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\begin{document}

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\section*{fundamental product theorem in topological K-theory}

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

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

[[!include cohomology - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{statement}{Statement}\dotfill \pageref*{statement} \linebreak
\noindent\hyperlink{bott_periodicity}{Bott periodicity}\dotfill \pageref*{bott_periodicity} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

For $X$ a [[compact Hausdorff space]] The \emph{fundamental product theorem in topological K-theory} identifies

\begin{enumerate}%
\item the [[topological K-theory]]-[[commutative ring|ring]] $K(X \times S^2)$ of the [[product topological space]] $X \times S^2$ with the [[2-sphere]] $S^2$;


\item the K-theory ring $K(X)$ of the original space $X$ with a generator $H$ for the [[basic line bundle on the 2-sphere]] adjoined:



\end{enumerate}
\begin{displaymath}
K(X) \otimes_{\mathbb{Z}} \mathbb{Z}[H]/(H-1)^2
    \overset{\simeq}{\longrightarrow}
  K(X \times S^2) 
  \,.
\end{displaymath}
This theorem in particular serves as a substantial step in a [[proof]] of [[Bott periodicity]] for [[topological K-theory]] (cor \ref{BottPeriodicity} below).

The usual [[proof]] proceeds by

\begin{enumerate}%
\item realizing all [[vector bundles]] on $X \times S^2$ via an $X$-parameterized [[clutching construction]];


\item showing that all the clutching functions are [[homotopy|homotopic]] to those that are Laurent polynomials as functions on $S^1$, hence products of a polynomial clutching $p$ functions with a monomial $z^{-n}$ of negative power;


\item observing that the bundle corresponding to a clutching function of the form $f z^n$ is equivalent to the bundle corresponding to $f$ and tensored with the $n$th [[tensor product of vector bundles]]-power of the [[basic complex line bundle on the 2-sphere]];


\item showing that some [[direct sum of vector bundles]] of the vector bundle corresponding to a polynomial clutching function with one coming from a trivial clutching function is given by a linear clutching function;


\item showing that bundles coming from linear clutching functions are [[direct sums of vector bundles|direct sums]] of one coming from a trivial clutching function with the one coming from the homogeneously linear part;



\end{enumerate}
Applying these steps to a vector bundle on $X \times S^2$ yields a virtual sum of [[external tensor products of vector bundles]] of bundles on $X$ with powers of the [[basic complex line bundle on the 2-sphere]]. This means that the function in the fundamental product theorem is surjective. By similar means one shows that it is also injective.

\hypertarget{statement}{}\subsection*{{Statement}}\label{statement}

For $S^2 \subset \mathbb{R}^3$ the [[2-sphere]] with its [[Euclidean space|Euclidean]] [[subspace topology]], write $h$ for the [[basic 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}
(by \href{basic+complex+line+bundle+on+the+2-sphere#TensorRelationForBasicLineBundleOn2Sphere}{this prop.}).

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 called the \emph{[[Bott element]]} of complex [[topological K-theory]].

It follows 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{
    \Phi \colon
    &
    K(X) \otimes \mathbb{Z}[h]/((h-1)^2)
      & \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 tensor product of vector bundles]].

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

For $X$ a [[compact Hausdorff space]], then ring homomorphism $\Phi \colon     K(X) \otimes \mathbb{Z}[h]/((h-1)^2) \longrightarrow K(X \times S^2)$ 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)}

For $X$ a [[compact Hausdorff space]] we have that the [[external tensor product of vector bundles]] 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{bott_periodicity}{}\subsection*{{Bott periodicity}}\label{bott_periodicity}

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

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

Then there is an [[isomorphism]] of [[reduced K-theory]]

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

\begin{example}
\label{TopologicalKTheoryRingOfThe2Sphere}\hypertarget{TopologicalKTheoryRingOfThe2Sphere}{}
\textbf{([[topological K-theory]] ring of the [[2-sphere]])}

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

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

This means that the relation $(h-1)^2 = 0$ satisfied by the [[basic 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(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(S^2) \simeq \mathbb{Z}
  \,.
\end{displaymath}
\end{example}
\hypertarget{references}{}\subsection*{{References}}\label{references}

Expositions include:

\begin{itemize}%
\item [[Klaus Wirthmüller]], section 6 (from p. 19 on) in \emph{Vector bundles and K-theory}, 2012 (\href{ftp://www.mathematik.uni-kl.de/pub/scripts/wirthm/Top/vbkt_skript.pdf}{pdf})


\item [[Allen Hatcher]], section 2.1 (from p. 45 on) in \emph{Vector bundles and K-theory} (\href{https://www.math.cornell.edu/~hatcher/VBKT/VBpage.html}{web})



\end{itemize}
[[!redirects external product theorem]] [[!redirects external product theorem in topological K-theory]]

[[!redirects fundamental product theorem in K-theory]]



\end{document}