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\section*{topological K-theory}

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

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

[[!include cohomology - contents]]

\hypertarget{topology}{}\paragraph*{{Topology}}\label{topology}

[[!include topology - 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{the_kgroup}{The K-group}\dotfill \pageref*{the_kgroup} \linebreak
\noindent\hyperlink{ReducedKGroup}{The reduced K-group}\dotfill \pageref*{ReducedKGroup} \linebreak
\noindent\hyperlink{TheRelativeKGroup}{The relative K-group}\dotfill \pageref*{TheRelativeKGroup} \linebreak
\noindent\hyperlink{TheGradedKGroups}{The graded K-groups}\dotfill \pageref*{TheGradedKGroups} \linebreak
\noindent\hyperlink{ForNonCompactSpaces}{For non-compact spaces}\dotfill \pageref*{ForNonCompactSpaces} \linebreak
\noindent\hyperlink{Examples}{Examples}\dotfill \pageref*{Examples} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{homotopy_invariance}{Homotopy invariance}\dotfill \pageref*{homotopy_invariance} \linebreak
\noindent\hyperlink{ExactSequences}{Exact sequences}\dotfill \pageref*{ExactSequences} \linebreak
\noindent\hyperlink{ExternalProducts}{External product}\dotfill \pageref*{ExternalProducts} \linebreak
\noindent\hyperlink{fundamental_product_theorem}{Fundamental product theorem}\dotfill \pageref*{fundamental_product_theorem} \linebreak
\noindent\hyperlink{BottPeriodicities}{Bott periodicity}\dotfill \pageref*{BottPeriodicities} \linebreak
\noindent\hyperlink{GradedRingStructure}{Graded-commutative ring structure}\dotfill \pageref*{GradedRingStructure} \linebreak
\noindent\hyperlink{ClassifyingSpace}{Classifying space}\dotfill \pageref*{ClassifyingSpace} \linebreak
\noindent\hyperlink{of_noncompact_spaces}{Of non-compact spaces}\dotfill \pageref*{of_noncompact_spaces} \linebreak
\noindent\hyperlink{AsAGeneralizedCohomologyTheory}{As a generalized cohomology theory}\dotfill \pageref*{AsAGeneralizedCohomologyTheory} \linebreak
\noindent\hyperlink{ComplexOrientationAndFormalGroupLaw}{Complex orientation and formal group law}\dotfill \pageref*{ComplexOrientationAndFormalGroupLaw} \linebreak
\noindent\hyperlink{spectrum}{Spectrum}\dotfill \pageref*{spectrum} \linebreak
\noindent\hyperlink{ring_spectrum}{Ring spectrum}\dotfill \pageref*{ring_spectrum} \linebreak
\noindent\hyperlink{chromatic_filtration}{Chromatic filtration}\dotfill \pageref*{chromatic_filtration} \linebreak
\noindent\hyperlink{as_the_shape_of_the_smooth_ktheory_spectrum}{As the shape of the smooth K-theory spectrum}\dotfill \pageref*{as_the_shape_of_the_smooth_ktheory_spectrum} \linebreak
\noindent\hyperlink{RelationToAlgebraicKTheory}{Relation to algebraic K-theory}\dotfill \pageref*{RelationToAlgebraicKTheory} \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_noncompact_spaces_2}{For non-compact spaces}\dotfill \pageref*{for_noncompact_spaces_2} \linebreak
\noindent\hyperlink{dbrane_charge}{D-brane charge}\dotfill \pageref*{dbrane_charge} \linebreak


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

What is called \emph{topological} [[K-theory]] is a collection of [[generalized (Eilenberg-Steenrod) cohomology]] theories whose cocycles in degree 0 on a [[topological space]] $X$ may be represented by pairs of [[vector bundles]], real or complex ones, on $X$ modulo a certain equivalence relation.

The following is the quick idea. For a detailed introduction see \emph{[[Introduction to Topological K-Theory]]}.

First, recall that for $k$ a [[field]] then a $k$-[[vector bundle]] over a [[topological space]] $X$ is a map $V \to X$ whose [[fibers]] are [[vector spaces]] which vary over $X$ in a controlled way. Explicitly this means that there exits an [[open cover]] $\{U_i \to X\}$ of $X$, a [[natural number]] $n \in \mathbb{N}$ (the \emph{[[rank of a vector bundle|rank]]} of the vector bundle) and a [[homeomorphism]] $U_i \times k^n \to V|_{U_i}$ over $U_i$ which is fiberwise a $k$-[[linear map]].

Vector bundles are of central interest in large parts of [[mathematics]] and [[physics]], for instance in [[Chern-Weil theory]] and [[cobordism theory]]. But the collection $Vect(X)_{/\sim}$ of [[isomorphism classes]] of vector bundles over a given space is in general hard to analyze. One reason for this is that these are classified in degree-1 \emph{[[nonabelian cohomology]]} with [[coefficients]] in the ([[nonabelian group|nonabelian]]) [[general linear group]] $GL(n,k)$. K-theory may roughly be thought of as the result of forcing vector bundles to be classified by an abelian [[cohomology theory]].

To that end, observe that all natural operations on [[vector spaces]] generalize to vector bundles by applying them [[fiber]]-wise. Notably there is the fiberwise [[direct sum of vector bundles]], also called the \emph{[[nLab:Whitney sum]]} operation. This operation gives the set $Vect(X)_{/\sim}$ of [[nLab:isomorphism classes]] of vector bundles the structure of an [[semi-group]] ([[monoid]]) $(Vect(X)_{/\sim},\oplus)$.

Now as under direct sum the [[nLab:dimension]] of vector spaces adds, similarly under [[nLab:direct sum of vector bundles]] their [[nLab:rank]] adds. Hence in analogy to how one passes from the additive [[semi-group]] ([[monoid]]) of [[natural numbers]] to the addtitive [[group]] of [[integers]] by adjoining formal additive inverses, so one may adjoin formal additive inverses to $(Vect(X)_{/\sim},\oplus)$. By a general prescription (``[[Grothendieck group of a commutative monoid]]'') this is achieved by first passing to the larger class of [[pairs]] $(V_+,V_-)$ of vector bundles (``[[virtual vector bundles]]''), and then [[quotient|quotienting]] out the [[equivalence relation]] given by

\begin{displaymath}
(V_+, V_-) \sim (V_+ \oplus W , V_- \oplus W)
\end{displaymath}
for all $W \in Vect(X)_{/\sim}$. The resulting set of [[equivalence classes]] is an [[abelian group]] with group operation given on representatives by

\begin{displaymath}
[V_+, V_-] \oplus [V'_+, V'_-] \coloneqq (V_+ \oplus V'_+, V_- \oplus V'_-)
\end{displaymath}
and with the [[inverse]] of $[V_+,V_-]$ given by

\begin{displaymath}
-[V_+, V_-] = [V_-, V_+]
  \,.
\end{displaymath}
This [[abelian group]] obtained from $(Vect(X)_{/\sim}, \oplus)$ is denoted $K(X)$ and often called \emph{the K-theory} of the space $X$. Here the letter ``K'' (due to [[Alexander Grothendieck]]) originates as a shorthand for the German word \emph{Klasse}, referring to the above process of forming [[equivalence classes]] of ([[isomorphism classes]]) of vector bundles.

This simple construction turns out to yield remarkably useful groups of [[homotopy]] [[invariants]]. A variety of deep facts in [[algebraic topology]] have fairly elementary proofs in terms of topolgical K-theory, for instance the [[Hopf invariant one]] problem (\hyperlink{AdamsAtiyah66}{Adams-Atiyah 66}).

One defines the ``higher'' K-groups of a topological space to be those of its higher [[reduced suspensions]]

\begin{displaymath}
K^{-n}(X) = K(\Sigma^n X)
  \,.
\end{displaymath}
The assignment $X \mapsto K^\bullet(X)$ turns out to share many properties of the assignment of [[ordinary cohomology]] groups $X \mapsto H^n(X,\mathbb{Z})$. One says that topological K-theory is a [[generalized (Eilenberg-Steenrod) cohomology]] theory. As such it is [[Brown representability theorem|represented]] by a [[spectrum]]. For $k = \mathbb{C}$ this is called [[KU]], for $k = \mathbb{R}$ this is called [[KO]]. (There is also the unification of both in [[KR]]-theory.)

One of the basic facts about topological K-theory, rather unexpected from the definition, is that these higher K-groups repeat \emph{periodically} in the degree $n$. For $k = \mathbb{R}$ the periodicity is 8, for $k = \mathbb{C}$ it is 2. This is called \emph{[[Bott periodicity]]}.

It turns out that an important source of [[virtual vector bundles]] representing classes in [[K-theory]] are [[index bundles]]: Given a [[Riemannian manifold|Riemannian]] [[spin structure|spin]] [[manifold]] $B$, then there is a [[vector bundle]] $S \to B$ called the \emph{[[spin bundle]]} of $B$, which carries a [[differential operator]], called the [[Dirac operator]] $D$. The [[index of a Dirac operator]] is the formal difference of its [[kernel]] by its [[cokernel]] $[ker D, coker D]$. Now given a continuous family $D_x$ of Dirac operators/Fredholm operators, parameterized by some topological space $X$, then these indices combine to a class in $K(X)$.

It is via this construction that topological K-theory connects to [[spin geometry]] (see e.g. [[Karoubi K-theory]]) and [[index theory]].

As the terminology indicates, both [[spin geometry]] and [[Dirac operator]] originate in [[physics]]. Accordingly, K-theory plays a central role in various areas of [[mathematical physics]], for instance in the theory of [[geometric quantization]] (``[[spin{\tt \symbol{94}}c quantization]]'') in the theory of [[D-branes]] (where it models [[D-brane charge]] and [[RR-fields]]) and in the theory of [[Kaluza-Klein compactification]] via [[spectral triples]] (see below).

All these geometric constructions have an [[operator algebra|operator algebraic]] incarnation: by the topological [[Serre-Swan theorem]] then [[vector bundles]] of finite rank are equivalently [[modules]] over the [[C\emph{-algebra]] of [[continuous functions]] on the base space. Using this relation one may express K-theory classes entirely operator algebraically, this is called \emph{[[operator K-theory]]}. Now [[Dirac operators]] are generalized to [[Fredholm operators]].}

There are more [[C\emph{-algebras]] than arising as [[algebras of functions]] of [[topological space]], namely non-commutative C}-algebras. One may think of these as defining [[non-commutative geometry]], but the definition of [[operator K-theory]] immediately generalizes to this situation (see also at \emph{[[KK-theory]]}).

While the [[C\emph{-algebra]] of a [[Riemannian manifold|Riemannian]] [[spin structure|spin]] [[manifold]] remembers only the underlying [[topological space]], one may algebraically encode the [[smooth structure]] and [[Riemannian structure]] by passing from [[Fredholm modules]] to ``[[spectral triples]]''. This may for instance be used to algebraically encode the spin physics underlying the [[standard model of particle physics]] and [[operator K-theory]] plays a crucial role in this.}

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

The following discussion of topological K-theory in terms of [[point-set topology]]. For more abstract perspectives see for instance \emph{[[Snaith's theorem]]} and other pointers at \emph{[[K-theory]]}.

Assumed background for the following is the content of

\begin{itemize}%
\item \emph{[[topological vector bundles]]}


\item \emph{[[Grothendieck group of a commutative monoid]]}


\item \emph{[[pointed topological spaces]]}



\end{itemize}
Throughout, let $k$ be a [[topological field]], usually the [[real numbers]] $\mathbb{R}$ or of [[complex numbers]] $\mathbb{C}$.

In the following we take

\begin{enumerate}%
\item \emph{[[vector space]]} to mean \emph{[[finite dimensional vector space]]} over $k$.


\item \emph{[[vector bundle]]} to mean \emph{[[topological vector bundle]] over $k$ of finite [[rank of a vector bundle|rank]]}.



\end{enumerate}
We say \emph{[[monoid]]} for \emph{[[semigroup]] with [[unitality|unit]]}.

For the most part below we will invoke the assumption that the base [[topological space]] $X$ is a [[compact Hausdorff space]]. Because then the following statement holds, which is crucial in some places:

\begin{lemma}
\label{DirectSumHasInverseUpToTrivialBundle}\hypertarget{DirectSumHasInverseUpToTrivialBundle}{}
\textbf{(over [[compact Hausdorff space]] every [[topological vector bundle]] is [[direct sum of vector bundles|direct summand]] of a [[trivial vector bundle]])}

For every [[topological vector bundle]] $E \to X$ over the [[compact Hausdorff space]] $X$ there exists a [[topological vector bundle]] $\tilde E \to X$ such that the [[direct sum of vector bundles]]

\begin{displaymath}
E \oplus_X \tilde E \simeq X \times \mathbb{R}^{n}
\end{displaymath}
is a [[trivial vector bundle]].

\end{lemma}
For \textbf{proof} see \href{topological+vector+bundle#TopologicalVectorbundleOverCompactHausdorffSpaceIsDirectSummandOfTrivialBundle}{this prop.} at \emph{[[topological vector bundle]]}.

\hypertarget{the_kgroup}{}\subsubsection*{{The K-group}}\label{the_kgroup}

The starting point is the simple observation that the operation of [[direct sum of vector bundles]] yields a [[monoid]] structure ([[semi-group]] with [[unitality|unit]]) on [[isomorphism classes]] of [[topological vector bundles]], which however is lacking [[inverse elements]] and hence is not an actual [[group]].

\begin{defn}
\label{SemigroupOfIsomorphismClassesOfTopologicalVectorBundlesOnX}\hypertarget{SemigroupOfIsomorphismClassesOfTopologicalVectorBundlesOnX}{}
\textbf{([[monoid]] of [[isomorphism classes]] of [[topological vector bundles]] on $X$)}

For $X$ a [[topological space]], write $Vect(X)_{/\sim}$ for the [[set]] of [[isomorphism classes]] of [[topological vector bundles]] over $X$. The operation of [[direct sum of vector bundles]]

\begin{displaymath}
(-)\oplus_X (-)
  \;\colon\;
  Vect(X) \times Vect(X)
    \longrightarrow
  Vect(X)
\end{displaymath}
descends to this [[quotient]] by [[isomorphism]]

\begin{displaymath}
[E_1] + [E_2]
  \;\coloneqq\;
  [E_1 \oplus_X E_2]
\end{displaymath}
to yield the structure of a [[monoid]] ([[semi-group]] with [[unitality|unit]])

\begin{displaymath}
\left(
     Vect(X)_{/\sim}, +
  \right)
  \,.
\end{displaymath}
\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
The operation of [[direct sum of vector bundles]] on [[isomorphism classes]] in def. \ref{SemigroupOfIsomorphismClassesOfTopologicalVectorBundlesOnX} is indeed not a [[group]]:

Let $x \in X$ be a chosen point of $x$ and write

\begin{displaymath}
rk_x
    \;\colon\;
  Vect(X)_{/\sim}
    \longrightarrow
  \mathbb{N}
\end{displaymath}
for the [[function]] which takes a [[topological vector bundle]] to the [[rank of a vector bundle|rank]] over the [[connected component]] of the point $x$.

Then under [[direct sum of vector bundles]] the rank is additive

\begin{displaymath}
rk_x(E_1 \oplus_X E_2)
   \,=\,
  rk_x(E_1) + rk_x(E_1)
  \,.
\end{displaymath}
Now since the [[natural numbers]] under [[addition]] are just a [[monoid]] ([[semi-group]] with [[unitality|unit]]), with no element except zero having an [[inverse element]] under the additive operation, it follows immediately that a necessary condition for the [[isomorphism class]] of a [[topological vector bundle]] to be invertible under [[direct sum of vector bundles]] is that its [[rank of a vector bundle]] be zero. But there is only one such class of vector bundles, in fact there is only one such vector bundle, namely the unique rank-zero bundle $X \times k^0$, necessarily a [[trivial vector bundle]].

Now for the [[monoid]] of [[natural numbers]] $(\mathbb{N},+)$ it is a time honored fact that it is interesting and useful to rectify its failure of being a [[group]] by [[universal construction|universally]] forcing it to become one. This is a process called \emph{[[group completion]]} and the group completion of the natural numbers is the additive group of [[integers]] $(\mathbb{Z},+)$.

The idea is hence to apply group completion also to the monoid $(Vect(X)_{/\sim}, +)$, and so that the [[rank of a vector bundle|rank]] operation above becomes a [[homomorphism]] of [[abelian groups]].

\end{remark}
An explicit construction of [[group completion]] of a [[commutative monoid]] is called the \emph{[[Grothendieck group of a commutative monoid]]}.

\begin{defn}
\label{KGroupByGrothendieckGroup}\hypertarget{KGroupByGrothendieckGroup}{}
\textbf{(K-group as the [[Grothendieck group of a commutative monoid|Grothendieck group]] of [[isomorphism classes]] of [[topological vector bundles]])}

For $X$ a [[topological space]], write

\begin{displaymath}
K(X) \;\coloneqq\; K(Vect(X)_{/\sim}, +)
\end{displaymath}
for the [[Grothendieck group of a commutative monoid|Grothendieck group]] of the [[commutative monoid]] (abelian [[semi-group]] with [[unitality|unit]]) of [[isomorphism classes]] of [[topological vector bundles]] on $X$ from def. \ref{SemigroupOfIsomorphismClassesOfTopologicalVectorBundlesOnX}.

This means that $K(X)$ is the [[group]] whose elements are [[equivalence classes]] of pairs

\begin{displaymath}
([E_+], [E_-]) \; \in Vect(X)_{/\sim} \times Vect(X)_{/\sim}
\end{displaymath}
of [[isomorphism classes]] of [[topological vector bundles]] on $X$, with respect to the [[equivalence relation]]

\begin{displaymath}
\big(
  \left(
    [E_+], [E_-]
  \right)
   \;\sim\;
  \left(
    [F_+, F_-]
  \right)
  \big)
   \;\Leftrightarrow\;
  \left(
    \underset{[G],[H] \in Vect(X)_{/\sim}}{\exists}
    \left(
      \left([E_+ \oplus_X G] , [E_- \oplus_X G]\right)
        \,=\,
      \left([F_+ \oplus_X H] , [F_- \oplus_X H]\right)
    \right)
  \right)
  \,.
\end{displaymath}
Here a pair $([E_+], [E_-])$ is also called a \emph{[[virtual vector bundle]]}, and its equivalence class under the above equivalence relation is also denoted

\begin{displaymath}
[E_+] - [E_-] \;\;\in K(X)
  \,.
\end{displaymath}
If $X$ is a [[pointed topological space]], hence equipped with a choice of point $x \in X$ then the difference of [[rank of a vector bundle|ranks]] $rk_x(-)$ of the representing vector bundles over the [[connected component]] of $x \in X$

\begin{displaymath}
rk_x( [E_+] - [E_-] )
    \;\coloneqq\;
  rk_x(E_+) -  rk_x(E_-) \in \mathbb{Z}
\end{displaymath}
is called the \emph{virtual rank} of the virtual vector bundle.

\end{defn}
\begin{example}
\label{KGroupOfThePoint}\hypertarget{KGroupOfThePoint}{}
\textbf{(K-group of the point is the [[integers]])}

Let $X = \ast$ be the [[point]]. Then a [[topological vector bundle]] on $X$ is just a [[vector space]]

\begin{displaymath}
Vect(\ast) \simeq Vect
\end{displaymath}
and an isomorphism of vector bundles is just a bijective [[linear map]].

Since [[finite dimensional vector spaces]] are [[isomorphism|isomorphic]] precisely if they have the same [[dimension]], the [[monoid]] ([[semi-group]] with [[unitality|unit]]) of isomorphism classes of vector bundles over the point (def. \ref{SemigroupOfIsomorphismClassesOfTopologicalVectorBundlesOnX}) is the [[natural numbers]]:

\begin{displaymath}
\left(
    Vect(\ast)_{/\sim}, +
  \right)
    \;\simeq\;
  \left(
    \mathbb{N}, +
  \right)
  \,.
\end{displaymath}
Accordingly the K-group of the point is the [[Grothendieck group of a commutative monoid|Grothendieck group]] of the [[natural numbers]], which is the additive group of [[integers]] (\href{Grothendieck+group+of+a+commutative+monoid#GrothendieckGroupOfNaturalNumbersUnderAdditionIsTheIntegers}{this example}):

\begin{displaymath}
K(\ast) \simeq (\mathbb{Z}, +)
\end{displaymath}
and this identification is the assignment of \emph{virtual rank} (def. \ref{KGroupByGrothendieckGroup}).

\end{example}
\begin{prop}
\label{OnCompactHausdorffVirtualVectorBundlesAreFormalDifferentcesWithATrivialBundle}\hypertarget{OnCompactHausdorffVirtualVectorBundlesAreFormalDifferentcesWithATrivialBundle}{}
\textbf{(on [[compact Hausdorff spaces]] all [[virtual vector bundles]] are formal difference by a [[trivial vector bundle]])}

If $X$ is a [[compact Hausdorff space]], then every [[virtual vector bundle]] on $X$ (def. \ref{KGroupByGrothendieckGroup}) is of the form

\begin{displaymath}
[E] - [X \times k^n]
\end{displaymath}
(i.e. with negative component represented by a [[trivial vector bundle]]).

\end{prop}
\begin{proof}
For $X$ compact Hausdorff then lemma \ref{DirectSumHasInverseUpToTrivialBundle} implies that for every [[topological vector bundle]] $E_-$ there exists a topological vector bundle $\tilde E_-$ with $E_- \oplus_X \tilde E_- \simeq X \times k^n$, and hence

\begin{displaymath}
[E_+] - [E_-]
  =
  \underset{[E]}{\underbrace{[E_+] + [\tilde E_-]}}
    -
  \underset{[X \times k^n]}{\underbrace{ \left( [E_-] + [\tilde E_-] \right) } }
  =
  [E] - [X \times k^n]
  \,.
\end{displaymath}
\end{proof}
\begin{remark}
\label{KTheoryRing}\hypertarget{KTheoryRing}{}
\textbf{([[commutative ring]] structure on $K(X)$ from [[tensor product of vector bundles]])}

Also the operation of [[tensor product of vector bundles]] over $X$ descends to [[isomorphism classes]] of [[topological vector bundles]] and makes $(Vect(X)_{\sim}, \oplus, \otimes )$ a [[semi-ring]] ([[rig]]).

(This is the shadow under passing to isomorphism classes of the fact that the [[category]] $Vect(X)$ is a [[distributive monoidal category]] under [[tensor product of vector bundles]].)

This multiplicative structure passes to the K-group (def. \ref{KGroupByGrothendieckGroup}) by the formula

\begin{displaymath}
[E_+, E_-] \cdot [F_+, F_-]
    \;\coloneqq\;
  [ (E_+ \otimes_X F_+) \oplus_X (E_- \otimes_X F_-) \,,\, (E_+ \otimes_X F_-) \oplus_X (E_- \otimes_X F_+)  ]
  \,.
\end{displaymath}
Accordingly the ring $(K(X), +,\cdot)$ is also called the \emph{K-theory ring} of $X$.

\end{remark}
\begin{remark}
\label{FunctorialityOfKGroup}\hypertarget{FunctorialityOfKGroup}{}
\textbf{([[functor|functoriality]] of the K-theory ring assignment)}

Let $f \colon X \longrightarrow Y$ be a [[continuous function]] between [[topological spaces]]. The operation of [[pullback of vector bundles]]

\begin{displaymath}
f^\ast \;\colon\; Vect(Y) \longrightarrow Vect(X)
\end{displaymath}
is compatible with [[direct sum of vector bundles]] as well as with [[tensor product of vector bundles]] and hence descends to a [[homomorphism]] of [[commutative rings]]

\begin{displaymath}
f^\ast \;\colon\; K(Y) \longrightarrow K(X)
\end{displaymath}
between the K-theory rings from remark \ref{KTheoryRing}. Moreover, for

\begin{displaymath}
X \overset{f}{\longrightarrow} Y \overset{g}{\longrightarrow} Z
\end{displaymath}
two consecutive [[continuous functions]], then the consecutive pullback of the vector bundle is isomorphic to the pullback along the composite map, which means that on K-group pullback preserves composition

\begin{displaymath}
(g \circ f)^\ast = f^\ast \circ g^\ast \;\colon\; K(Z) \longrightarrow K(X)
  \,.
\end{displaymath}
Finally, of course pullback along an [[identity function]] $id_X \colon X \to X$ is the identity group homomorphism.

In summary this says that the assignment of K-groups to topological spaces is a [[functor]]

\begin{displaymath}
K(-)
   \;\colon\;
  Top^{op}
    \longrightarrow
  CRing
\end{displaymath}
from the [[opposite category]] of the [[category]] [[Top]] of [[topological space]] to the category [[CRing]] of [[commutative rings]].

\end{remark}
We consider next the image of plain vector bundles in [[virtual vector bundles]]:

\begin{prop}
\label{StableEquivalenceOfVectorBundles}\hypertarget{StableEquivalenceOfVectorBundles}{}
\textbf{([[stable equivalence of vector bundles]])}

Let $X$ be a [[topological space]]. Define an [[equivalence relation]] $\sim_{stable}$ on [[topological vector bundles]] over $X$ by declaring two vector bundles $E_1 E_2 \in Vect(X)$ to be equivalent if there exists a [[trivial vector bundle]] $X \times k^n$ of some [[rank]] $n$ such that after [[tensor product of vector bundles]] with this [[trivial vector bundle]], both bundles become [[isomorphism|isomorphic]]

\begin{displaymath}
\left(
    E_1
      \sim_{stable}
    E_2
  \right)
    \;\Leftrightarrow\;
  \underset{n \in \mathbb{N}}{\exists}
  \left(
     E_1 \otimes_X (X \times k^n)
       \;\simeq\;
     E_2 \otimes_X (X \times k^n)
  \right)
  \,.
\end{displaymath}
If $E_1 \sim_{stable} E_2$ we say that $E_1$ and $E_2$ are \emph{stably equivalent vector bundles}.

\end{prop}
\begin{prop}
\label{}\hypertarget{}{}
\textbf{([[image]] of plain [[vector bundles]] in [[virtual vector bundles]])}

Let $X$ be a [[topological space]]. There is a [[homomorphism]] of [[semigroups]]

\begin{displaymath}
\itexarray{
    Vect(X)_{/\sim}
      & \longrightarrow &
    K(X)
    \\
    [E_1] &\overset{\phantom{AAA}}{\mapsto}& \left( [E_1], [X \times k^0] \right)
  }
\end{displaymath}
from the [[isomorphism classes]] of [[topological vector bundles]] (def. \ref{SemigroupOfIsomorphismClassesOfTopologicalVectorBundlesOnX}) to the K-group of $X$ (def. \ref{KGroupByGrothendieckGroup} ).

If $X$ is a [[compact Hausdorff space]], then the [[image]] of this function is the [[stable equivalence classes of vector bundles]] (def. \ref{StableEquivalenceOfVectorBundles}), hence this function factors as an [[epimorphism]] onto $Vect(X)_{/\sim_{stable}}$ followed by an [[injection]]

\begin{displaymath}
Vect(X)_{/\sim}
    \longrightarrow
  Vect(X)_{/\sim_{stable}}
    \hookrightarrow
  K(X)
  \,.
\end{displaymath}
\end{prop}
\begin{proof}
The homomorphism of [[commutative monoids]] $Vect(X)_{/\sim} \to K(X)$ is the one given by the [[universal property]] of the [[Grothendieck group of a commutative monoid|Grothendieck group]] construction (\href{Grothendieck+group+of+a+commutative+monoid#UniversalProperty}{this prop.}).

By definition of the [[Grothendieck group of a commutative monoid|Grothendieck group]] (\href{Grothendieck+group+of+a+commutative+monoid#GrothendieckGroupViaQuotientOfCartesianProduct}{this def.}), two elements of the form

\begin{displaymath}
\left(
    [E_1], [X \times k^0]
  \right)
  \phantom{AA}
  \text{and}
  \phantom{AA}
  \left(
    [E_2], [X \times k^0]
  \right)
\end{displaymath}
are equivalent precisely if there exist vector bundles $F_1$ and $F_2$ such that

\begin{displaymath}
\left(
     [ E_1 \oplus_X F_1 ], [ F_1]
  \right)
   \;=\;
  \left(
    [E_2 \oplus_X F_2], [F_2]
  \right)
  \,.
\end{displaymath}
First of all this means that $F_1 \simeq F_2$, hence is equivalent to the existence of a vector bundle $F$ such that

\begin{displaymath}
[E_1 \oplus F] \;=\; [E_2 \oplus F]
  \,.
\end{displaymath}
Now, by the assumption that $X$ is compact Hausdorff, lemma \ref{DirectSumHasInverseUpToTrivialBundle} implies that there exists a vector bundle $\tilde F$ such that

\begin{displaymath}
F \oplus_X \tilde F \simeq X \times k^n
\end{displaymath}
is the [[trivial vector bundle]] of some [[rank of a vector bundle|rank]] $n \in \mathbb{N}$. This means that the above is equivalent already to the existence of an $n \in \mathbb{N}$ such that

\begin{displaymath}
[E_1 \oplus (X \times k^n)]
   \;=\;
 [E_2 \oplus (X \times k^n)]
 \,.
\end{displaymath}
This is the definition of stable equivalence from def. \ref{StableEquivalenceOfVectorBundles}.

\end{proof}
\hypertarget{ReducedKGroup}{}\subsubsection*{{The reduced K-group}}\label{ReducedKGroup}

\begin{defn}
\label{KernelReducedKGroup}\hypertarget{KernelReducedKGroup}{}
\textbf{([[reduced K-theory]])}

Let $X$ be a [[pointed topological space]], hence a [[topological space]] equipped with a choice of point $x \in X$, hence with a [[continuous function]] $const_x \colon \ast  \to X$ from the [[point space]].

By the functoriality of the K-groups (remark \ref{FunctorialityOfKGroup}) this induces a group homomorphism

\begin{displaymath}
const_x^\ast \;\colon\; K(X) \longrightarrow K(\ast)
\end{displaymath}
given by restricting a virtual vector bundle to the basepoint.

The [[kernel]] of this map is called the \emph{[[reduced K-theory]] group} of $(X,x)$, denoted

\begin{displaymath}
\tilde K(X) \;\coloneqq\; ker(const_x^\ast)
  \,.
\end{displaymath}
\end{defn}
\begin{example}
\label{ExpressingPlainKTHeoryGroupInTermsOfReducedKTheoryGroup}\hypertarget{ExpressingPlainKTHeoryGroupInTermsOfReducedKTheoryGroup}{}
\textbf{(expressing plain K-groups as reduced K-groups)}

Let $X$ be a [[topological space]]. Write $X_* \coloneqq X \sqcup \ast$ for its [[disjoint union space]] with the [[point space]], and regard this as a [[pointed topological space]] with base point the adjoined point.

Then the reduced K-theory of $X_+$ is the plain K-theory of $X$:

\begin{displaymath}
\tilde K(X_+)
    \simeq
  K(X)
  \,.
\end{displaymath}
Because every [[topological vector bundle]] on $X \sqcup \ast$ is the [[direct sum of vector bundles]] of one that has [[rank of a vector bundle|rank]] zero on $\ast$ and one that has rank zero on $X$ (\href{direct+sum+of+vector+bundles#DirectSumOnDisjointUnionSpace}{this example.})

\end{example}
\begin{remark}
\label{RestrictionInKTheoryToPointComputesVirtualRank}\hypertarget{RestrictionInKTheoryToPointComputesVirtualRank}{}
\textbf{(restriction in K-theory to the point computes virtual rank)}

By example \ref{KGroupOfThePoint} we have that

\begin{enumerate}%
\item $K(\ast) \simeq \mathbb{Z}$;


\item under this identification the function $const_x^\ast$ is the assignment of \emph{virtual rank}

\begin{displaymath}
\itexarray{
     K(X) &\overset{const_x^\ast}{\longrightarrow}& K(\ast) &\overset{\simeq}{\to}& \mathbb{Z}
     \\
     [E]- [F] &\overset{\phantom{AAA}}{\mapsto}& [E_x] - [F_x] &\mapsto& rk_x(E) - rk_x(F)
  }
\end{displaymath}


\end{enumerate}
\end{remark}
\begin{defn}
\label{VanishingAtInfinity}\hypertarget{VanishingAtInfinity}{}
\textbf{([[vanishing at infinity]])}

If $X$ is a [[locally compact topological space|locally compact]] [[Hausdorff space]], then a [[continuous function]]

\begin{displaymath}
f \;\colon\; X \longrightarrow \mathbb{R}
\end{displaymath}
is said to [[vanishing at infinity|vanish at infinity]] if it [[extension|extends]] by zero to the [[one-point compactification]] $X^* \coloneqq (X \sqcup \{\infty\}, \tau_{cpt})$

\begin{displaymath}
\itexarray{
    X &\overset{f}{\longrightarrow}& \mathbb{R}
    \\
    \downarrow & \nearrow_{\mathrlap{ x \mapsto \left\{ \itexarray{ f(x) &\vert& x \in X \\ 0 &\vert& x = \infty }  \right. }}
    \\
    X^\ast
  }
\end{displaymath}
Now the [[one-point compactification]] $X^\ast$ is a [[compact Hausdorff space]] (by \href{one-point+compactification#OnePointExtensionIsCompact}{this prop.} and \href{one-point+compactification#HausdorffOnePointCompactification}{this prop.}) and canonically a [[pointed topological space]] with basepoint the element $\infty \in X^\ast$.

Moreover, every [[compact Hausdorff space]] $X$ arises this way as the [[one-point compactification]] of the [[complement]] [[subspace]] of any of its points: $X \simeq (X \setminus \{x\})^\ast$ (by \href{one-point+compactification#CompactHausdorffSpaceIsCompactificationOfComplementOfAnyPoint}{this remark}).

Since [[open subspaces of compact Hausdorff spaces are locally compact]], this complement [[subspace]] $X \setminus \{x\} \subset X$ is a [[locally compact topological space|locally compact]] [[Hausdorff space]], and every locally compact Hausdorff spaces arises this way (by \href{one-point+compactification#InclusionIntoOnePointExtensionIsOpenEmbedding}{this prop.}).

Therefore one may think of the [[reduced K-groups]] $\tilde K(X)$ (def. \ref{KernelReducedKGroup}) of compact Hausdorff spaces as the those K-groups of locally compact Hausdorff spaces which ``vanish at infinity''.

\end{defn}
\begin{remark}
\label{FunctorialityOfReducedKGroups}\hypertarget{FunctorialityOfReducedKGroups}{}
\textbf{([[functor|functoriality]] of the reduced K-groups)}

By the functoriality of the unreduced K-groups (remark \ref{FunctorialityOfKGroup}) on (the [[opposite category|opposite]] of) the [[category]] [[Top]] of all topological spaces, the reduced K-groups (def. \ref{KernelReducedKGroup}) becomes functorial on the category $Top^{\ast/}$ of \emph{[[pointed topological spaces]]} (whose [[morphisms]] are the [[continuous functions]] that preserve the base-point):

\begin{displaymath}
\tilde K \;\colon\; (Top^{\ast/})^{op} \longrightarrow Ab
  \,.
\end{displaymath}
This follows by the functoriality of the [[kernel]] construction (which in turn follows by the [[universal property]] of the kernel):

For $(X,x)$ and $(Y,y)$ [[pointed topological spaces]] and $f \colon X \longrightarrow Y$ a continuous function which preserves basepoints $f(x) = y$ then

\begin{displaymath}
\itexarray{
     ker(const_x^\ast)
       &\longrightarrow&
     K(X)
       &\overset{const_x^\ast}{\longrightarrow}&
     K(\{x\})
     \\
     \uparrow^{\mathrlap{\exists !}} && \uparrow^{\mathrlap{f^\ast}} && \uparrow^{\mathrlap{f^\ast}}
     \\
     ker(const_y^\ast)
       &\longrightarrow&
     K(Y)
       &\overset{const_x^\ast}{\longrightarrow}&
     K(\{y\})
  }
  \,.
\end{displaymath}
\end{remark}
\begin{prop}
\label{KGrupDirectSummandReducedKGroup}\hypertarget{KGrupDirectSummandReducedKGroup}{}
\textbf{(over [[compact Hausdorff spaces]] $\tilde K(X)$ is a [[direct sum|direct summand]] of $K(X)$)}

If $(X,x)$ is a [[pointed topological space|pointed]] [[compact Hausdorff space]] then the defining [[short exact sequence]] of [[reduced K-theory]] groups (def. \ref{KernelReducedKGroup})

\begin{displaymath}
0 \to \tilde K(X) \overset{\phantom{AAA}}{\hookrightarrow} K(X) \overset{const_x^\ast}{\longrightarrow} K(\ast) \simeq \mathbb{Z}
   \to 0
\end{displaymath}
[[split exact sequence|splits]] and thus yields an [[isomorphism]], which is given by

\begin{displaymath}
\itexarray{
    K(X)
      &\overset{\phantom{A}\simeq \phantom{A}}{\longrightarrow}&
    \tilde K(X) \oplus \mathbb{Z}
    \\
    [E] - [X \times k^n]
      &\overset{\phantom{AAA}}{\mapsto}&
    ([E], rk_x(E) - n)
  }
  \,.
\end{displaymath}
Here on the left we are using prop. \ref{OnCompactHausdorffVirtualVectorBundlesAreFormalDifferentcesWithATrivialBundle} to represent any element of the K-group as a [[virtual vector bundle|virtual]] difference of a vector bundle $E$ by a [[trivial vector bundle]], and $rk_x(E) \in \mathbb{N}$ denotes the [[rank of a vector bundle|rank]] of this vector bundle over the [[connected component]] of $x \in X$.

Equivalently this means that every element of $K(X)$ decomposes as follows into a piece that has vanishing virtual rank over the connected component of $x$ and a virtual [[trivial vector bundle]].

\begin{displaymath}
[E]- [X \times k^n]
    =
  \underset{\in \tilde K(X) \subset K(X)}{\underbrace{\left( [E] - [X \times k^{rk_x(E)}] \right)}}
    -
  \underset{\in \mathbb{Z} \subset K(X) }{\underbrace{[X \times k^{n-rk_x(E)}]}}
  \,.
\end{displaymath}
\end{prop}
\begin{proof}
By remark \ref{RestrictionInKTheoryToPointComputesVirtualRank} the kernel of $const_x^\ast$ is identified with the [[virtual vector bundles]] of vanishing virtual rank. By prop. \ref{OnCompactHausdorffVirtualVectorBundlesAreFormalDifferentcesWithATrivialBundle} this kernel is identified with the elements of the form

\begin{displaymath}
[E] - [X \times k^{rk_x(E)}]
  \,.
\end{displaymath}
\end{proof}
\begin{example}
\label{BottElement}\hypertarget{BottElement}{}
\textbf{([[Bott element]])}

For $S^2$ the [[Euclidean space|Euclidean]] [[2-sphere]], write

\begin{displaymath}
h \in Vect_{\mathbb{C}}(S^2) \longrightarrow K_{\mathbb{C}}(S^2)
\end{displaymath}
for the complex topological K-theory class of the [[basic complex line bundle on the 2-sphere]]. By prop. \ref{KGrupDirectSummandReducedKGroup} its image in [[reduced K-theory]] is the [[virtual vector bundle]]

\begin{displaymath}
\beta \;\coloneqq \; h-1 \in \tilde K_{\mathbb{C}}(S^2)
  \,.
\end{displaymath}
This is known as the \emph{[[Bott element]]}, due to its key role in the [[Bott periodicity]] of complex topological K-theory, discussed \hyperlink{BottPeriodicities}{below}.

\end{example}
In order to describe $\tilde K(X)$ itself as an equivalence class, we consider the followign refinement of [[stable equivalence of vector bundles]] (def. \ref{StableEquivalenceOfVectorBundles}):

\begin{defn}
\label{EquivalenceRelationForReducedKTheory}\hypertarget{EquivalenceRelationForReducedKTheory}{}
\textbf{([[equivalence relation]] for [[reduced K-theory]] on [[compact Hausdorff spaces]])}

For $X$ a [[topological space]], define an [[equivalence relation]] on the [[set]] of [[topological vector bundles]] $E \to X$ over $X$ by declaring that $E_1 \sim E_2$ if there exists $k_1, k_2 \in \mathbb{N}$ such that there is an [[isomorphism]] of [[topological vector bundles]] between the [[direct sum of vector bundles]] of $E_1$ with the [[trivial vector bundle]] $X \times \mathbb{R}^{k_1}$ and of $E_2$ with $X \times \mathbb{R}^{k_2}$

\begin{displaymath}
(E_1 \sim_{red} E_2)
   \Leftrightarrow
  \left(
    \underset{k_1,k_2 \in \mathbb{N}}{\exists}
      \left(
         (E_1 \oplus_X (X \times \mathbb{R}^{k_1})
           \;\simeq\;
         (E_2 \oplus_X (X \times \mathbb{R}^{k_2})
      \right)
  \right)
  \,.
\end{displaymath}
The operation of [[direct sum of vector bundles]] descends to these quotients

\begin{displaymath}
[E_1] + [E_2]
  \;\coloneqq\;
  [ E_1 \oplus_X E_2 ]
\end{displaymath}
to yield a commutative [[semi-group]]

\begin{displaymath}
\left(Vect(X)_{/\sim_{red}}, +\right)
  \,.
\end{displaymath}
\end{defn}
\begin{prop}
\label{ReducedKEquivalenceRelationVerified}\hypertarget{ReducedKEquivalenceRelationVerified}{}
For $X$ a [[compact Hausdorff space]] then the commutative [[monoid]] $(Vect(X)_{/\sim_{red}}, +)$ from def. \ref{EquivalenceRelationForReducedKTheory} is already an [[abelian group]] and is in fact [[natural isomorphism|naturally isomorphic]] to the [[reduced K-theory]] group $\tilde K(X)$ (def. \ref{KernelReducedKGroup}):

\begin{displaymath}
\tilde K(X) \simeq (Vect(X)_{/\sim_{red}}, +)
  \,.
\end{displaymath}
\end{prop}
\begin{proof}
By prop. \ref{KGrupDirectSummandReducedKGroup} $\tilde K(X)$ is the subgroup of the [[Grothendieck group of a commutative monoid|Grothendieck group]] $K(X)$ on the elements of the form $[E]- [X \times k^{rk_x(E)}]$, which are clearly entirely determined by $[E] \in Vect(X)_{/\sim}$. Hence we need to check if the equivalence relation of the Gorthendieck goup coincides with $\sim_{red}$ on these representatives.

The relation in the Grothendieck group is given by

\begin{displaymath}
\left(
    [E_1]
      \sim
    [E_2]
  \right)
    \Leftrightarrow
  \left(
    \underset{[G], [H] \in Vect(X)_{/\sim}}{\exists}
    \left(
      ( [E_1]+ [G], [X \times k^{rk_x(E_1)}] + [G] )
      \;=\;
      ( [E_2] + [H], [X \times k^{rk_x(E_2)}] + [H] )
    \right)
  \right)
\end{displaymath}
As before, in prop. \ref{OnCompactHausdorffVirtualVectorBundlesAreFormalDifferentcesWithATrivialBundle} we may assume without restriction that $G = X \times k^{n_1}$ and $H = X \times k^{n_2}$ are [[trivial vector bundles]]. Then the above equality on the first component

\begin{displaymath}
[E_1] + [X \times k^{n_1}]
  =
  [E_2] + [X \times k^{n_2}]
\end{displaymath}
is the one that defines $\sim_{red}$, and since isomorphic vector bundles necessarily have the same rank, it implies the equality of the second component.

\end{proof}
\begin{remark}
\label{}\hypertarget{}{}
\textbf{([[non-unital ring|non-unital]] [[commutative ring]]-structure on $\tilde K(X)$)}

In view of the [[commutative ring]] structure on the K-group $K(X)$ from remark \ref{KTheoryRing}, the reduced K-group $\tilde K(X)$ from def. \ref{KernelReducedKGroup}, being the [[kernel]] of a ring [[homomorphism]] (remark \ref{FunctorialityOfKGroup}) is an ideal in $K(X)$, hence itself a [[non-unital ring|non-unital]] [[commutative ring]].

(The ring unit of $K(X)$ is the class $[X \times k^1, X \times k^0]$ of the [[trivial vector bundle|trivial]] [[line bundle]] on $X$, which has virtual rank 1, and hence is not in $\tilde K(x)$.)

\end{remark}
\hypertarget{TheRelativeKGroup}{}\subsubsection*{{The relative K-group}}\label{TheRelativeKGroup}

\begin{defn}
\label{RelativeKTheory}\hypertarget{RelativeKTheory}{}
\textbf{([[relative K-theory]])}

Let

\begin{enumerate}%
\item $X$ be a [[compact Hausdorff space]];


\item $A \subset X$ a [[closed subspace]].



\end{enumerate}
Then the \emph{[[relative K-theory]] group of the pair $(X,A)$}, denoted $K(X,A)$ is the [[reduced K-theory]] group (def. \ref{KernelReducedKGroup}) of the [[quotient space]] $X/A$ (\href{quotient+space#QuotientBySubspace}{this def.}):

\begin{displaymath}
K(X,A)
   \;\coloneqq\;
  \tilde K(X/A)
  \,.
\end{displaymath}
\end{defn}
\begin{defn}
\label{RelatveKTheoryReducesToBareKTheoryAndToReducedKTheory}\hypertarget{RelatveKTheoryReducesToBareKTheoryAndToReducedKTheory}{}
\textbf{(expressing plain and reduced K-theory in terms of relative K-theory)}

The [[relative K-theory]] construction from def. \ref{RelativeKTheory} reduces in special cases to the plain K-theory group and to the [[reduced K-theory]] group.

Recall that for the case that $A = \emptyset \subset X$ then $X/\emptyset = X_+ = X \sqcup \ast$ (by \href{quotient+space#QuotientBySubspace}{this example}). Therefore:

\begin{enumerate}%
\item for $A = \emptyset \subset X$ we have $K(X,\emptyset) = \tilde K(X \sqcup \ast) \simeq K(X)$ (example \ref{ExpressingPlainKTHeoryGroupInTermsOfReducedKTheoryGroup});


\item for $A = \{x\} \subset X$ we have $K(X, \{x\}) = \tilde K(X/\{x\}) = \tilde K(X)$.



\end{enumerate}
\end{defn}
\hypertarget{TheGradedKGroups}{}\subsubsection*{{The graded K-groups}}\label{TheGradedKGroups}

The (reduced) K-theory groups of reduced suspensions of pointed space are called the ``K-group in degree 1'':

\begin{defn}
\label{GradedKGroups}\hypertarget{GradedKGroups}{}
\textbf{(graded K-groups)}

For $X$ a [[pointed topological space]] write

\begin{displaymath}
\tilde K^1(X)
    \;\coloneqq\;
  \tilde K(\Sigma X)
\end{displaymath}
for the [[reduced K-theory]] of the [[reduced suspension]] of $X$.

For $X$ a [[compact Hausdorff space]] and $A \subset X$ a [[closed subspace]], write

\begin{displaymath}
K^1(X,A)
    \coloneqq
  \tilde K( \Sigma(X/A) )
\end{displaymath}
for the [[reduced K-theory]] of the [[reduced suspension]] of the [[quotient space]].

We say these are the K(-cohomology)-groups in degree 1. For emphasis one says that the original K-groups are in degree zero and writes

\begin{displaymath}
K^0(X) \coloneqq K(X)
  \phantom{AAAA}
  \tilde K^0(X) \coloneqq \tilde K(X)
    \phantom{AAA}
  K^0(X,A) \coloneqq K(X,A)
  \,.
\end{displaymath}
The groups are collected to the \emph{graded K-groups}, which are the [[direct sums]]

\begin{displaymath}
\tilde K^\bullet(X)
    \coloneqq
  \tilde K^0(X) \oplus \tilde K^1(X)
\end{displaymath}
and

\begin{displaymath}
K^\bullet(X,A)
    \coloneqq
  K^0(X,A) \oplus K^1(X.A)
\end{displaymath}
regarded as $\mathbb{Z}/2$-graded groups.

Under [[tensor product of vector bundles]] this becomes a non-unital $\mathbb{Z}/2$- [[graded-commutative ring]] (discussed \hyperlink{GradedRingStructure}{below}).

\end{defn}
Recall from example \ref{ExpressingPlainKTHeoryGroupInTermsOfReducedKTheoryGroup} and from example \ref{RelatveKTheoryReducesToBareKTheoryAndToReducedKTheory} the identifications of plain, reduced and relative K-groups, which with the degree-zero notation from def. \ref{GradedKGroups} read:

\begin{displaymath}
K^0(X, \emptyset)
    \simeq
  \tilde K^0(X_+)
    \simeq
  K^0(X)
\end{displaymath}
The analogue is true for the K-groups in degree 1 from def. \ref{GradedKGroups}, though this is no longer completely trivial:

\hypertarget{ForNonCompactSpaces}{}\subsubsection*{{For non-compact spaces}}\label{ForNonCompactSpaces}

By prop. \ref{OnCompactHausdorffVirtualVectorBundlesAreFormalDifferentcesWithATrivialBundle} the topological K-theory groups of [[compact topological spaces]] are [[representable functor|represended]] by [[homotopy classes]] of [[continuous functions]] into the [[classifying spaces]] $B O \times \mathbb{Z}$ and $B U \times \mathbb{Z}$, respectively (def. \ref{BUn}).

\begin{displaymath}
\left(
    X \text{compact}
  \right)
    \;\Rightarrow\;
  \left(
    K_{\mathbb{C}}(X)
      \simeq
    [X \to B U \times \mathbb{Z}]
    \phantom{AAAAA}
    K_{\mathbb{R}}(X)
      \simeq
    [X, B O \times \mathbb{Z}]
  \right)
  \,.
\end{displaymath}
There are various ways of generalizing this situation to non-compact spaces:

\begin{defn}
\label{GrothendieckGroupKTheory}\hypertarget{GrothendieckGroupKTheory}{}
\textbf{(Grothendieck group topological K-theory)}

The [[Grothendieck group]] construction on the [[monoid]] $(Vect(X)/_\sim, \oplus)$ of [[isomorphism classes]] of [[topological vector bundles]] makes sense for every [[topological space]] $X$. For non-compact $X$ this is usually just called that ``the Grothendieck group of vector bundles on $X$'', sometimes denoted

\begin{displaymath}
\mathbb{K}(X) \coloneqq GrothGroup( Vect(X)/_\sim, \oplus )
     \,.
\end{displaymath}
\end{defn}
\begin{defn}
\label{RepresntableTopologicalKTheory}\hypertarget{RepresntableTopologicalKTheory}{}
\textbf{(representable topological K-theory)}

The group of [[homotopy classes]] of [[continuous functions]] into a (classifying) space is of course well defined for any domain space, hence for any topological space $X$ one may set

\begin{displaymath}
K_{\mathbb{C}}(X)_{rep}
      \;\coloneqq\;
     [X,B U \times \mathbb{Z}]
     \phantom{AAAAA}
     K_{\mathbb{R}}(X)_{rep}
      \;\coloneqq\;
     [X,B O \times \mathbb{Z}]
\end{displaymath}
This is called \emph{representable K-theory}.

\end{defn}
Representable K-theory over [[paracompact topological spaces]] was discussed in (\hyperlink{Karoubi70}{Karoubi 70}).

\begin{defn}
\label{InverseLimitTopologicalKTheory}\hypertarget{InverseLimitTopologicalKTheory}{}
\textbf{(inverse limit topological K-theory)}

Let $X$ be a [[topological space]] with the structure of a [[CW-complex]], hence a [[colimit]] (``[[direct limit]]'') $X \simeq \underset{\longrightarrow}{\lim}_n X_n$ such that each $X_n$ is a [[finite cell complex]], hence in particular a [[compact topological space]]. Then the [[limit]] ([[inverse limit]]) of the corresponding K-group

\begin{displaymath}
K(X)_{invl} \coloneqq \underset{\longleftarrow}{\lim}_n K(X_n)
\end{displaymath}
is called the \emph{inverse limit K-theory} of $X$.

\end{defn}
$\,$

No two of these definitions are equivalent to each other on all of their domain of defintion (e.g. \hyperlink{AndersonHodgkin68}{Anderson-Hodgkin 68}, \hyperlink{JackowskiOliver}{Jackowski-Oliver}).

Representable and direct limit K-theory of spaces that are [[sequential colimits]] of [[compact spaces]] differ in general by a [[lim{\tt \symbol{94}}1]]-term (\hyperlink{SegalAtiyah69}{Segal-Atiyah 69, prop. 4.1}).

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

\begin{example}
\label{}\hypertarget{}{}
\textbf{(topological K-theory ring of the [[point space]])}

We have already seen in example \ref{KGroupOfThePoint} that

\begin{displaymath}
K(\ast) \simeq \mathbb{Z}
  \,.
\end{displaymath}
\end{example}
\begin{example}
\label{ComplexTopologicalKTheoryOfTheCircle}\hypertarget{ComplexTopologicalKTheoryOfTheCircle}{}
\textbf{(complex topological K-theory ring of the [[circle]])}

Since the complex [[general linear group]] $GL(n,\mathbb{C})$ is [[path-connected topological space|path-connected]] (\href{general+linear+group#ConnectednessOfGeneralLinearGroup}{this prop.}) and hence the [[classifying space]] $B GL(n,\mathbb{C})$ is [[simply connected topological space|simply]]-connected, hence its [[fundamental group]] is trivial $\pi_1(B GL(n,\mathbb{C})) \simeq [S^1, B GL(n,\mathbb{C})] = 1$. Accordingly, all [[complex vector bundles]] on $S^1$ are [[isomorphism|isomorphic]] toa [[trivial vector bundle]].

It follows that

\begin{displaymath}
K_{\mathbb{C}}(S^1) \simeq \mathbb{Z}
  \phantom{AA} \text{and} \phantom{AA}
  \tilde K_{ \mathbb{C} }(S^1) \simeq 0
  \,.
\end{displaymath}
\end{example}
\begin{example}
\label{TopologicalKTheoryRingOfThe2Sphere}\hypertarget{TopologicalKTheoryRingOfThe2Sphere}{}
\textbf{(complex topological K-theory ring of the [[2-sphere]])}

For $X = \ast$ the [[point space]], the [[fundamental product theorem in topological K-theory]] \ref{FundamentalProductTheorem} states that the homomorphism

\begin{displaymath}
\itexarray{
    \mathbb{Z}[h]/((h-1)^2)
      &\longrightarrow&
    K_{\mathbb{C}}(S^2)
    \\
    h &\mapsto& h
  }
\end{displaymath}
is an [[isomorphism]].

This means that the relation $(h-1)^2 = 0$ satisfied by the [[basic complex line bundle on the 2-sphere]] (\href{basic+complex+line+bundle+on+the+2-sphere#TensorRelationForBasicLineBundleOn2Sphere}{this prop.}) is the \emph{only} relation is satisfies in topological K-theory.

Notice that the underlying [[abelian group]] of $\mathbb{Z}[h]/((h-1)^2)$ is two [[direct sum]] copies of the [[integers]],

\begin{displaymath}
K_{\mathbb{C}}(S^2) \simeq \mathbb{Z} \oplus \mathbb{Z} = \langle 1, h\rangle
\end{displaymath}
one copy spanned by the [[trivial vector bundle|trivial]] [[complex line bundle]] on the 2-sphere, the other spanned by the [[basic complex line bundle on the 2-sphere]]. (In contrast, the underlying abelian group of the [[polynomial ring]] $\mathbb{R}[h]$ has infinitely many copies of $\mathbb{Z}$, one for each $h^n$, for $n \in \mathbb{N}$).

It follows (by \href{topological+K-theory#KGrupDirectSummandReducedKGroup}{this prop.}) that the [[reduced K-theory]] group of the 2-sphere is

\begin{displaymath}
\tilde K_{\mathbb{C}}(S^2) \simeq \mathbb{Z}
  \,.
\end{displaymath}
\end{example}
\begin{example}
\label{}\hypertarget{}{}
\textbf{(complex topological K-theory of the [[torus]])}

Consider the [[torus]] $S^1 \times S^1$, the [[product topological space]] of the [[circle]] with itself (with the [[Euclidean space|Euclidean]] [[subspace topology]]).

By example \ref{ReducedKTheoryOfProductSpace} we have

\begin{displaymath}
\tilde K(S^1 \times S^1)
    \simeq
  \tilde K(\underset{S^2}{\underbrace{S^1 \wedge S^1}})
    \oplus
  \underset{= 0}{
  \underbrace{
  \tilde K(S^1)
    \oplus
  \tilde K(S^1)
  }}
  \,.
\end{displaymath}
Since the [[smash product]] of the circle with itself is the [[2-sphere]], and since, the complex K-theory of the circle vanishes by example \ref{ComplexTopologicalKTheoryOfTheCircle}, this shows that the topological K-theory of the torus coincides with that of the 2-sphere:

\begin{displaymath}
K(S^1 \times S^1) \simeq K(S^2)
  \,.
\end{displaymath}
\end{example}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{homotopy_invariance}{}\subsubsection*{{Homotopy invariance}}\label{homotopy_invariance}

\begin{prop}
\label{KGroupsHomotopyInvariance}\hypertarget{KGroupsHomotopyInvariance}{}
\textbf{([[homotopy invariance]] of K-groups)}

Let $X$ and $Y$ be [[paracompact Hausdorff spaces]], and let

\begin{displaymath}
f \;\colon\; X \longrightarrow Y
\end{displaymath}
be a [[continuous function]] which is a [[homotopy equivalence]]. Then the pullback operation on (reduced) K-groups along $f$ (remark \ref{FunctorialityOfKGroup}, remark \ref{FunctorialityOfReducedKGroups}) is an [[isomorphism]]:

\begin{displaymath}
f^\ast
  \;\colon\;
  K(Y)
    \overset{\simeq}{\longrightarrow}
  K(X)
\end{displaymath}
and

\begin{displaymath}
f^\ast
  \;\colon\;
  \tilde K(Y)
    \overset{\simeq}{\longrightarrow}
  \tilde K(X)
  \,.
\end{displaymath}
\end{prop}
\begin{proof}
This is an immediate consequence of the fact that over paracompact Hausdorff spaces isomorphism classes of topological vector bundles are homotopy invariant (\href{topological+vector+bundle#HomotopyInvarianceOfIsomorphismClassesOfVectorBundles}{this example.})

\end{proof}
\hypertarget{ExactSequences}{}\subsubsection*{{Exact sequences}}\label{ExactSequences}

We discuss the [[long exact sequences in cohomology]] for topological K-theory.

What makes these work in prop. \ref{ExactSequenceInReducedTopologicalKTheory} below, turns out to be the following property of [[topological vector bundles]]:

\begin{lemma}
\label{VectorBundleOnClosedSubsetOfCompactHausdorffSpaceIsPullbackOfBundeOnQuotientSpace}\hypertarget{VectorBundleOnClosedSubsetOfCompactHausdorffSpaceIsPullbackOfBundeOnQuotientSpace}{}
\textbf{([[topological vector bundle]] [[trivial vector bundle|trivial]] over [[closed subspace]] of [[compact Hausdorff space]] is [[pullback bundle|pullback]] of bundle on [[quotient space]])}

Let $X$ be a [[compact Hausdorff space]] and let $A \subset X$ be a [[closed subspace]].

If a [[topological vector bundle]] $E \overset{p}{\to} X$ is such that its restriction $E\vert_A$ is [[trivializable vector bundle|trivializable]], then $E$ is [[isomorphism|isomorphic]] to the [[pullback bundle]] $q^\ast E'$ of a topological vector bundle $E' \to X/A$ over the [[quotient space]]..

\end{lemma}
The \textbf{proof} of lemma \ref{VectorBundleOnClosedSubsetOfCompactHausdorffSpaceIsPullbackOfBundeOnQuotientSpace} is given at \emph{[[topological vector bundle]]} \href{topological+vector+bundle+VectorBundleOnClosedSubsetOfCompactHausdorffSpaceIsPullbackOfBundeOnQuotientSpace}{here}. What makes that proof work, in turn, is the [[Tietze extension theorem]], via \href{topological+vector+bundle#IsomorphismOfVectorBundlesOnClosedSubsetOfCompactHausdorffSpaceExtendsToOpenNeighbourhoods}{this lemma}.

\begin{prop}
\label{ExactSequenceInReducedTopologicalKTheory}\hypertarget{ExactSequenceInReducedTopologicalKTheory}{}
\textbf{(exact sequence in reduced topological K-theory)}

Let

\begin{itemize}%
\item $X$ be a [[pointed topological space|pointed]] [[compact Hausdorff space]];


\item $A \subset X$ a [[pointed topological space|pointed]] [[closed subset|closed]] [[subspace]]



\end{itemize}
Write $X/A$ for the corresponding [[quotient space]] (\href{quotient+space#QuotientBySubspace}{this def.}). Denote the [[continuous functions]] of [[subspace]] inclusion and of [[quotient space]] co-projection by

\begin{displaymath}
A
    \overset{i}{\longrightarrow}
  X
    \overset{q}{\longrightarrow}
  X/A
  \,,
\end{displaymath}
respectively. Then the induced sequence of [[reduced K-theory]] groups (def. \ref{KernelReducedKGroup}, remark \ref{FunctorialityOfReducedKGroups})

\begin{displaymath}
\tilde K_{\mathbb{C}}(X/A)
    \overset{q^\ast}{\longrightarrow}
  \tilde K_{\mathbb{C}}(X)
    \overset{i^\ast}{\longrightarrow}
  \tilde K_{\mathbb{C}}(A)
\end{displaymath}
is [[exact sequence|exact]], meaning that they induce an [[isomorphism]]

\begin{displaymath}
im(q^\ast) \simeq ker(i^\ast)
\end{displaymath}
between the [[image]] of $g^\ast$ and the [[kernel]] of $i^\ast$.

Similarly the sequence of unreduced and [[relative K-groups]] (def. \ref{RelativeKTheory}) is exact:

\begin{displaymath}
\tilde K_{\mathbb{C}}(X/A)
    \overset{q^\ast}{\longrightarrow}
  K_{\mathbb{C}}(X)
    \overset{i^\ast}{\longrightarrow}
  K_{\mathbb{C}}(A)
\end{displaymath}
\end{prop}
(e.g. \href{Wirthmuller12}{Wirthmuller 12, p. 32 (34 of 67)}, \hyperlink{Hatcher}{Hatcher, prop. 2.9})

\begin{proof}
First observe that both statements are equivalent to each other: By [[natural transformation|naturality]] of the construction of [[kernels]] the following [[commuting diagram|diagram commutes]]:

\begin{displaymath}
\itexarray{
    \tilde K_{\mathbb{C}}(X/A)
      &\overset{q^\ast}{\longrightarrow}&
    \tilde K_{\mathbb{C}}(X)
      &\overset{i^\ast}{\longrightarrow}&
    \tilde K_{\mathbb{C}}(A)
    \\
    {}^{\mathllap{id}}\downarrow && \downarrow && \downarrow
    \\
    \tilde K_{\mathbb{C}}(X/A)
      &\overset{q^\ast}{\longrightarrow}&
    K_{\mathbb{C}}(X)
      &\overset{i^\ast}{\longrightarrow}&
    K_{\mathbb{C}}(A)
    \\
    &&
    \downarrow && \downarrow
    \\
    && K_{\mathbb{C}}(\ast) &\underset{id}{\longrightarrow}& K_{\mathbb{C}}(\ast)
  }
  \,.
\end{displaymath}
Here the top vertical morphisms, the [[kernel]] inclusions, are [[monomorphisms]], and hence the top horizonal row is exact precisely if the middle horizontal row is.

Hence it is sufficient to consider the top row. First of all the composite function $A \overset{i}{\to} X \overset{q}{\to} X/A$ is a [[constant function]], constant on the basepoint, and hence $i^\ast \circ q^\ast$ is a constant function, constant on zero. This says that we have an inclusion

\begin{displaymath}
im(q^\ast) \subset ker(i^\ast)
  \,.
\end{displaymath}
Hence it only remains to see for $x \in \tilde K(X)$ a class with $i^\ast(x) = 0$ that $x = q^\ast(y)$ comes from a class on the quotient $X/A$. But by compactness, the class $x$ is given by a [[virtual vector bundle]] of the form $E - rk(E)$ (prop. \ref{OnCompactHausdorffVirtualVectorBundlesAreFormalDifferentcesWithATrivialBundle}, prop. \ref{KGrupDirectSummandReducedKGroup}). and by prop. \ref{ReducedKEquivalenceRelationVerified} the triviality of $i^\ast(E- rk(E))$ means that there is $n \in \mathbb{N}$ such that $i \ast(E) \oplus_A (A \times \mathbb{C}^n)$ is a [[trivializable vector bundle]]. Therefore lemma \ref{VectorBundleOnClosedSubsetOfCompactHausdorffSpaceIsPullbackOfBundeOnQuotientSpace} gives that $E \oplus_X (X \times \mathbb{C}^n)$ is isomorphic to the [[pullback bundle]] of a vector bundle $E'$ on $X/A$. This proves the claim.

\end{proof}
\begin{cor}
\label{LongExactSequenceInReducedTopologicalKTheory}\hypertarget{LongExactSequenceInReducedTopologicalKTheory}{}
\textbf{([[long exact sequence]] in [[reduced K-theory|reduced topological K-theory]])}

For $X$ a [[compact Hausdorff space]] and for $A \subset X$ a [[closed subset|closed]] [[subspace]] inclusion, there is a [[long exact sequence]] of [[reduced K-theory]] groups of the form

\begin{displaymath}
\cdots
    \longrightarrow
  \tilde K_{\mathbb{C}}(\Sigma (X/A))
    \longrightarrow
  \tilde K_{\mathbb{C}}(\Sigma X)
    \longrightarrow
  \tilde K_{\mathbb{C}}( \Sigma A )
    \longrightarrow
  \tilde K_{\mathbb{C}}(X/A)
   \longrightarrow
  \tilde K_{\mathbb{C}}(X)
    \longrightarrow
  \tilde K_{\mathbb{C}}(A)
  \,,
\end{displaymath}
where $\Sigma(-)$ denotes [[reduced suspension]].

\end{cor}
\begin{proof}
The sequence is induced by functoriality (remark \ref{FunctorialityOfReducedKGroups}) from the long [[cofiber sequence]]

\begin{displaymath}
A
    \overset{i}{\longrightarrow}
  X
    \longrightarrow
  X \cup Cone(A)
    \longrightarrow
  (X \cup Cone(A)) \cup Cone(X)
    \longrightarrow
  ((X \cup Cone(A)) \cup Cone(X)) \cup (X \cup Cone(A))
    \longrightarrow
  \cdots
\end{displaymath}
obtained by consecutively forming [[mapping cones]]. By the discussion at \emph{[[topological cofiber sequence]]} this may be rearranged as

\begin{displaymath}
\itexarray{
    A
      &\overset{i}{\longrightarrow}&
    X
      &\overset{j}{\longrightarrow}&
    Cone(i)
      &\longrightarrow&
    Cone(j)
    \\
    && &&
    \downarrow {\mathrlap{\text{homotopy} \atop \text{equivalence}}}
    &&
    \downarrow^{\mathrlap{\text{homotopy} \atop \text{equivalence}}}
    \\
    && &&
    X/A
    &&
    S A
      &\overset{S i}{\longrightarrow}&
    S X
      &\longrightarrow& \cdots
  }
\end{displaymath}
(\href{topological+cofiber+sequence#HomotopyEquivalenceSuspensionWithMappingConeOfMappingCone}{this prop.} and \href{topological+cofiber+sequence}{this}).

The claim hence follows by the [[homotopy invariance]] of the K-groups (prop. \ref{KGroupsHomotopyInvariance}).

\end{proof}
$\,$

We discuss some useful consequences of the [[long exact sequences in cohomology]].

\begin{prop}
\label{DirectSumOfKTheoryGroupsOverRetracts}\hypertarget{DirectSumOfKTheoryGroupsOverRetracts}{}
\textbf{([[direct sum]] decomposition of K-theory groups over [[retractions]])}

Let $X$ be a ([[pointed topological space|pointed]]) [[compact topological space]] and $A \subset X$ a ([[pointed topological space|pointed]]) [[closed subspace]], such that the [[subspace]] inclusion $A \overset{i}{\to}$ X as a [[retraction]], i.e. a [[continuous function]] $r \colon X \to A$ such that the [[composition|composite]]

\begin{displaymath}
id_A
    \;\colon\;
  A
    \overset{i}{\longrightarrow}
  X
    \overset{r}{\longrightarrow}
  A
\end{displaymath}
is the [[identity function]].

Then there is a splitting of the K-theory group of $X$ as a [[direct sum]] of the K-theory of $A$ and the [[relative K-theory]] of the [[quotient space]] $X/A$:

\begin{displaymath}
K(X)
    \;\simeq\;
  K(A)
    \oplus
  K(X,A)
\end{displaymath}
and in the pointed case a splitting of the [[reduced K-theory]] groups

\begin{displaymath}
\tilde K(X)
    \;\simeq\;
  \tilde K(A)
    \oplus
  K(X,A)
  \,.
\end{displaymath}
\end{prop}
\begin{proof}
The long exact sequence from cor \ref{LongExactSequenceInReducedTopologicalKTheory} together with the retraction yields

\begin{displaymath}
\tilde K(A)
    \underoverset
      {\underset{r^\ast}{\longrightarrow}}
      {\overset{i^\ast}{\longleftarrow}}
      {}
  \tilde K(X)
    \overset{}{\longleftarrow}
  K(X,A)
    \longleftarrow
  \tilde K(\Sigma A)
    \underoverset
      {\underset{}{\longrightarrow}}
      {\overset{}{\longleftarrow}}
      {}
  \tilde K(\Sigma X)
  \,.
\end{displaymath}
The splitting makes the morphisms $i^\ast$ and its suspension be [[surjections]], so that the long exact sequence decomposes into [[short exact sequences]] which are [[split exact sequence|split exact]]:

\begin{displaymath}
0
    \longleftarrow
  \tilde K(A)
    \underoverset
     {\underset{r^\ast}{\longrightarrow}}
     {\overset{i^\ast}{\longleftarrow}}
     {}
  \tilde K(X)
    \longleftarrow
  K(X,A)
    \longleftarrow
  0
  \,.
\end{displaymath}
\end{proof}
\begin{example}
\label{FiniteWedgePropertyForReducedTopologicalKTheory}\hypertarget{FiniteWedgePropertyForReducedTopologicalKTheory}{}
\textbf{(finite [[wedge axiom]])}

Let $(X,x)$ and $(Y,y)$ be two [[pointed topological space|pointed]] [[compact Hausdorff spaces]] with [[wedge sum]]

\begin{displaymath}
X \vee Y \;\coloneqq\; (X \sqcup Y)/(x \sim y)
\end{displaymath}
(i.e. the [[quotient topological space|quotient]] of their [[disjoint union space]] by re-identifying the base points).

Then there is an [[isomorphism]]

\begin{displaymath}
\tilde K(X \vee Y) \simeq \tilde K(X) \oplus \tilde K(Y)
  \,.
\end{displaymath}
\end{example}
\begin{proof}
We have [[retracts]]

\begin{displaymath}
X = X \times \{y\} \hookrightarrow X \times Y
\end{displaymath}
and

\begin{displaymath}
Y = \{x\} \times Y \hookrightarrow (X \times Y) / (X \times \{y\})
  \,.
\end{displaymath}
Applying prop. \ref{DirectSumOfKTheoryGroupsOverRetracts} to each of these consecutively yields an isomorphism that establishes the claim:

\begin{displaymath}
\tilde K(X \times Y)
    \simeq
  \tilde K(X)
    \oplus
  \tilde K( (X \times Y)/(X \times \{y\}) )
    \simeq
  \tilde K(X) \oplus \tilde K(Y) \oplus \tilde K(X \wedge Y)
  \,.
\end{displaymath}
This proves the claim.

Alternatively, we may again argue directly from the long exact sequence:

Consider the subspace inclusion

\begin{displaymath}
X \subset X \vee Y
  \,.
\end{displaymath}
This is a [[closed subspace]] because its [[complement]] is $(X \vee Y) \setminus X = Y \setminus \{y\}$ which is open because all points in a [[Hausdorff space]] (which is in particular $T_1$-[[separation axiom|separated]]) are closed. Moreover, by definition of [[wedge sum]] the corresponding [[quotient space]] is $Y$:

\begin{displaymath}
(X \vee Y) / X \simeq Y
  \,.
\end{displaymath}
Similary for the inclusion of $Y$. Hence in particular these inclusions and quotients are [[retractions]] in that they factor the [[identity maps]] as

\begin{displaymath}
id_X \;\colon\; X \longrightarrow X \vee Y \longrightarrow X
  \phantom{AA}
    \text{and}
  \phantom{AA}
  id_Y \;\colon\; Y \longrightarrow X \vee Y \longrightarrow Y
  \,.
\end{displaymath}
By functoriality (remark \ref{FunctorialityOfReducedKGroups}) this implies that similarly

\begin{displaymath}
id_{\tilde K(X)}
    \;\colon\;
  \tilde K(X) \longrightarrow \tilde K(X \vee Y) \longrightarrow \tilde K(X)
  \phantom{AA}
   \text{and}
  \phantom{AA}
  id_{\tilde K(Y)}
    \;\colon\;
  \tilde K(Y) \longrightarrow \tilde K(X \vee Y) \longrightarrow \tilde K(Y)
  \,.
\end{displaymath}
In particular these maps are [[injective function|injections]] and [[surjective function|surjections]], respectively.

Therefore by prop. \ref{ExactSequenceInReducedTopologicalKTheory} there are [[short exact sequences]] of the form

\begin{displaymath}
0
    \to
  \tilde K(X)
    \longrightarrow
  \tilde K(X \vee Y)
    \longrightarrow
  \tilde K(Y)
    \to
  0
\end{displaymath}
which are [[split exact sequences|split exact]]. This implies the claim.

\end{proof}
\hypertarget{ExternalProducts}{}\subsubsection*{{External product}}\label{ExternalProducts}

\begin{defn}
\label{ExternalTensorProductInKTheory}\hypertarget{ExternalTensorProductInKTheory}{}
\textbf{(external product in K-theory)}

Let $X$ and $Y$ be [[topological spaces]]. Then the [[external tensor product of vector bundles|external tensor product]] of [[topological vector bundles]] over $X$ and $Y$

\begin{displaymath}
\boxtimes
    \;\colon\;
  Vect(X) \times Vect(Y)
    \longrightarrow
  Vect(X \times Y)
\end{displaymath}
induces on K-groups an \emph{external product}

\begin{displaymath}
\boxtimes
    \;\colon\;
  K(X) \oplus K(Y)
    \longrightarrow
  K(X \times Y)
\end{displaymath}
\end{defn}
We want to see that this restricts to an operation on [[reduced K-theory]]. To this end we need the following proposition:

\begin{prop}
\label{ReducedKTheoryOfProductSpace}\hypertarget{ReducedKTheoryOfProductSpace}{}
\textbf{(reduced K-theory of product space)}

Let $(X,x_0)$ $(Y,y_0)$ be two [[pointed topological space|pointed]] [[compact Hausdorff spaces]] with $X \times Y$ their [[product topological space]] and $X \wedge Y$ their [[smash product]]. Then there is an isomorphism of [[reduced K-theory]] groups

\begin{displaymath}
\tilde K(X \times Y)
    \;\simeq\;
  \tilde K(X \wedge Y)
    \oplus
  \tilde K(X)
    \oplus
  \tilde K(Y)
  \,.
\end{displaymath}
\end{prop}
\begin{proof}
Be definition, the [[smash product]] is the [[quotient topological space]] of the [[product topological space]] by the [[wedge sum]]:

\begin{displaymath}
X \wedge Y
  \;=\;
  (X \times Y) / (X \vee Y)
\end{displaymath}
for the inclusion

\begin{displaymath}
\itexarray{
    X \vee Y &\overset{i}{\longrightarrow}& X \times Y
    \\
    x &\mapsto& (x, y_0)
    \\
    y &\mapsto& (x_0, y)
  }
  \,.
\end{displaymath}
This quotient is still a [[compact topological space]] because [[continuous images of compact spaces are compact]] and and it is still [[Hausdorff topological space]] because [[compact subspaces in Hausdorff spaces are separated by neighbourhoods from points]], so that the point $(X \vee Y)/ (X \vee Y) \in (X \times Y)/(X \vee Y)$ is separated by open neighbourhoods from points in $(X \times Y) \setminus (X \vee Y)$.

Hence corollary \ref{LongExactSequenceInReducedTopologicalKTheory} yields a [[long exact sequence]] of the form

\begin{displaymath}
\tilde K(\Sigma(X \times Y))
    \overset{\Sigma i^\ast}{\longrightarrow}
  \tilde K( (\Sigma X) \vee (\Sigma Y))
    \longrightarrow
  \tilde K( X \wedge Y )
    \longrightarrow
  \tilde K( X \times Y )
    \overset{i^\ast}{\longrightarrow}
  \tilde K(X \vee Y)
  \,.
\end{displaymath}
By example \ref{FiniteWedgePropertyForReducedTopologicalKTheory} the two terms involving reduced topological K-theory of a wedge sum are direct sums of the reduced K-theory of the wedge summands:

\begin{displaymath}
\tilde K(\Sigma(X \times Y))
    \overset{\Sigma i^\ast}{\longrightarrow}
  \tilde K(\Sigma X) \oplus \tilde K(\Sigma Y)
    \longrightarrow
  \tilde K( X \wedge Y )
    \longrightarrow
  \tilde K( X \times Y )
    \overset{i^\ast}{\longrightarrow}
  \tilde K(X) \oplus \tilde K(Y)
  \,.
\end{displaymath}
Now observe that, via example \ref{FiniteWedgePropertyForReducedTopologicalKTheory}, the morphisms $i^\ast$ and $\Sigma i^\ast$ are [[split epimorphisms]], with [[section]] given by ``external direct sum''

\begin{displaymath}
\itexarray{
    \tilde K(X) \oplus \tilde K(Y)
      &\longrightarrow&
    \tilde K(X \times Y)
    \\
    (E_X, E_Y)
      &\mapsto&
    p_X^\ast(E_X) + p_Y^\ast(E_Y)
  }
  \,.
\end{displaymath}
This means that the long exact sequence decomposes into [[short exact sequences]]

\begin{displaymath}
0
   \to
  \tilde K(X \wedge Y)
    \longrightarrow
  \tilde K(X \times Y)
    \overset{i^\ast}{\longrightarrow}
  \tilde K(X) \oplus \tilde K(Y)
    \to
  0
\end{displaymath}
which moreover are [[split exact sequence|split exact]]. This yields the claim.

\end{proof}
It follows that:

\begin{prop}
\label{ExternalTensorProductOnReducedKGroups}\hypertarget{ExternalTensorProductOnReducedKGroups}{}
\textbf{(external product on reduced K-groups)}

Let $X$ and $Y$ be [[pointed topological space|pointed]] [[compact Hausdorff spaces]]. Then the external product on K-groups (def. \ref{ExternalTensorProductInKTheory}) restricts to [[reduced K-groups]] under the inclusion $\tilde K(-) \hookrightarrow K(-)$ from prop. \ref{KGrupDirectSummandReducedKGroup} and the inclusion $\tilde K(-\wedge -) \hookrightarrow K(-\times -)$ from prop. \ref{ReducedKTheoryOfProductSpace}, in that there is a morphism $\tilde \boxtimes$ that makes the following [[commuting diagram|diagram commute]]:

\begin{displaymath}
\itexarray{
    \tilde K(X) \oplus \tilde K(Y)
      &\hookrightarrow&
    K(X) \oplus K(Y)
    \\
    {}^{\mathllap{\tilde \boxtimes}}\downarrow && \downarrow^{\mathrlap{\boxtimes}}
    \\
    \tilde K(X \wedge Y) &\hookrightarrow& K(X \times Y)
  }
  \,.
\end{displaymath}
\end{prop}
\begin{proof}
By prop. \ref{KGrupDirectSummandReducedKGroup} the elements in $\tilde K(X)$ and $\tilde K(Y)$ are represented by [[virtual vector bundles]] which vanish when restricted to the base points $x \in X$ and $y \in Y$, respectively. But this implies that their [[external tensor product of vector bundles]] vanishes over $X \times \{y\}$ and $\{x\} \times Y$. From the proof of prop. \ref{ReducedKTheoryOfProductSpace} it is the restriction of the product to to these subspaces that gives the map

\begin{displaymath}
K(X \times Y) \simeq \tilde K(X \times Y) \oplus \tilde K(X) \oplus \tilde K(Y)
    \longrightarrow
  \tilde K(X) \oplus \tilde K(Y)
\end{displaymath}
and hence on these element this component vanishes.

\end{proof}
\hypertarget{fundamental_product_theorem}{}\subsubsection*{{Fundamental product theorem}}\label{fundamental_product_theorem}

In order to compute K-classes, one needs the computation of some basic cases, such as that of the K-theory groups of [[n-spheres]] and of [[product spaces]] with $n$-spheres. The \emph{[[fundamental product theorem in K-theory]]} determines these K-theory groups. Its result is most succinctly summarized by the statement of \emph{[[Bott periodicity]]}, to which we turn \hyperlink{BottPeriodicities}{below}.

Before discussing the product theorem, it is useful to recall the analogous situation in [[ordinary cohomology]] $H^\bullet(-) \coloneqq H^\bullet(\mathbb{Z})$. Here it is immediate to determine the cohomology groups of the [[n-spheres]], in particular one finds that for the [[2-sphere]] is $H^\bullet(S^2) = \mathbb{Z}\langle e\rangle \oplus \mathbb{Z}\langle h\rangle$, for $h \in \tilde H^2(S^2)$ the [[first Chern class]] of the [[basic complex line bundle on the 2-sphere]]. As a ring this has the trivial product $h^2 = 0$, since by degree-reasons the [[cup product]] goes $H^2(S^2) \otimes H^2(S^2) \to H^4(S^2) = 0$.

Therefore me may write the ordinary [[cohomology ring]] of the 2-sphere as the following [[quotient ring]] of the [[polynomial ring]] in the generator $h$:

\begin{displaymath}
H^\bullet(S^2) \simeq \mathbb{Z}[h]/\left( (h)^2 \right)
  \,.
\end{displaymath}
Notice that in ordinary cohomology $h$ is also the generator of the [[reduced cohomology]] group $\tilde H^\bullet(S^2) \simeq \mathbb{Z}\langle h\rangle$. Now as an element of $K_{\mathbb{C}}(S^2)$ the [[basic complex line bundle on the 2-sphere]] is not reduced, but its image in [[reduced K-theory]] is the [[Bott element]] [[virtual vector bundle]] $\beta = h-1$ (def. \ref{BottElement}). The [[fundamental product theorem in topological K-theory]] says, in particular, that the complex topological K-theory of the 2-sphere behaves in just the same way as the ordinary cohomology, if only one replaces the generator $h$ by $\beta = h-1$.

First of all, the Bott element also squares to zero:

\begin{prop}
\label{BottElementNilotentcy}\hypertarget{BottElementNilotentcy}{}
\textbf{(nilpotency of the [[Bott element]])}

For $S^2 \subset \mathbb{R}^3$ the [[2-sphere]] with its [[Euclidean space|Euclidean]] [[subspace topology]], write $h \in Vect_{\mathbb{C}}(S^2)_{/\sim}$ for the [[basic complex line bundle on the 2-sphere]]. Its image in the [[topological K-theory]] ring $K(S^2)$ satisfies the relation

\begin{displaymath}
2 h = h^2 + 1
  \;\;\Leftrightarrow\;\;
  (h-1)^2 = 0
\end{displaymath}
\end{prop}
A \textbf{proof} of this may be obtained by analysis of the relevant [[clutching function]], see \emph{\href{basic+complex+line+bundle+on+the+2-sphere#TensorRelationForBasicLineBundleOn2Sphere}{here}}.

Notice that $h-1$ is the image of $h$ in the [[reduced K-theory]] $\tilde K(X)$ of $S^2$ under the splitting $K(X) \simeq \tilde K(X) \oplus \mathbb{Z}$ (by \href{topological+K-theory#KGrupDirectSummandReducedKGroup}{this prop.}). This element

\begin{displaymath}
h - 1 \in \tilde K_{\mathbb{C}}(S^2)
\end{displaymath}
is the \emph{[[Bott element]]} of complex [[topological K-theory]] (def. \ref{BottElement}).

It follows from prop. \ref{BottElementNilotentcy} that there is a [[ring homomorphism]] of the form

\begin{displaymath}
\itexarray{
    \mathbb{Z}[h]/\left( (h-1)^2 \right)
      &\overset{}{\longrightarrow}&
    K(S^2)
    \\
    h &\overset{\phantom{AAA}}{\mapsto}& h
  }
\end{displaymath}
from the [[polynomial ring]] in one abstract generator, [[quotient ring|quotiented]] by this relation, to the [[topological K-theory]] ring.

More generally, for $X$ a [[topological space]], then this induces the composite ring homomorphism

\begin{displaymath}
\itexarray{
    K(X) \otimes \mathbb{Z}[h]/\left((h-1)^2 \right)
      & \longrightarrow &
    K(X) \otimes K(S^2)
      & \overset{\boxtimes}{\longrightarrow} &
    K(X \times S^2)
    \\
    (E, h)
      &\overset{\phantom{AAA} }{\mapsto}&
    (E,H)
      &\overset{\phantom{AAA}}{\mapsto}&
    (\pi_{X}^\ast E) \cdot (\pi_{S^2}^\ast H)
  }
\end{displaymath}
to the topological K-theory ring of the [[product topological space]] $X \times S^2$, where the second map $\boxtimes$ is the external product from def. \ref{ExternalTensorProductInKTheory}.

\begin{prop}
\label{FundamentalProductTheorem}\hypertarget{FundamentalProductTheorem}{}
\textbf{([[fundamental product theorem in topological K-theory]])}

For $X$ a [[compact Hausdorff space]], then this ring homomorphism is an [[isomorphism]].

\end{prop}
(e.g. \hyperlink{Hatcher}{Hatcher, theorem 2.2})

\begin{remark}
\label{}\hypertarget{}{}
More generally, for $L\to X$ a [[complex line bundle]] with class $l \in K(X)$ and with $P(1 \oplus L)$ denoting its [[projective bundle]] then

\begin{displaymath}
K(X)[h]/((h-1)(l \cdot h -1))
   \simeq
  K(P(1 \oplus L))
\end{displaymath}
\end{remark}
(e.g. \hyperlink{Wirthmuller12}{Wirthmuller 12, p. 17})

As a special case this implies the first statement above:

For $X = \ast$ the product theorem prop. \ref{FundamentalProductTheorem} says in particular that the first of the two morphisms in the composite is an [[isomorphism]] (example \ref{TopologicalKTheoryRingOfThe2Sphere} below) and hence by the [[two-out-of-three]]-property for [[isomorphisms]] it follows that:

\begin{cor}
\label{ExternalProductTheorem}\hypertarget{ExternalProductTheorem}{}
\textbf{([[external product theorem in topological K-theory]])}

For $X$ a [[compact Hausdorff space]] we have that the external product in K-theory $\boxtimes$ (def. \ref{ExternalTensorProductInKTheory}) with vector bundles on the [[2-sphere]]

\begin{displaymath}
\boxtimes
    \;\colon\;
  K(X) \otimes K(S^2) \overset{\simeq}{\longrightarrow} K(X \times S^2)
\end{displaymath}
is an [[isomorphism]] in [[topological K-theory]].

\end{cor}
\hypertarget{BottPeriodicities}{}\subsubsection*{{Bott periodicity}}\label{BottPeriodicities}

When restricted to [[reduced K-theory]] then the external product theorem (cor. \ref{ExternalProductTheorem}) yields the statement of [[Bott periodicity]] of topological K-theory:

\begin{prop}
\label{BottPeriodicity}\hypertarget{BottPeriodicity}{}
\textbf{([[Bott periodicity]])}

Let $X$ be a [[pointed topological space|pointed]] [[compact Hausdorff space]].

Then the external product $\tilde X$ in reduced K-theory (prop. \ref{ExternalTensorProductInKTheory}) with the image of the [[basic complex line bundle on the 2-sphere]] in reduced K-theory yields an [[isomorphism]] of [[reduced K-groups]]

\begin{displaymath}
(h-1)
    \widetilde \boxtimes
  (-)
    \;\colon\;
  \tilde K(X) \overset{\simeq}{\longrightarrow} \tilde K(\Sigma^2 X)
\end{displaymath}
from that of $X$ to that of its double [[suspension]] $\Sigma^2 X$.

\end{prop}
\hyperlink{Wirthmuller12}{e.g. Wirthmuller 12, p. 34 (36 of 67)}

\begin{proof}
By \href{topologica+K-theory#ReducedKTheoryOfProductSpace}{this example} there is for any two pointed compact Hausdorff spaces $X$ and $Y$ an [[isomorphism]]

\begin{displaymath}
\tilde K(Y \times X)
    \simeq
  \tilde K(Y \wedge X)
    \oplus
  \tilde K(Y)
    \oplus
  \tilde K(X)
\end{displaymath}
relating the reduced K-theory of the [[product topological space]] with that of the [[smash product]].

Using this and the fact that for any pointed compact Hausdorff space $Z$ we have $K(Z) \simeq \tilde K(Z) \oplus \mathbb{Z}$ (\href{topological+K-theory#KGrupDirectSummandReducedKGroup}{this prop.}) the isomorphism of the external product theorem (cor. \ref{ExternalProductTheorem})

\begin{displaymath}
K(S^2) \otimes K(X)
   \underoverset{\simeq}{\boxtimes}{\longrightarrow}
  K(S^2 \times X)
\end{displaymath}
becomes

\begin{displaymath}
\left(
    \tilde K(S^2) \oplus \mathbb{Z}
  \right)
    \otimes
  \left(
     \tilde K(X) \oplus \mathbb{Z}
  \right)
   \;\simeq\;
  \left(
    \tilde K(S^2 \times X)
     \oplus
     \mathbb{Z}
  \right)
   \simeq
  \left(
    \tilde K(S^2 \wedge X)
      \oplus
    \tilde K(S^2)
      \oplus
    \tilde K(X)
      \oplus
    \mathbb{Z}
  \right)
  \,.
\end{displaymath}
Multiplying out and chasing through the constructions to see that this reduces to an isomorphism on the common summand $\tilde K(S^2) \oplus \tilde K(X) \oplus \mathbb{Z}$, this yields an isomorphism of the form

\begin{displaymath}
\tilde K(S^2) \otimes \tilde K(X)
    \underoverset{\simeq}{\widetilde \boxtimes}{\longrightarrow}
  \tilde K(S^2 \wedge X)
    =
  \tilde K(\Sigma^2 X)
  \,,
\end{displaymath}
where on the right we used that [[smash product]] with the 2-sphere is the same as double [[suspension]].

Finally there is an [[isomorphism]]

\begin{displaymath}
\itexarray{
    \mathbb{Z}
      &\underoverset{\simeq}{ \beta }{\longrightarrow}&
    \tilde K_{\mathbb{C}}(S^2)
    \\
    1 &\overset{\phantom{AAA}}{\mapsto}& (h-1)
  }
\end{displaymath}
(example \ref{TopologicalKTheoryRingOfThe2Sphere}). The composite

\begin{displaymath}
\itexarray{
    \tilde K_{\mathbb{C}}(X)
      &
      \simeq
    \mathbb{Z} \otimes \tilde K_{\mathbb{C}}(X)
      \overset{ \beta \otimes id }{\longrightarrow}
    \tilde K_{\mathbb{C}}(S^2) \otimes \tilde K_{\mathbb{C}}(X)
      \underoverset{\simeq}{\widetilde \boxtimes}{\longrightarrow}
      &
    \tilde K_{\mathbb{C}}(S^2 \wedge X)
      =
    \tilde K_{\mathbb{C}}(\Sigma^2 X)
    \\
    E - rk_x(E) &\overset{\phantom{AAAA}}{\mapsto}& (h-1) \widetilde \boxtimes (E - rk_x(E))
  }
\end{displaymath}
is the isomorphism to be established.

\end{proof}
\hypertarget{GradedRingStructure}{}\subsubsection*{{Graded-commutative ring structure}}\label{GradedRingStructure}

The external product on reduced K-groups from prop. \ref{ExternalTensorProductOnReducedKGroups} allows to extend the [[commutative ring]] structure from the plain K-groups (remark \ref{KTheoryRing}) to a ring structure on the graded K-groups from def. \ref{GradedKGroups}. This is def. \ref{ProductOnGradedKGroups} below.

To state this definition, recall that

\begin{enumerate}%
\item for $X$ a [[pointed topological space]] then the [[diagonal]] map to its [[product topological space]] $X \times X$ induced a diagonal to the [[smash product]] $X \wedge X = (X \times X)/(X \vee X)$

\begin{displaymath}
X \overset{\Delta_X}{\longrightarrow} X \times X \overset{q}{\longrightarrow} X\wedge X
\end{displaymath}

\item since [[reduced suspension]] is equivalently [[smash product]] with the [[circle]] $\Sigma X \simeq S^1 \wedge X$, there are induced ``partial diagonal maps'' of the form

\begin{displaymath}
\Sigma (q \circ \Delta_X)
   \;\colon\;
  \Sigma X \simeq S^1 \wedge X \overset{S^1 \wedge (q\circ \Delta_X)}{\longrightarrow} S^1 \wedge X \wedge X \simeq (\Sigma X) \wedge X
\end{displaymath}
etc.



\end{enumerate}
\begin{defn}
\label{ProductOnGradedKGroups}\hypertarget{ProductOnGradedKGroups}{}
\textbf{(product on graded K-groups)}

For $X$ a [[pointed topological space|pointed]] [[compact Hausdorff space]], the \emph{product on graded K-groups}

\begin{displaymath}
(-)\cdot (-)
    \;\colon\;
  K^\bullet(X) \otimes K^\bullet(X)
    \longrightarrow
  K^\bullet(X)
\end{displaymath}
is the [[linear map]] which on the direct summands $\tilde K^0(X) \coloneqq \tilde K(X)$ and $\tilde K^1(X) \coloneqq \tilde K(\Sigma X)$ is given by the following morphisms, which are [[composition|composites]] of the external product $\tilde \boxtimes$ on reduced K-groups from prop. \ref{ExternalTensorProductOnReducedKGroups} with pullbacks along the above suspended diagonal maps:

\begin{displaymath}
\tilde K(X) \otimes \tilde K(X)
    \overset{\tilde \boxtimes}{\longrightarrow}
  \tilde K(X)
\end{displaymath}
\begin{displaymath}
\tilde K(X) \otimes \tilde K(\Sigma X)
    \overset{\tilde \boxtimes}{\longrightarrow}
  \tilde K(X \wedge (\Sigma X))
    \overset{ (\Sigma(q \circ \Delta_X))^\ast }{\longrightarrow}
  \tilde K(\Sigma X)
\end{displaymath}
\begin{displaymath}
\tilde K(\Sigma X) \otimes \tilde K(\Sigma X)
    \overset{\tilde \boxtimes}{\longrightarrow}
  \tilde K( (\Sigma X) \wedge (\Sigma X))
    \overset{ (\Sigma^2(q \circ \Delta_X))^\ast }{\longrightarrow}
  \tilde K(\Sigma^2 X)
    \simeq
  \tilde K(X)
  \,,
\end{displaymath}
where the last isomorphism on the right is [[Bott periodicity]] isomorphism (prop. \ref{BottPeriodicity}).

\end{defn}
\hypertarget{ClassifyingSpace}{}\subsubsection*{{Classifying space}}\label{ClassifyingSpace}

We discuss how the [[classifying space]] for $\tilde K^0$ is the [[delooping]] of the [[stable unitary group]].

\begin{defn}
\label{BUn}\hypertarget{BUn}{}
\textbf{([[classifying space]] of the [[stable unitary group]])}

For $n \in \mathbb{N}$ write $U(n)$ for the [[unitary group]] in dimension $n$ and $O(n)$ for the [[orthogonal group]] in dimension $n$, both regarded as [[topological groups]] in the standard way. Write $B U(n) , B O(n)\in$ [[Top]] for the corresponding [[classifying space]].

Write

\begin{displaymath}
[X, B O(n)] := \pi_0 Top(X, B O(n))
\end{displaymath}
and

\begin{displaymath}
[X, B U(n)] := \pi_0 Top(X, B U(n))
\end{displaymath}
for the set of [[homotopy]]-classes of [[continuous function]]s $X \to B U(n)$.

\end{defn}
\begin{prop}
\label{BUnClassifyingSpace}\hypertarget{BUnClassifyingSpace}{}
This is equivalently the set of [[isomorphism]] classes of $O(n)$- or $U(n)$-[[principal bundle]]s on $X$ as well as of rank-$n$ real or complex [[vector bundle]]s on $X$, respectively:

\begin{displaymath}
[X, B O(n)] \simeq O(n) Bund(X) \simeq Vect_{\mathbb{R}}(X,n)
  \,,
\end{displaymath}
\begin{displaymath}
[X, B U(n)] \simeq U(n) Bund(X) \simeq Vect_{\mathbb{C}}(X,n)
  \,.
\end{displaymath}
\end{prop}
\begin{defn}
\label{InclusionOfUns}\hypertarget{InclusionOfUns}{}
For each $n$ let

\begin{displaymath}
U(n) \to U(n+1)
\end{displaymath}
be the inclusion of [[topological group]]s given by inclusion of $n \times n$ [[matrices]] into $(n+1) \times (n+1)$-matrices given by the block-diagonal form

\begin{displaymath}
\left[g\right] \mapsto
  \left[
    \itexarray{
      1 & [0]
      \\
      [0] & [g]
    }
  \right]
  \,.
\end{displaymath}
This induces a corresponding sequence of morphisms of classifying spaces, def. \ref{BUn}, in [[Top]]

\begin{displaymath}
B U(0) \hookrightarrow B U(1) \hookrightarrow B U(2) \hookrightarrow \cdots
  \,.
\end{displaymath}
Write

\begin{displaymath}
B U := {\lim_{\to}}_{n \in \mathbb{N}} B U(n)
\end{displaymath}
for the [[homotopy colimit]] (the ``homotopy [[direct limit]]'') over this diagram (see at \emph{[[homotopy colimit]]} the section \emph{\href{homotopy+limit#SequentialHocolims}{Sequential homotopy colimits}}).

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
The [[topological space]] $B U$ is \textbf{not} equivalent to $B U(\mathcal{H})$, where $U(\mathcal{H})$ is the [[unitary group]] on a separable infinite-dimensional [[Hilbert space]] $\mathcal{H}$. In fact the latter is [[contractible]], hence has a [[weak homotopy equivalence]] to the point

\begin{displaymath}
B U(\mathcal{H}) \simeq *
\end{displaymath}
while $B U$ has nontrivial [[homotopy group]]s in arbitrary higher degree (by [[Kuiper's theorem]]).

But there is the group $U(\mathcal{H})_{\mathcal{K}} \subset U(\mathcal{H})$ of unitary operators that differ from the [[identity]] by a [[compact operator]]. This is essentially $U = \Omega B U$. See \hyperlink{Uk}{below}.

\end{remark}
\begin{prop}
\label{}\hypertarget{}{}
Write $\mathbb{Z}$ for the set of [[integer]]s regarded as a [[discrete topological space]].

The product spaces

\begin{displaymath}
B O \times \mathbb{Z}\,,\;\;\;\;\;B U \times \mathbb{Z}
\end{displaymath}
are [[classifying spaces]] for real and complex $K$-theory, respectively: for every compact Hausdorff topological space $X$, we have an isomorphism of groups

\begin{displaymath}
\tilde K(X)
  \simeq
  [X, B U ]
  \,.
\end{displaymath}
\begin{displaymath}
K(X)
  \simeq
  [X, B U \times \mathbb{Z}]
  \,.
\end{displaymath}
\end{prop}
See for instance (\hyperlink{Friedlander}{Friedlander, prop. 3.2}) or (\hyperlink{Karoubi}{Karoubi, prop. 1.32, theorem 1.33}).

\begin{proof}
First consider the statement for reduced cohomology $\tilde K(X)$:

Since a [[compact topological space]] is a [[compact object]] in [[Top]] (and using that the [[classifying spaces]] $B U(n)$ are (see there) [[paracompact topological space]]s, hence normal, and since the inclusion morphisms are closed inclusions (\ldots{})) the [[hom-functor]] out of it commutes with the [[filtered colimit]]

\begin{displaymath}
\begin{aligned}
    Top(X, B U)
    &=
    Top(X, {\lim_\to}_n B U(n))
    \\
    & \simeq {\lim_\to}_n Top(X, B U (n))
  \end{aligned}
  \,.
\end{displaymath}
Since $[X, B U(n)] \simeq U(n) Bund(X)$, in the last line the colimit is over [[vector bundle]]s of all ranks and identifies two if they become isomorphic after adding a trivial bundle of some finite rank.

For the full statement use that by prop. \ref{missing} we have

\begin{displaymath}
K(X) \simeq H^0(X, \mathbb{Z}) \oplus \tilde K(X)
  \,.
\end{displaymath}
Because $H^0(X,\mathbb{Z}) \simeq [X, \mathbb{Z}]$ it follows that

\begin{displaymath}
H^0(X, \mathbb{Z}) \oplus \tilde K(X)
  \simeq
  [X, \mathbb{Z}] \times [X, B U]
  \simeq
  [X, B U \times \mathbb{Z}]
  \,.
\end{displaymath}
\end{proof}
There is another variant on the classifying space

\begin{defn}
\label{Uk}\hypertarget{Uk}{}
Let

\begin{displaymath}
U_{\mathcal{K}}
  =
  \left\{
    g \in U(\mathcal{H}) | g - id \in \mathcal{K}
  \right\}
\end{displaymath}
be the group of unitary operators on a [[separable Hilbert space]] $\mathcal{H}$ which differ from the identity by a [[compact operator]].

\end{defn}
Palais showed that

\begin{prop}
\label{}\hypertarget{}{}
$U_\mathcal{K}$ is a [[homotopy equivalence|homotopy equivalent]] model for $B U$. It is in fact the [[norm closure]] of the evident model of $B U$ in $U(\mathcal{H})$.

Moreover $U_{\mathcal{K}} \subset U(\mathcal{H})$ is a [[Banach Lie group|Banach Lie]] [[normal subgroup]].

\end{prop}
Since $U(\mathcal{H})$ is [[contractible]], it follows that

\begin{displaymath}
B U_{\mathcal{K}} \coloneqq U(\mathcal{H})/U_{\mathcal{K}}
\end{displaymath}
is a model for the [[classifying space]] of reduced K-theory.

\hypertarget{of_noncompact_spaces}{}\subsubsection*{{Of non-compact spaces}}\label{of_noncompact_spaces}

For $G$ a compact Lie group with [[classifying space]] $B G$ (in general non-compact) then the map from the [[Grothendieck group]] $\mathbb{K}(B G) \coloneqq Grp(Vect(B G)/_\sim, \oplus)$ (def. \ref{GrothendieckGroupKTheory}) to the representable K-theory $K(B G)_{rep} \coloneqq [X, B U \times\mathbb{Z}]$ (def. \ref{RepresntableTopologicalKTheory}) is [[injective function|injective]]

\begin{displaymath}
\mathbb{K}(B G) \hookrightarrow K(B G)_{rep}
   \,.
\end{displaymath}
\hyperlink{JackowskiOliver}{Jackowski-Oliver}

\hypertarget{AsAGeneralizedCohomologyTheory}{}\subsubsection*{{As a generalized cohomology theory}}\label{AsAGeneralizedCohomologyTheory}

Topological K-theory satisfies the axioms of a [[generalized (Eilenberg-Steenrod) cohomology theory]] (\hyperlink{AtiyahHirzebruch61}{Atiyah-Hirzebruch 61, 1.8}).

This is essentially the statement of the long exact sequences \hyperlink{ExactSequences}{above}.

\hypertarget{ComplexOrientationAndFormalGroupLaw}{}\subsubsection*{{Complex orientation and formal group law}}\label{ComplexOrientationAndFormalGroupLaw}

A [[multiplicative cohomology theory|multiplicative]] [[generalized (Eilenberg-Steenrod) cohomology]] theory $E$ is called \emph{[[complex oriented cohomology|complex orientable]]} if the element $1 \in E(\ast) \simeq \tilde E(S^0)$ is in the image of the pullback morphism

\begin{displaymath}
\tilde i^\ast \;\colon\; \tilde E^2(B U(1)) \longrightarrow \tilde E^2(S^2) \simeq \tilde E^0(S^0)
  \,.
\end{displaymath}
If so, then a choice of pre-image $c^E_1 \in E^2(B U(1))$ is a choice of \emph{complex orientation} (\href{complex+oriented+cohomology+theory#ComplexOrientedCohomologyTheory}{this def.}).

Now for $E = K_{\mathbb{C}}$ being complex topological K-theory regarded as a generalized cohomology theory as \hyperlink{AsAGeneralizedCohomologyTheory}{above}, then by [[Bott periodicity]] (prop. \ref{BottPeriodicity}) and by $\tilde K_{\mathbb{Z}}(S^2) \simeq \mathbb{Z} \cdot (h-1)$ (example \ref{ComplexTopologicalKTheoryOfTheCircle}) this reduces to the statement that there is an element $c^K_1 \in \tilde K_{\mathbb{C}}(B U(1))$ such that its image under

\begin{displaymath}
\tilde i^\ast
    \;\colon\;
  \tilde K_{\mathbb{C}}(B U(1))
    \longrightarrow
  \tilde K_{\mathbb{C}}(S^2)
    \simeq
  \mathbb{Z} \cdot (h-1)
\end{displaymath}
is the [[Bott element]] $h-1$, the [[virtual vector bundle]] difference between the [[basic complex line bundle on the 2-sphere]] and the [[trivial vector bundle|trivial]] [[complex line bundle]].

By the very nature of the [[basic complex line bundle on the 2-sphere]] $h$, it is the restriction of the [[universal complex line bundle]] $\mathcal{O}(1)$ on $B U(1) \simeq \mathbb{C}P^\infty$ along the defining cell inclusion $i \colon S^2 \hookrightarrow \mathbb{C}P^\infty \simeq B U(1)$. Hence if we set

\begin{displaymath}
c_K^1 \;\coloneqq\; \mathcal{O}(1)-1 \; \in \tilde K_{\mathbb{C}}(B U(1))
\end{displaymath}
then this is a [[complex oriented cohomology|complex orientation]] for complex topological K-theory.

From this we obtain the [[formal group law]] associated with topological K-theory (from \href{complex+oriented+cohomology+theory#ComplexOrientedCohomologyTheoryFormalGroupLaw}{this prop.}):

By the nature of the [[classifying space]] $B U(1)$ we have that for

\begin{displaymath}
\mu \;\colon\; B U(1) \times B U(1) \longrightarrow B U(1)
\end{displaymath}
the group product operation, which classifies the [[tensor product of vector bundles|tensor product of line bundles]], that

\begin{displaymath}
\mu^\ast \mathcal{O}(1) \simeq pr_1^\ast \mathcal{O}(1) \otimes_{B U(1)} pr_2^\ast \mathcal{O}(1)
  \,,
\end{displaymath}
where

\begin{displaymath}
pr_i\colon B U(1) \times B U(1) \to B U(1)
\end{displaymath}
are the two [[projections]] out of the [[Cartesian product]]. Hence

\begin{displaymath}
\begin{aligned}
    \mu^\ast c_1^K
    &\coloneqq
    \mu^\ast (\mathcal{O}(1) - 1)
    \\
      & =
    \left(pr_1^\ast \mathcal{O}(1)\right)  \cdot  \left(pr_2^\ast \mathcal{O}(1)\right) - 1
    \\
      &  =
    \left(pr_1^\ast (\mathcal{O}(1) -1)\right) \cdot \left(pr_2^\ast( \mathcal{O}(1) -1)\right)
    + pr_1^\ast \mathcal{O}(1)
    + pr_2^\ast \mathcal{O}(2)
    - 2
    \\
     & =
    \left(pr_1^\ast (\mathcal{O}(1) -1)\right) \cdot \left(pr_2^\ast( \mathcal{O}(1) -1)\right)
     +
    \left(pr_1^\ast \mathcal{O}(1) - 1\right)
     +
    \left(pr_2^\ast \mathcal{O}(1) - 1\right)
  \end{aligned}
\end{displaymath}
This shows that the [[formal group law]] associated with the complex orientation of complex topological K-theory is that of the \emph{[[formal multiplicative group]]} given by

\begin{displaymath}
f(x,y) = x + y + x y
  \,.
\end{displaymath}
\hypertarget{spectrum}{}\subsubsection*{{Spectrum}}\label{spectrum}

Being a [[generalized (Eilenberg-Steenrod) cohomology]] theory, topological K-theory is represented by a [[spectrum]]: the \emph{[[K-theory spectrum]]}.

e.g. \hyperlink{Switzer75}{Switzer 75, p. 216}

The degree-0 part of this spectrum, i.e. the classifying space for degree 0 topological $K$-theory is modeled in particular by the space $Fred$ of [[Fredholm operator]]s.

\hypertarget{ring_spectrum}{}\subsubsection*{{Ring spectrum}}\label{ring_spectrum}

This [[K-theory]] spectrum has the structure of a [[ring spectrum]]

(e.g. \hyperlink{Switzer75}{Switzer 75, section 13.90, around p. 300},

see also p. 205 (213 of 251) in \emph{[[A Concise Course in Algebraic Topology]]})

(\ldots{})

\hypertarget{chromatic_filtration}{}\subsubsection*{{Chromatic filtration}}\label{chromatic_filtration}

[[!include chromatic tower examples - table]]

\hypertarget{as_the_shape_of_the_smooth_ktheory_spectrum}{}\subsubsection*{{As the shape of the smooth K-theory spectrum}}\label{as_the_shape_of_the_smooth_ktheory_spectrum}

See at \emph{[[differential cohomology diagram]]}.

\hypertarget{RelationToAlgebraicKTheory}{}\subsubsection*{{Relation to algebraic K-theory}}\label{RelationToAlgebraicKTheory}

The topological K-theory over a space $X$ is not identical with the \emph{[[algebraic K-theory]]} of the ring of functions on $X$, but the two are closely related. See for instance (\hyperlink{Paluch}{Paluch}, \hyperlink{Rosenberg}{Rosenberg}). See at \emph{[[comparison map between algebraic and topological K-theory]]}.

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

\begin{itemize}%
\item [[virtual vector bundle]]


\item [[K-theory]]


\item \textbf{topological K-theory}

\begin{itemize}%
\item [[Atiyah-Bott-Shapiro isomorphism]]


\item [[topological index]]


\item [[KR-theory]]


\item [[vectorial bundle]]



\end{itemize}

\item [[groupoid K-theory]]


\item [[twisted K-theory]]


\item [[differential K-theory]]


\item [[twisted differential K-theory]]


\item [[semi-topological K-theory]]


\item [[algebraic K-theory]]



\end{itemize}
[[!include orientations in higher quantization - table]]

[[!include string theory and cohomology theory -- table]]

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

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

The ``ring of complex vector bundles'' $K(X)$ was introduced in

\begin{itemize}%
\item [[M. F. Atiyah]], [[F. Hirzebruch]], \emph{Riemann-Roch theorems for differentiable manifolds}, Bull. Amer. Math Soc. vol. 65 (1959) pp. 276-281.

\end{itemize}
and shown to give a [[generalized (Eilenberg-Steenrod) cohomology]] theory in

\begin{itemize}%
\item [[M. F. Atiyah]], [[F. Hirzebruch]], \emph{Vector bundles and homogeneous spaces}, 1961, Proc. Sympos. Pure Math., Vol. III pp. 7--38 American Mathematical Society, Providence, R.I. (\href{http://hirzebruch.mpim-bonn.mpg.de/87/}{web}, \href{http://www.ams.org/mathscinet-getitem?mr=0139181}{MR 0139181})

\end{itemize}
Representable K-theory over non-compact spaces was considered in

\begin{itemize}%
\item [[Max Karoubi]], \emph{Espaces Classifiants en K-Th\'e{}orie}, Transactions of the American Mathematical Society Vol. 147, No. 1 (Jan., 1970), pp. 75-115 (\href{http://www.jstor.org/stable/1995218}{jstor})

\end{itemize}
and (over [[classifying spaces]] in the context of [[equivariant K-theory]]) in

\begin{itemize}%
\item [[Graeme Segal]], [[Michael Atiyah]], section 4 of \emph{Equivariant K-theory and completion}, J. Differential Geometry 3 (1969), 1--18. MR 0259946 (41 \#4575

\end{itemize}
Early lecture notes on topological K-theory in a general context of [[stable homotopy theory]] and [[generalized cohomology theory]] includes

\begin{itemize}%
\item [[Frank Adams]], part III, section 2 of \emph{[[Stable homotopy and generalised homology]]}, 1974

\end{itemize}
Textbook accounts on topological K-theory include

\begin{itemize}%
\item [[M. F. Atiyah]], \emph{K-theory}, Benjamin New-York, 1967, (\href{http://www.maths.ed.ac.uk/~aar/papers/atiyahk.pdf}{pdf})


\item [[Robert Switzer]], sections 11 and 13.90 of \emph{Algebraic Topology - Homotopy and Homology}, Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen, Vol. 212, Springer-Verlag, New York, N. Y., 1975.


\item [[Max Karoubi]], \emph{K-theory: an introduction}, Grundlehren der Math. Wissen. 226 Springer 1978, Reprinted in Classics in Mathematics (2008)


\item [[Allen Hatcher]], \emph{Vector bundles and K-theory}, 2003/2009 (\href{http://www.math.cornell.edu/~hatcher/VBKT/VBpage.html}{web})



\end{itemize}
Further introductions include

\begin{itemize}%
\item [[H. Blaine Lawson]], [[Marie-Louise Michelsohn]], \emph{[[Spin geometry]]}, Princeton University Press (1989)


\item Marcelo Aguilar, [[Samuel Gitler]], Carlos Prieto, section 9 of \emph{Algebraic topology from a homotopical viewpoint}, Springer (2002) (\href{http://tocs.ulb.tu-darmstadt.de/106999419.pdf}{toc pdf})


\item [[Max Karoubi]], \emph{K-theory. An elementary introduction}, lectures given at the Clay Mathematics Academy (\href{https://arxiv.org/abs/math/0602082}{arXiv:math/0602082})


\item [[Eric Friedlander]], \emph{An introduction to K-theory} (emphasis on [[algebraic K-theory]]), 2007 (\href{http://users.ictp.it/~pub_off/lectures/lns023/Friedlander/Friedlander.pdf}{pdf})


\item [[Varvara Karpova]], \emph{Complex topological K-theory}, 2009 (\href{http://infoscience.epfl.ch/record/162450/files/karpova.semestre.hess2.pdf}{pdf})


\item Chris Blair, \emph{Some K-theory examples}, 2009 (\href{http://www.maths.tcd.ie/~cblair/notes/kex.pdf}{pdf})


\item [[Klaus Wirthmüller]], \emph{Vector bundles and K-theory}, 2012 (\href{ftp://www.mathematik.uni-kl.de/pub/scripts/wirthm/Top/vbkt_skript.pdf}{pdf})


\item Aderemi Kuku, \emph{Introduction to K-theory and some applications} (\href{https://www.math.ksu.edu/~zlin/kuku/Intro-Kthy.pdf}{pdf})



\end{itemize}
A textbook account of topological K-theory with an eye towards [[operator K-theory]] is section 1 of

\begin{itemize}%
\item [[Bruce Blackadar]], \emph{[[K-Theory for Operator Algebras]]}

\end{itemize}
The [[comparison map between algebraic and topological K-theory]] is discussed for instance in

\begin{itemize}%
\item Michael Paluch, \emph{Algebraic $K$-theory and topological spaces} K-theory 0471 (\href{http://www.math.uiuc.edu/K-theory/0471/}{web})


\item [[Jonathan Rosenberg]], \emph{Comparison Between Algebraic and Topological K-Theory for Banach Algebras and $C^*$-Algebras}, (\href{http://www2.math.umd.edu/~jmr/algtopK.pdf}{pdf})



\end{itemize}
Discussion from the point of view of [[smooth stacks]] and [[differential K-theory]] is in

\begin{itemize}%
\item [[Ulrich Bunke]], [[Thomas Nikolaus]], [[Michael Völkl]], \emph{Differential cohomology theories as sheaves of spectra}, Journal of Homotopy and Related Structures October 2014 (\href{http://arxiv.org/abs/1311.3188}{arXiv:1311.3188})

\end{itemize}
The proof of the [[Hopf invariant one]] theorem in terms of topological K-theory is due to

\begin{itemize}%
\item [[Frank Adams]], [[Michael Atiyah]], \emph{K-theory and the Hopf invariant}, Quart. J. Math. Oxford (2), 17 (1966), 31-38 (\href{http://www.maths.ed.ac.uk/~aar/papers/adamatiy.pdf}{pdf})

\end{itemize}
\hypertarget{for_noncompact_spaces_2}{}\subsubsection*{{For non-compact spaces}}\label{for_noncompact_spaces_2}

Topological K-theory of [[Eilenberg-MacLane spaces]] is discussed in

\begin{itemize}%
\item D.W. Anderson, Luke Hodgkin \emph{The K-theory of Eilenberg-Maclane complexes}, Topology, Volume 7, Issue 3, August 1968, Pages 317-329 (\href{https://doi.org/10.1016/0040-9383(68}{doi:10.1016/0040-9383(68)90009-8}90009-8))

\end{itemize}
Topological topological K-theory of [[classifying spaces]] of [[Lie groups]] is in

\begin{itemize}%
\item Stefan Jackowski and Bob Oliver, \emph{Vector bundles over classifying spaces of compact Lie groups} (\href{http://hopf.math.purdue.edu/Jackowski-Oliver/bg-bu.pdf}{pdf})

\end{itemize}
\hypertarget{dbrane_charge}{}\subsubsection*{{D-brane charge}}\label{dbrane_charge}

Discussion of [[twisted K-theory|twisted]] [[differential K-theory|differential]] [[topological K-theory|K-theory]] and its relation to [[D-brane charge]] in [[type II string theory]] (see also \href{D-brane#ReferencesKTheoryDescription}{there}):

\begin{itemize}%
\item Daniel Grady, [[Hisham Sati]], \emph{Ramond-Ramond fields and twisted differential K-theory} (\href{https://arxiv.org/abs/1903.08843}{arXiv:1903.08843})

\end{itemize}
Discussion of [[twisted K-theory|twisted]] [[differential K-theory|differential]] [[KO-theory|orthogonal]] [[topological K-theory|K-theory]] and its relation to [[D-brane charge]] in [[type I string theory]] (on [[orientifolds]]):

\begin{itemize}%
\item Daniel Grady, [[Hisham Sati]], \emph{Twisted differential KO-theory} (\href{https://arxiv.org/abs/1905.09085}{arXiv:1905.09085})

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

[[!redirects periodic complex K-theory]]

[[!redirects complex topological K-theory]]



\end{document}