A,B)$ is naturally an [[abelian group]] under [[direct sum]] of bimodules and operators.

\end{prop}
\begin{prop}
\label{KasparovProduct}\hypertarget{KasparovProduct}{}
There is a composition operation

\begin{displaymath}
KK(A,B) \times KK(B,C) \to KK(A,C)
\end{displaymath}
such that (\ldots{}). This is called the \textbf{Kasparov product}.

\end{prop}
A streamlined version of the definition of the Kasparov product is in (\hyperlink{Skandalis84}{Skandalis 84}).

\begin{remark}
\label{}\hypertarget{}{}
From the point of view of [[E-theory]] the Kasparov product is equivalently just the composition of homotopy classes of completely poistive [[asymptotic C\emph{-homomorphisms]]. See at \emph{[[E-theory]]} for more on this.}

\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
On the other hand, at least between $C^\ast$-algebras which are [[algebras of functions]] on [[smooth manifolds]] $A_i = C(X_i)$ , KK-classes are presented by [[correspondences]] $X_1 \leftarrow Z \to X_2$ and the Kasparov product is given just by the [[fiber product]]-composition operation on correspondences (\hyperlink{ConnesSkandalis84}{Connes-Skandalis 84, theorem 3.2}, \hyperlink{BlockWeinberger99}{Block-Weinberger 99, section 3}).

\end{remark}
\hypertarget{UniversalCharacterization}{}\subsubsection*{{Universal category-theoretic characterization}}\label{UniversalCharacterization}

\begin{prop}
\label{KasparovProductIsAssociative}\hypertarget{KasparovProductIsAssociative}{}
The Kasparov product, def. \ref{KasparovProduct}, is [[associativity|associative]]. Thus under the Kasparov product

\begin{displaymath}
KK(-,-) \;\colon\; C^\ast Alg \times C^\ast Alg \to C^\ast Alg
\end{displaymath}
is the [[hom-functor]] of an [[additive category]].

\end{prop}
(\hyperlink{Higson}{Higson 87, theorem 4.1})

The category $KK$ is a kind of [[localization]] of the category of [[C-star-algebras]]:

\begin{theorem}
\label{KKIsAdditiveSplitExactLocalizationAtCompacts}\hypertarget{KKIsAdditiveSplitExactLocalizationAtCompacts}{}
The canonical functor

\begin{displaymath}
Q \colon C^\ast Alg \to KK
\end{displaymath}
exhibits $KK$ as the [[universal property|universal category]] receiving a functor from [[C\emph{-algebras]] such that}

\begin{enumerate}%
\item $KK$ is an [[additive category]];


\item $Q$ is homotopy-invariant;


\item $Q$ [[isomorphism|inverts]] the [[tensor product]] with the [[C\emph{-algebra]] of [[compact operators]]}

(for all $C^\ast$-homomorphisms of the form $id \otimes e \langle e,- \rangle \;\colon A\; \to A \otimes \mathcal{K}$ the morphism $Q(id \otimes e \langle e)$ is an [[isomorphism]]).


\item $Q$ preserves [[split short exact sequences]].



\end{enumerate}
\end{theorem}
This is due to (\hyperlink{Higson}{Higson 87, theorem 4.5}). The generalization to the equivariant case is due to (\hyperlink{Thomsen98}{Thomsen 98}).

\begin{remark}
\label{}\hypertarget{}{}
The localization conditions here are analogous to those that define the localization of [[stable (∞,1)-categories]] to [[noncommutative motives]] (see there for more).

\end{remark}
\begin{cor}
\label{}\hypertarget{}{}
The minimal [[tensor product of C-star-algebras]]

\begin{displaymath}
\otimes \colon C^\ast Alg \times C^\ast Alg \to C^\ast Alg
\end{displaymath}
extends uniquely to a [[tensor product]] $\otimes_{KK}$ on $KK$ such that there is a [[commuting diagram]] of [[functors]]

\begin{displaymath}
\itexarray{
    C^\ast Alg \times C^\ast Alg &\stackrel{Q \times Q}{\to}& KK
    \\
    \downarrow^{\mathrlap{\otimes}} && \downarrow^{\mathrlap{\otimes_{KK}}}
    \\
    C^\ast Alg &\stackrel{Q}{\to}& KK
  }
  \,.
\end{displaymath}
\end{cor}
(\hyperlink{Higson}{Higson 87, theorem 4.8})

For more discussion of more explicit presentations of this [[localization]] process for obtaining KK-theory see at \emph{[[homotopical structure on C\emph{-algebras]]\_ and also at \emph{[[model structure on operator algebras]]}.}}

\hypertarget{RelationToHomotopyClassesOfStarHomomorphisms}{}\subsubsection*{{In terms of homotopy-classes of $\ast$-homomorphisms}}\label{RelationToHomotopyClassesOfStarHomomorphisms}

Theorem (Cuntz)

If $A,B$ are [[C-star-algebras]] with $A$ separable and $B$ $\sigma$-unital, then

\begin{displaymath}
KK(A,B) \simeq [q A, B \otimes \mathcal{K}]
  \,,
\end{displaymath}
where

\begin{itemize}%
\item $q A$ is the [[kernel]] of the [[codiagonal]] $A \star A \to A$,


\item $\mathcal{K}$ is the $C^\ast$-algebra of [[compact operators]].


\item $[-,-]$ is the set of [[homotopy]] [[equivalence classes]] of $\ast$-[[homomorphisms]].



\end{itemize}
(reviewed in (\hyperlink{JoachimJohnson07}{Joachim-Johnson07})).

\hypertarget{InTermsOfCorrespondences}{}\subsubsection*{{In terms of correspondences/spans of groupoids}}\label{InTermsOfCorrespondences}

At least to some extent, KK-classes between [[C\emph{-algebras]] of [[continuous functions]] on manifolds/spaces, and maybe more generally between [[groupoid convolution algebras]] can be represented by certain [[equivalence classes]] of [[spans]]/[[correspondences]]}

\begin{displaymath}
X \leftarrow (Z,E) \to Y
\end{displaymath}
of such [[spaces]].

See the corresponding \hyperlink{ReferencesInTermsOfCorrespondences}{references below}.

Such a description by abelianizations of [[correspondences]] is reminiscent of similar constructions of [[motivic cohomology]]. See \hyperlink{AsAnAnalogOfMotives}{below}. For more on this see also the pointers at at \emph{[[motivic quantization]]}.

(\ldots{})

\begin{itemize}%
\item category of equivariant correspondences equipped with cocycle: $\hat F_{\mathcal{G}}^\ast$ (theorem 2.26);


\item specifically for K-theory cocycles: $\widehat {KK}_{\mathcal{G}}^\ast$ (section 4, page 27)


\item pull-push from correspondences to KK in proof of theorem 4.2, bottom of p. 27



\end{itemize}
(\ldots{})

\hypertarget{AsAnAnalogOfMotives}{}\subsubsection*{{As an analog of motives in noncommutative topology}}\label{AsAnAnalogOfMotives}

To some extent [[KK-theory]]/[[E-theory]] look like an analogue in [[noncommutative topology]] of what in [[algebraic geometry]] is the category of [[motives]]. (\hyperlink{ConnesConsaniMarcolli05}{Connes-Consani-Marcolli 05}). (\hyperlink{Meyer06}{Meyer 06}).

Specifically the characterization in terms of spans/correspondences \hyperlink{InTermsOfCorrespondences}{above} is reminiscent to the definition of [[pure motives]], see the rferences below: \emph{\hyperlink{ReferencesInTermsOfCorrespondences}{In terms of correspondences}. A relation between bivariant [[algebraic K-theory]] and [[motivic cohomology]] is discussed in (\hyperlink{GarkushaPanin11}{Garkusha-Panin 11}).}

A universal [[functor]] from KK-theory to [[noncommutative motives]]

\begin{displaymath}
KK \longrightarrow NCC_{dg}
\end{displaymath}
was given in (\hyperlink{Mahanta13}{Mahanta 13}). This sends a [[C\emph{-algebra]] to the [[dg-category]] of [[perfect complexes]] over (the [[unitalization]] of) its underlying [[associative algebra]].}

\hypertarget{EquivariantKKTheory}{}\subsubsection*{{Equivariant KK-theory}}\label{EquivariantKKTheory}

Pretty much all of KK-theory has a generalization to [[equivariant cohomology]] where all algebras and modules are equipped with [[actions]] of a given [[topological group]] or more generally [[topological groupoid]] $\mathcal{G}$, and all operators are suitably [[invariant|invariant]]/[[equivariance|equivariant]] under this action. See at \emph{[[equivariant KK-theory]]} for more.

The [[Baum-Connes conjecture]] and the [[Green-Julg theorem]] assert that under some conditions $\mathcal{G}$-equivariant KK-theory is equivalent to the plain KK-theory of the [[groupoid convolution algebras]] of the corresponding [[action groupoids]]. See at \emph{[[Green-Julg theorem]]} for details.

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

\hypertarget{basic_examples}{}\subsubsection*{{Basic examples}}\label{basic_examples}

\begin{example}
\label{}\hypertarget{}{}
For $f \colon A \to B$ a [[homomorphism]] of $\mathbb{Z}_2$ [[graded C\emph{-algebras]], take $B$ as a right [[Hilbert module]] over itself and equip it with the left [[action]] of $A$ induced by $f$. This makes it a [[Hilbert bimodule]]. Together with the 0-[[Fredholm operator]], this represents an element}

\begin{displaymath}
(B, f, 0) \in KK(A,B)
  \,.
\end{displaymath}
\end{example}
For instance (\hyperlink{Blackadar99}{Blackadar 99, example 17.1.2 a)}).

\begin{example}
\label{}\hypertarget{}{}
For

\begin{displaymath}
(H_i, F_i) \in KK(A,B)
\end{displaymath}
a [[Fredholm module|Fredholm]] $(A_i,B)$-[[Hilbert bimodule]] for $i \in \{1,2\}$, the [[direct sum]] is

\begin{displaymath}
(H_1 \oplus H_2, F_1 \oplus F_2) \in KK(A_1\oplus A_2, B)
  \,.
\end{displaymath}
\end{example}
For instance (\hyperlink{Blackadar99}{Blackadar 99, example 17.1.2 c)}).

\hypertarget{the_archetypical_examples}{}\subsubsection*{{The archetypical examples}}\label{the_archetypical_examples}

\begin{example}
\label{}\hypertarget{}{}
Let $(X,g)$ be a [[closed manifold|closed]] [[smooth manifold|smooth]] [[Riemannian manifold]], and let $V_0, V_1$ be two smooth [[vector bundles]] over $X$ with Hermitian strucure ([[associated bundle|associated]] to a chosen [[unitary group]]-[[principal bundle]]).

Then given an [[elliptic operator|elliptic]] [[pseudodifferential operator]]

\begin{displaymath}
P \colon \Gamma(V_0) \to \Gamma(V_1)
\end{displaymath}
on smooth [[sections]] it extends to an essentially [[unitary operator|unitary]] [[Fredholm operator]] on [[square integrable function|square integrable]] sections $L^2(V_i)$.

Consider then the $\mathbb{Z}_2$-graded [[Hilbert space]]

\begin{displaymath}
H \coloneqq L^2(V_0) \oplus L^2(V_1)
\end{displaymath}
equipped with the evident action of $C(X)$ (by ``[[multiplication operators]]''). Then with $P$ a [[parametrix]] for $Q$, the operator

\begin{displaymath}
F \coloneqq 
  \left[
    \itexarray{
       0 & Q
       \\
       P & 0
    }
  \right]
\end{displaymath}
is a [[Fredholm operator]] on $H$, so that

\begin{displaymath}
\left(
    L^2(V_1) \oplus L^2(V_2), 
  \left[
    \itexarray{
       0 & Q
       \\
       P & 0
    }
  \right]
  \right)
  \in 
  KK(C(X),\mathbb{C})
  \,.
\end{displaymath}
\end{example}
\begin{example}
\label{}\hypertarget{}{}
Let $(X,g)$ be an [[almost complex manifold]] and let $D \colon \overline{\partial} + \overline{\partial}^\ast$ be the [[Dolbeault-Dirac operator]]. This extends to an operator on

\begin{displaymath}
H \coloneqq L^2(\Omega^{0,\bullet})
\end{displaymath}
and

\begin{displaymath}
F \coloneqq \frac{D}{\sqrt{1 + D^2}}
\end{displaymath}
(defined by [[functional calculus]]) is then a [[Fredholm operator]] on that. Then

\begin{displaymath}
\left(
    L^2(\Omega^{0,\bullet}),
    \frac{\overline{\partial} + \overline{\partial}^\ast}{\sqrt{1+ (\overline{\partial} + \overline{\partial}^\ast)^2}}
  \right)
  \in
  KK(C(X), \mathbb{C})
  \,.
\end{displaymath}
\end{example}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{RelationToKCohomologyAndTwistedKTheory}{}\subsubsection*{{Relation to operator K-cohomology, K-homology, twisted K-theory}}\label{RelationToKCohomologyAndTwistedKTheory}

KK-theory is a joint generalization of [[operator K-theory]], hence also of [[topological K-theory]], as well as of [[K-homology]] and of [[twisted K-theory]].

For $A \in$ [[C\emph{Alg]] we have that}

\begin{itemize}%
\item $KK(\mathbb{C}, A) \simeq K_0(A)$

\end{itemize}
is the [[operator K-theory]] group of $A$ in degree 0 and

\begin{itemize}%
\item $KK(C(\mathbb{R}^1),A) \simeq K_1(A)$

\end{itemize}
is the [[operator K-theory]] group of $A$ in degree 1. (e.g. (\hyperlink{Introduction}{Introduction, p. 20}). If here $A = C(X)$ is the [[C\emph{-algebra]] of functions on a suitable [[topological space]] $X$, then this is the [[topological K-theory]] of that space}

\begin{itemize}%
\item $KK(\mathbb{C}, C(X)) \simeq K^0(X)$


\item $KK(C(\mathbb{R}), C(X)) \simeq K^1(X)$.



\end{itemize}
More generally, if $A = C_r(\mathcal{G}_\bullet)$ is the reduced [[groupoid convolution algebra]] of a [[Lie groupoid]], then

\begin{itemize}%
\item $KK(\mathbb{C}, C_r(\mathcal{G}_\bullet)) \simeq K^0(\mathcal{G})$

\end{itemize}
is the K-theory of the corresponding [[differentiable stack]]. If moreover $c \colon \mathcal{G} \to \mathbf{B}^2 U(1)$ is a [[circle 2-group]]-[[principal 2-bundle]] ($U(1)$-[[bundle gerbe]]) over $\mathcal{X}$ and if $A = C(\mathcal{X}_\bullet, c)$ is the [[twisted groupoid convolution algebra]] of the corresponding [[centrally extended Lie groupoid]], then

\begin{itemize}%
\item $KK(\mathbb{C}, C_r(\mathcal{X}_\bullet,x)) = K^0(\mathcal{X}, c)$

\end{itemize}
is the corresponding [[twisted K-theory]] (\hyperlink{TXLG}{Tu, Xu, Laurent-Gengoux 03}).

On the other hand, with $A$ in the first argument and the complex numbers in the second we have that

\begin{itemize}%
\item $K(A,\mathbb{C}) \simeq K^0(A)$

\end{itemize}
ar equivalence classes of $A$-[[Fredholm modules]] and hence the [[K-homology]] of $A$.

(\ldots{})

\hypertarget{relation_to_extensions}{}\subsubsection*{{Relation to extensions}}\label{relation_to_extensions}

There is an [[isomorphism]]

$KK(A,B) \simeq Ext^1(A,B)$

to a suitable group of suitable [[extensions]] of $A$ by $B$. (\hyperlink{Kasparov80}{Kasparov 80}, reviewed in \hyperlink{Inassaridze}{Inassaridze}).

\hypertarget{TriangulatedAndSpectrumEnrichedStructure}{}\subsubsection*{{Triangulated structure and $KU$-module structure}}\label{TriangulatedAndSpectrumEnrichedStructure}

\begin{prop}
\label{}\hypertarget{}{}
$KK$ is naturally a stable [[triangulated category]].

\end{prop}
(\hyperlink{Meyer}{Meyer 07}, \hyperlink{Uuye10}{Uuye 10, theorem 2.29}).

\begin{prop}
\label{}\hypertarget{}{}
There is a [[functor]]

\begin{displaymath}
\mathbb{K}(-) \;\colon\; C^\ast Alg \to Ho(Spectra)
\end{displaymath}
to the [[stable homotopy category]] such that

\begin{enumerate}%
\item $\pi_n(\mathbb{K})(A) \simeq K_n(A)$, for all $A \in C^\ast Alg$, (hence the spectrum is a cohomology spectrum for the [[operator K-theory]] of $A$);


\item $\mathbb{K}(\mathbb{C})$ is naturally a [[ring spectrum]];


\item $\mathbb{K}(A)$ is naturally a [[symmetric spectrum|symmetric]] $\mathbb{K}(\mathbb{C})$-[[module spectrum]]


\item $\mathbb{K}$ lifts to a [[lax monoidal functor]]

\begin{displaymath}
\mathbb{K} \;\colon\; C^\ast Alg \to Ho(\mathbb{K}(\mathbb{C}) Mod)
\end{displaymath}
to the [[homotopy category]] of [[module spectra]], and this in turn extends to a [[lax monoidal functor]] on the KK-category

\begin{displaymath}
\mathbb{K} \;\colon\; KK \to Ho(\mathbb{K}(\mathbb{C}) Mod)
  \,.
\end{displaymath}

\item $\mathbb{K}$ restricts to a [[fully faithful functor]] on the [[thick subcategory]] of the [[triangulated category]] $KK$ generated by the [[tensor unit]] (the ``[[bootstrap category]]'').



\end{enumerate}
\end{prop}
This is the main result of (\hyperlink{DEKM11}{DEKM 11, section 3}).

\begin{remark}
\label{}\hypertarget{}{}
Since $\mathbb{K}$ is a [[lax monoidal functor]] in particular it preserves [[dual objects]] and [[dual morphisms]], hence [[Poincaré duality algebras]] and their [[Umkehr maps]].

\end{remark}
\hypertarget{knneth_theorem}{}\subsubsection*{{K\"u{}nneth theorem}}\label{knneth_theorem}

The [[thick subcategory]] of the [[triangulated category]] $KK$ generated from the [[tensor unit]] is called the \emph{[[bootstrap category]]} $Boot \hookrightarrow KK$. For $A \in Boot \hookrightarrow KK$ one has that $KK(A,B)$ satisfies a [[Künneth theorem]]. See at \emph{[[bootstrap category]]} for more.

\hypertarget{excision_and_relation_to_etheory}{}\subsubsection*{{Excision and relation to E-theory}}\label{excision_and_relation_to_etheory}

\begin{defn}
\label{}\hypertarget{}{}
Given a [[short exact sequence]] of [[C\emph{-algebras]] one says that $KK$ satisfies \textbf{[[excision]]} or that it is \textbf{excisive} for this sequence if it preserves its [[exact sequence|exactness]] in the middle.}

\end{defn}
\begin{example}
\label{}\hypertarget{}{}
By theorem \ref{KKIsAdditiveSplitExactLocalizationAtCompacts}, $KK$ is excisive over [[split exact sequences]].

\end{example}
\begin{prop}
\label{}\hypertarget{}{}
$KK$ is excisive for [[nuclear C\emph{-algebras]] in the first argument.}

\end{prop}
This is discussed (\hyperlink{Kasparov80}{Kasparov 80, section 7}), (\hyperlink{CuntzSkandalis86}{Cuntz-Skandalis 86}).

More generally:

\begin{prop}
\label{}\hypertarget{}{}
$KK$ is excisive for [[K-nuclear C\emph{-algebras]] in the first argument.}

\end{prop}
(\hyperlink{Skandalis88}{Skandalis 88})

\begin{remark}
\label{}\hypertarget{}{}
It is not expected that excision is satisfied fully generally by $KK$. Instead, the [[universal property|universal]] improvement of $KK$-theory under excision can be constructed. This is called \emph{[[E-theory]]}. See there for more.

\end{remark}
\hypertarget{PoincareDualityAndThomIsomorphism}{}\subsubsection*{{Poincar\'e{} duality and Thom isomorphism}}\label{PoincareDualityAndThomIsomorphism}

\begin{defn}
\label{PoincareDualityAlgebra}\hypertarget{PoincareDualityAlgebra}{}
A [[C\emph{-algebra]] is a [[Poincaré duality algebra]] if it is a [[dualizable object]] in the [[symmetric monoidal category]] $KK$ with dual its [[opposite algebra]].}

\end{defn}
(\hyperlink{BrodzkiMathaiRosenbergSzabo07}{Brodzki-Mathai-Rosenberg-Szabo 07, def. 2.1})

\begin{prop}
\label{PoincareDuality}\hypertarget{PoincareDuality}{}
Let $X$ be a [[smooth manifold]] which is [[compact topological space|compact]]. Then the [[C\emph{-algebra]] $C(X) \otimes C_0(T^\ast X)$ (the tensor product of the algebra of functions of [[compact support]] on $X$ and on its [[cotangent bundle]]) is isomorphic, in $KK$, to $\mathbb{C}$:}

\begin{displaymath}
d \colon C(X) \otimes C_0(T^\ast X) \stackrel{\simeq}{\to}  
  \mathbb{C}
  \,.
\end{displaymath}
\end{prop}
(\hyperlink{Kasparov80}{Kasparov 80})

\begin{cor}
\label{}\hypertarget{}{}
For $X$ a [[compact topological space|compact]] [[smooth manifold]], there is a [[natural isomorphism]] ([[Thom isomorphism]])

\begin{displaymath}
K_0( C_0(T^\ast X))
    \simeq
  KK(\mathbb{C}, C_0(T^\ast X))
    \stackrel{KK(C,(-)\otimes C(X))}{\to}
  KK(C(X), C(X) \otimes C_0(T^\ast X) )
    \underoverset{\simeq}{KK(C(X), d)}{\to}
  KK(C(X), \mathbb{C} )
  \,.
\end{displaymath}
\end{cor}
For more discussion see at \emph{[[Poincaré duality algebra]]}.

\hypertarget{UmkehrMap}{}\subsubsection*{{Push-forward in KK-theory}}\label{UmkehrMap}

[[Umkehr map]] in KK-theory (\hyperlink{BrodzkiMathaiRosenbergSzabo07}{Brodzki-Mathai-Rosenberg-Szabo 07, section 3.3})

If $A$, $B$ are [[Poincaré duality algebras]], def. \ref{PoincareDualityAlgebra}, then for $f \colon A \to B$ a morphism, the corresponding [[Umkehr map]] is (postcomposition) with the [[dual morphism]] of its [[opposite algebra]] version:

\begin{displaymath}
f! \coloneqq (f^op)^\ast
  \,.
\end{displaymath}
(\hyperlink{BrodzkiMathaiRosenbergSzabo07}{Brodzki-Mathai-Rosenberg-Szabo 07, p. 14})

For more and a discussion of [[twisted Umkehr maps]] see at \emph{[[Poincaré duality algebra]]} and at \emph{[[Freed-Witten-Kapustin anomaly cancellation]]}.

\hypertarget{further_theorems}{}\subsection*{{Further Theorems}}\label{further_theorems}

\begin{itemize}%
\item The [[Baum-Connes conjecture]] is naturally formulated within KK-theory.


\item The [[Novikov conjecture]] has been verified in many cases using KK-theory. (see for instance \hyperlink{Rosenberg80}{Rosenberg 80}).


\item The [[Atiyah-Singer index theorem]] is naturally formutaled in KK-theory/[[E-theory]]. (See (\hyperlink{HigsonRoe}{Higson-Roe})).



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

\begin{itemize}%
\item [[KK-bootstrap category]]


\item [[equivariant KK-theory]]


\item [[index theory]]


\item [[bivariant cohomology theory]]


\item [[bivariant algebraic K-theory]], [[K-motive]]


\item [[operator K-theory]]


\item [[E-theory]]


\item [[homotopical structure on C\emph{-algebras]], [[model structure on operator algebras]]}


\item [[assembly map]], [[Baum-Connes conjecture]]



\end{itemize}
[[!include noncommutative motives - table]]

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

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

KK-theory was introduced by [[Gennady Kasparov]] in

\begin{itemize}%
\item [[Gennady Kasparov]], \emph{The operator $K$-functor and extensions of $C^{\ast}$-algebras}, Izv. Akad. Nauk SSSR Ser. Mat. \textbf{44} (1980), no. 3, 571--636, 719, \href{http://www.ams.org/mathscinet-getitem?mr=582160}{MR81m:58075}, \href{http://www.zentralblatt-math.org/zmath/search/?an=Zbl%200464.46054%7CZbl%200448.46051}{Zbl}, \href{http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=im&paperid=1739&option_lang=eng}{abstract}, \href{http://dx.doi.org/10.1070%2FIM1981v016n03ABEH001320}{english doi}, free Russian original: \href{http://www.mathnet.ru/php/getFT.phtml?jrnid=im&paperid=1739&volume=44&year=1980&issue=3&fpage=571&what=fullt&option_lang=eng}{pdf}

\end{itemize}
prompted by the advances in [[Brown-Douglas-Fillmore theory]], especially in the last 1977 article.

Some streamlining of the definitions appeared in

\begin{itemize}%
\item [[Georges Skandalis]], \emph{Some remarks on Kasparov theory}, J. Funct. Anal. 59 (1984) 337-347.

\end{itemize}
A textbook account is in

\begin{itemize}%
\item [[Bruce Blackadar]], \emph{[[K-Theory for Operator Algebras]]}, 2nd ed. Cambridge University Press, Cambridge (1999)

\end{itemize}
Introductions and surveys include

\begin{itemize}%
\item [[Gennady Kasparov]], \emph{Operator K-theory and its applications: elliptic operators, group representations, higher signatures $C^\ast$-extensions}, Proceedings ICM 1983 Warszawa, PWN-Elsevier (1984) 987-1000.


\item [[Nigel Higson]], \emph{A primer on KK-theory}. Proc. Sympos. Pure Math. 51, Part 1, 239--283. (1990) (\href{http://www.math.psu.edu/higson/math/Papers_files/Higson%20-%201990%20-%20A%20primer%20on%20KK-theory.pdf}{pdf})


\item [[Georges Skandalis]], \emph{Kasparov's bivariant K-theory and applications} Exposition. Math. 9, 193--250 (1991) (\href{http://www.math.vanderbilt.edu/~bisch/ncgoa08/talks/skandalis2.pdf}{pdf slides})


\item \emph{Introduction to KK-theory and E-theory}, Lecture notes (Lisbon 2009) (\href{http://oaa.ist.utl.pt/files/cursos/courseD_Lecture4_KK_and_E1.pdf}{pdf slides})


\item [[Heath Emerson]], [[R. Meyer]] (notes taken by S. Hong), \emph{KK-theory and Baum-Connes conjecture}, Lectures at \emph{Summer school on operator algebras and noncommutative geometry} (June 2010) (\href{http://www.pims.math.ca/files/EM_%20KK-theory_BC_Conjecture.pdf}{pdf})


\item [[R. Meyer]], \emph{How analysis and topology interact in bivariant K-theory}, 2006 (\href{http://mate.dm.uba.ar/~gcorti/article.pdf}{pdf})



\end{itemize}
\hypertarget{excision}{}\subsubsection*{{Excision}}\label{excision}

Excision for KK-theory is further studied in

\begin{itemize}%
\item [[Joachim Cuntz]], [[Georges Skandalis]], \emph{Mapping cones and exact sequences in KK-theory}, J. Operator Theory 15 (1986) 163-180.


\item [[Georges Skandalis]], \emph{Une notion de nuclearit\'e{} en K-theorie}, K-Theory 1 (1988) 549-574.



\end{itemize}
\hypertarget{in_category_theory_and_homotopy_theory}{}\subsubsection*{{In Category theory and Homotopy theory}}\label{in_category_theory_and_homotopy_theory}

KK-theory is naturally understood in terms of [[universal properties]] in [[category theory]] and in [[homotopy theory]].

That $KK(A,B)$ is naturally thought of as a collection of ``generalized [[homomorphisms]]'' of $C^\ast$-algebras was amplified in

\begin{itemize}%
\item [[Joachim Cuntz]], \emph{Generalized Homomorphisms Between $C^\ast$-algebras and KK-theory,} Springer Lecture Notes in Mathematics, 1031 (1983), 31-45. doi:\href{https://doi.org/10.1007/BFb0072109}{10.1007/BFb0072109}


\item [[Joachim Cuntz]], \emph{K-theory and C\emph{-algebras,\_ Springer Lecture Notes in Mathematics, 1046 (1984), 55-79.}}



\end{itemize}
That under the Kasparov product these are indeed the [[hom-objects]] in a [[category]] was first observed in

\begin{itemize}%
\item [[Nigel Higson]], \emph{A characterization of KK-theory}, Pacific J. Math. Volume 126, Number 2 (1987), 253-276. (\href{http://projecteuclid.org/euclid.pjm/1102699804}{EUCLID})

\end{itemize}
where moreover this category is realized as the [[universal property|universal]] additive and split exact ``[[localization]]'' of $C^\ast Alg$ at the $C^\ast$-algebra of [[compact operators]].

The generalization of this statement to [[equivariant cohomology|equivariant]] KK-theory is in

\begin{itemize}%
\item Klaus Thomsen, \emph{The universal property of equivariant KK-theory} (\href{http://www.math.uiuc.edu/K-theory/0249/}{K-theory preprint})

\end{itemize}
Characterization of KK-theory as the [[satellites]] of a functor is in

\begin{itemize}%
\item [[Hvedri Inassaridze]], \emph{Universal property of Kasparov bivariant K-theory} ([[InassaridzeOnKK.pdf:file]], \href{http://www.math.uni-bielefeld.de/~bak/kasp.ps}{ps})

\end{itemize}
A [[triangulated category]] structure for KK-theory is discussed in

\begin{itemize}%
\item [[Ralf Meyer]], \emph{Categorical aspects of bivariant K-theory}, (\href{http://arxiv.org/abs/math/0702145}{arXiv:math/0702145})


\item [[Ralf Meyer]], [[Ryszard Nest]], \emph{Homological algebra in bivariant K-theory and other triangulated categories} (\href{http://arxiv.org/abs/math/0702146}{arXiv:math/0702146})


\item [[Ralf Meyer]], \emph{KK-theory as a triangulated category}, Lecture notes (2009) (\href{http://wwwmath.uni-muenster.de/u/echters/Focused-Semester/lecturenotes/Meyer_--_KK-theory_as_a_triangulated_category.pdf}{pdf})



\end{itemize}
A [[model category]] realization of KK-theory is discussed in

\begin{itemize}%
\item [[Michael Joachim]], [[Mark Johnson]], \emph{Realizing Kasparov's KK-theory groups as the homotopy classes of maps of a Quillen model category} (\href{http://arxiv.org/abs/0705.1971}{arXiv:0705.1971})

\end{itemize}
A [[category of fibrant objects]]-structure on [[C\emph{Alg]] which unifies the above homotopical pictures is discussed in}

\begin{itemize}%
\item [[Otgonbayar Uuye]], \emph{Homotopy theory for $C^\ast$-algebras} (\href{http://arxiv.org/abs/1011.2926}{arXiv:1011.2926})

\end{itemize}
More on this is at \emph{[[homotopical structure on C\emph{-algebras]]\_.}}

Further discussion in the context of [[stable homotopy theory]] and [[E-theory]] is in

\begin{itemize}%
\item Martin Grensing, \emph{Noncommutative stable homotopy theory} (\href{http://arxiv.org/abs/1302.4751}{arXiv:1302.4751})


\item [[Snigdhayan Mahanta]], \emph{Higher nonunital Quillen $K'$-theory, KK-dualities and applications to topological $\mathbb{T}$-duality}, Journal of Geometry and Physics, Volume 61, Issue 5 2011, p. 875-889. (\href{http://wwwmath.uni-muenster.de/u/snigdhayan.mahanta/papers/KQ.pdf}{pdf})



\end{itemize}
Refinement of [[operator K-theory]] to cohomology [[spectra]] is discussed in

\begin{itemize}%
\item [[Ulrich Bunke]], [[Michael Joachim]], [[Stephan Stolz]], \emph{Classifying spaces and spectra representing the K-theory of a graded $C^\ast$-algebra}, High-dimensional manifold topology, World Sci. Publ., River Edge, NJ, 2003, pp. 80--102

\end{itemize}
This construction is functorial (only) for \emph{essential} $\ast$-homomorphisms of [[C\emph{-algebras]].}

A refinement of the KK-category to a [[spectrum]]-[[enriched category]] ($\sim$ [[stable (∞,1)-category]]) is claimed in

\begin{itemize}%
\item [[Michael Joachim]], [[Stephan Stolz]], \emph{An enrichment of $KK$-theory over the category of symmetric spectra} M\"u{}nster J. of Math. 2 (2009), 143--182 (\href{http://www3.nd.edu/~stolz/KKenrich.pdf}{pdf})

\end{itemize}
and the generalization of this to [[equivariant K-theory]] over geometrically discrete groupoids is discussed in

\begin{itemize}%
\item [[Paul Mitchener]], \emph{$KK$-theory spectra for $C^\ast$-categories and discrete groupoid $C^\ast$-algebras} (\href{http://arxiv.org/abs/0711.2152}{arXiv:0711.2152})

\end{itemize}
but this construction is stated to be mistaken on p. 3 of

\begin{itemize}%
\item [[Ivo Dell'Ambrogio]], [[Heath Emerson]], [[Tamaz Kandelaki]], [[Ralf Meyer]], \emph{A functorial equivariant K-theory spectrum and an equivariant Lefschetz formula} (\href{http://arxiv.org/abs/1104.3441}{arXiv:1104.3441})

\end{itemize}
This article in turn considers a variant of the construction in (\hyperlink{BunkeJoachimStolz03}{Bunke-Joachim-Stolz 03}) which gives operator K-theory spectra that are functorial for general $\ast$-homomorphisms.

Observations relating to a genuine [[stable (∞,1)-category]] structure maybe at least of [[E-theory]] are in

\begin{itemize}%
\item [[Snigdhayan Mahanta]], \emph{Noncommutative stable homotopy and semigroup $C^*$-algebras}, ESI preprint 2394 (\href{http://arxiv.org/abs/1211.6576}{arXiv:1211.6576})

\end{itemize}
\hypertarget{in_the_context_of_the_novikov_conjecture}{}\subsubsection*{{In the context of the Novikov conjecture}}\label{in_the_context_of_the_novikov_conjecture}

\begin{itemize}%
\item [[Jonathan Rosenberg]], \emph{Group C\emph{-algebras and Topological Invariants\_ , Proc. Conf. in Neptun, Romania, 1980, Pitman (London, 1985)}}

\end{itemize}
\hypertarget{in_the_context_of_the_atiyahsinger_index_theorem}{}\subsubsection*{{In the context of the Atiyah-Singer index theorem}}\label{in_the_context_of_the_atiyahsinger_index_theorem}

The classical [[Atiyah-Singer index theorem]] is reviewed in [[operator K-theory]] (with some hints towards KK-theory) in

\begin{itemize}%
\item [[Nigel Higson]], [[John Roe]], \emph{Lectures on operator K-theory and the Atiyah-Singer Index Theorem} (2004) (\href{http://folk.uio.no/rognes/higson/Book.pdf}{pdf})

\end{itemize}
Generalization to the relative case in [[KK-theory]], hence for indices of fiberwise [[elliptic operators]] on [[Hilbert C\emph{-module]]-[[fiber bundles]] is in}

\begin{itemize}%
\item Jody Trout, \emph{Asymptotic Morphisms and Elliptic Operators over $C^\ast$-algebras}, K-theory, 18 (1999) 277-315 (\href{http://arxiv.org/abs/math/9906098}{arXiv:math/9906098})

\end{itemize}
\hypertarget{for_convolution_algebras_and_in_geometric_quantization}{}\subsubsection*{{For convolution algebras and In geometric quantization}}\label{for_convolution_algebras_and_in_geometric_quantization}

Discussion of KK-theory with an eye towards [[representation of a C-star algebra|C-star representations]] of [[groupoid convolution

algebras]] in the context of [[geometric quantization]] [[geometric quantization by push-forward|by push-forward]] is in

\begin{itemize}%
\item [[Klaas Landsman]], \emph{Quantization as a functor} (\href{http://arxiv.org/abs/math-ph/0107023}{arXiv:math-ph/0107023})


\item [[Klaas Landsman]], \emph{Functorial quantization and the Guillemin-Sternberg conjecture} , Proc. Bialowieza 2002 (\href{http://arxiv.org/abs/math-ph/0307059}{arXiv:math-ph/0307059})


\item [[Rogier Bos]], \emph{Groupoids in geometric quantization} PhD Thesis (2007) (\href{http://www.math.ist.utl.pt/~rbos/ProefschriftA4.pdf}{pdf})



\end{itemize}
with a summary/exposition in

\begin{itemize}%
\item [[Klaas Landsman]], \emph{Functoriality of quantization: a KK-theoretic approach}, talk at ECOAS, Dartmouth College, October 2010 (\href{http://www.academia.edu/1992202/Functoriality_of_quantization_a_KK-theoretic_approach}{web})

\end{itemize}
See also the related references at \emph{[[Guillemin-Sternberg geometric quantization conjecture]]}.

The KK-theory of twisted convolution algebras and its relation to [[twisted K-theory]] of [[differentiable stacks]] is discussed in

\begin{itemize}%
\item [[Jean-Louis Tu]], [[Ping Xu]], [[Camille Laurent-Gengoux]], \emph{Twisted K-theory of differentiable stacks} (\href{http://arxiv.org/abs/math/0306138}{arXiv:math/0306138})

\end{itemize}
Discussion of groupoid 1-[[cocycles]] and their effect on the [[groupoid algebra]] KK-theory is discussed in

\begin{itemize}%
\item Bram Mesland, \emph{Groupoid cocycles and K-theory} (\href{http://arxiv.org/abs/1005.3677}{arXiv:1005.3677})

\end{itemize}
\hypertarget{ReferencesInTermsOfCorrespondences}{}\subsubsection*{{In terms of correspondences/spans}}\label{ReferencesInTermsOfCorrespondences}

\hypertarget{for_plain_kktheory}{}\paragraph*{{For plain KK-theory}}\label{for_plain_kktheory}

KK-classes between [[algebras of functions]] on [[smooth manifolds]] are described in terms of [[equivalence classes]] of [[correspondence]] manifolds carrying a [[vector bundle]] in section 3 of

\begin{itemize}%
\item [[Alain Connes]], [[Georges Skandalis]], \emph{The longitudinal index theorem for foliations}. Publ. Res. Inst. Math. Sci. 20, no. 6, 1139--1183 (1984) (\href{http://www.alainconnes.org/docs/longitudinal.pdf}{pdf})

\end{itemize}
This generalizes the [[Baum-Douglas geometric cycles]] from [[K-homology]] to KK-theory.

A further generalization of this, where one algebra $C(Y)$ is generalized to $C(Y) \otimes A$ for $A$ a unital separable $C^\ast$-algebra, is in section 3 of

\begin{itemize}%
\item [[Jonathan Block]], [[Shmuel Weinberger]], \emph{Arithmetic manifolds of positive scalar curvature.}, J. Diff. Geom. 52, no. 2, 375--406 (1999). (\href{http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.30.297}{web})

\end{itemize}
In section 5 of

\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}
this is reviewed and then a characterization in terms of [[co-spans]] of [[C\emph{-algebras]] is given. This version is effectively a restatement of \hyperlink{RelationToHomotopyClassesOfStarHomomorphisms}{the characterization by Cuntz} as reproduced in (\hyperlink{Blackadar99}{Blackadar 99, corollary 17.8.4}).}

For similar structures see also at \emph{[[motive]]} in the section \emph{\href{motive#RelationToKKTheory}{Relation to bivariant K-theory}}.

A `generators and relations' description of KK-theory in terms of spans is given in

\begin{itemize}%
\item Bernhard Burgstaller, \emph{The generators and relations picture of KK-theory}, (2016) arXiv:\href{http://arxiv.org/abs/1602.03034}{1602.03034}

\end{itemize}
\hypertarget{ReferencesCorrespondencesEquivariant}{}\paragraph*{{For equivariant KK-theory}}\label{ReferencesCorrespondencesEquivariant}

Generalization of such [[correspondence]]-presentation to [[equivariant KK-theory]] (and hence, by the [[Green-Julg theorem]] essentially to KK-theory of [[groupoid algebras]] of [[action groupoids]] of [[compact topological groups]]) -- was introduced 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})

\end{itemize}
based on

\begin{itemize}%
\item [[Heath Emerson]], [[Ralf Meyer]] \emph{Equivariant embedding theorems and topological index maps}, Adv. Math. 225 (2010), 2840-2882 (\href{http://arxiv.org/abs/0908.1465}{arXiv:0908.1465})


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



\end{itemize}
based on technical aspects of the construction of pushforward along and comoposition of equivariant correspondences in

\begin{itemize}%
\item [[Paul Baum]], [[Jonathan Block]], \emph{Equivariant bicycles on singular spaces}. C.R. Acad. Sci. Paris, t. 311 Serie I, 1990 (\href{http://www.math.upenn.edu/~blockj/papers/BlockEquivariantbi.pdf}{pdf})


\item [[Heath Emerson]], [[Ralf Meyer]], \emph{Equivariant embedding theorems and topological index maps}, Adv. Math. 225 (2010), 2840-2882 (\href{http://arxiv.org/abs/0908.1465}{arXiv:0908.1465})



\end{itemize}
Further developments of this are in

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


\item [[Heath Emerson]], Robert Yuncken, \emph{Equivariant correspondences and the Borel-Bott-Weil theorem} (\href{http://arxiv.org/abs/0905.1153}{arXiv:0905.1153})



\end{itemize}
\hypertarget{RelationToMotives}{}\subsubsection*{{Relation to motives and algebraic KK-theory}}\label{RelationToMotives}

The general analogy between KK-cocycles and [[motives]] is noted explicitly in

\begin{itemize}%
\item [[Alain Connes]], Caterina Consani, [[Matilde Marcolli]], \emph{Noncommutative geometry and motives: the thermodynamics of endomotives} (\href{http://arxiv.org/abs/math/0512138}{arXiv:math/0512138})


\item [[Alain Connes]], [[Matilde Marcolli]], \emph{[[Noncommutative Geometry, Quantum Fields and Motives]]}



\end{itemize}
and also very briefly in (\hyperlink{Meyer06}{Meyer 06}).

A relation between [[motivic cohomology]] and bivariant [[algebraic K-theory]] is discussed in

\begin{itemize}%
\item [[Guillermo Cortiñas]], [[Andreas Thom]], \emph{Bivariant algebraic K-theory}. J. Reine Angew. Math. 510 (2007), 71--124. (\href{http://arxiv.org/abs/math/0603531}{arXiv:math/0603531})


\item [[Snigdahayan Mahanta]], \emph{Noncommutative correspondence categories, simplicial sets and pro $C^\ast$-algebras} (\href{http://arxiv.org/abs/0906.5400}{arXiv:0906.5400})


\item [[Snigdahayan Mahanta]], \emph{Higher nonunital Quillen $K'$-theory, KK-dualities and applications to topological $\mathbb{T}$-duality}, Journal of Geometry and Physics, Volume 61, Issue 5 2011, p. 875-889. (\href{http://wwwmath.uni-muenster.de/u/snigdhayan.mahanta/papers/KQ.pdf}{pdf})


\item [[Grigory Garkusha]], Ivan Panin, \emph{K-motives of algebraic varieties} (\href{http://arxiv.org/abs/1108.0375}{arXiv:1108.0375})


\item [[Grigory Garkusha]], \emph{Algebraic Kasparov K-theory. II} (\href{http://arxiv.org/abs/1206.0178}{arXiv:1206.0178})



\end{itemize}
For a collection of literature see also paragraph 1.5 in

\begin{itemize}%
\item [[Andrew Blumberg]], [[David Gepner]], [[Gonçalo Tabuada]], \emph{A universal characterization of higher algebraic K-theory} (\href{http://arxiv.org/abs/1001.2282}{arXiv:1001.2282})

\end{itemize}
(in the context of [[noncommutative motives]]).

In

\begin{itemize}%
\item [[Snigdhayan Mahanta]], \emph{Higher nonunital Quillen $K'$-theory, KK-dualities, and applications to topological T-duality}, J. Geom. Phys., 61 (5), 875-889, 2011 (\href{http://wwwmath.uni-muenster.de/u/snigdhayan.mahanta/papers/KQ.pdf}{pdf}, \href{http://wwwmath.uni-muenster.de/u/snigdhayan.mahanta/papers/KKTD.pdf}{talk notes})

\end{itemize}
it is shown that there is a universal functor $KK \longrightarrow NCC_{dg}$ from [[KK-theory]] to the category of [[noncommutative motives]], which is the category of [[dg-categories]] and dg-[[profunctors]] up to homotopy between them. This is given by sending a [[C\emph{-algebra]] to the [[dg-category]] of [[perfect complexes]] of (the unitalization of) its underlying [[associative algebra]].}

See also at \emph{[[motivic quantization]]} and \emph{[[motives in physics]]}.

\hypertarget{in_dbrane_theory}{}\subsubsection*{{In D-brane theory}}\label{in_dbrane_theory}

KK-theory also describes [[RR-field]] [[charges]] and sources in [[D-brane]] theory.

A review is in

\begin{itemize}%
\item [[Richard Szabo]], \emph{D-branes and bivariant K-theory}, Noncommutative Geometry and Physics 3 1 (2013): 131. (\href{http://arxiv.org/abs/0809.3029}{arXiv:0809.3029})

\end{itemize}
based on

\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})

\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})

\emph{D-branes, KK-theory and duality on noncommutative spaces}, J. Phys. Conf. Ser. 103:012004,2008 (\href{http://arxiv.org/abs/0709.2128}{arXiv:0709.2128})



\end{itemize}
\hypertarget{smooth_refinement_and_spectral_triples}{}\subsubsection*{{Smooth refinement and spectral triples}}\label{smooth_refinement_and_spectral_triples}

Discussion of KK-theory for [[spectral triples]] is discussed in

\begin{itemize}%
\item [[Bram Mesland]], \emph{Spectral triples and KK-theory: A survey} (\href{http://arxiv.org/abs/1304.3802}{arXiv:1304.3802})

\end{itemize}
[[!redirects Kasparov K-theory]] [[!redirects bivariant K-theory]]

[[!redirects KK-group]] [[!redirects KK-groups]]

[[!redirects Kasparov KK-group]] [[!redirects Kasparov KK-groups]]

[[!redirects Kasparov KK-groups]]



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