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\begin{document}

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\section*{basic complex line bundle on the 2-sphere}

[[!redirects basic line bundle on the 2-sphere]]

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

\hypertarget{bundles}{}\paragraph*{{Bundles}}\label{bundles}

[[!include bundles - 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{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

The \emph{basic line bundle on the 2-sphere} is the [[complex line bundle]] on the [[2-sphere]] whose [[first Chern class]] is a generator of $H^2(S^2, \mathbb{Z})$.

This is the [[pullback bundle]] of the map $S^2 \to B U(1) \simeq B^2 \mathbb{Z}$ to the [[classifying space]]/[[Eilenberg-MacLane space]] which itself represents a generator of the [[homotopy group]] $\pi_2(S^2) \simeq \mathbb{Z}$.

Beware that this basic line bundle is sometimes called the ``canonical line bundle on the 2-sphere'', but it is \emph{not} [[isomorphism|isomorphic]] to what in complex geometry is called the [[canonical bundle]] of the 2-sphere regarded as a [[Riemann surface]]. Instead it is ``one half'' of the latter, its [[theta characteristic]]. See also at \emph{[[geometric quantization of the 2-sphere]]}.

The basic line bundle is the canonically [[associated bundle]] to basic [[circle principal bundle]]: the [[complex Hopf fibration]]. Another name for it is the [[tautological line bundle]] for the complex [[projective space|projective line]] $\mathbb{P}^1(\mathbb{C})$ (the [[Riemann sphere]]), namely the map $\mathbb{C}^2 \setminus \{(0, 0)\} \to \mathbb{P}^1(\mathbb{C})$ mapping $(x, y)$ to $[x; y]$.

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

The [[classifying space]] for [[circle principal bundles]], or equivalently (via forming [[associated bundles]]) that of [[complex line bundles]] is $B U(1)$, which as a [[Grassmannian]] is the infinite [[complex projective space]] $\mathbb{C}P^\infty$. The [[homotopy type]] of this space is that of the [[Eilenberg-MacLane space]] $K(\mathbb{Z},2)$. This means that $K(\mathbb{Z},2)$ is in particular [[path-connected topological space|path-connected]] and has second [[homotopy group]] the [[integers]]: $\pi_2(K(\mathbb{Z},2)) \simeq \mathbb{Z}$.

It being the [[classifying space]] for complex line bundles means that

\begin{displaymath}
\left\{
     \itexarray{ 
        \text{isomorphism classes of}
        \\
        \text{complex line bundles}
        \\
        \text{on}\,\, S^2 
     }
  \right\}
  \;\simeq\;
  \left\{
    \itexarray{
      \text{continuous functions}
      \\
      S^2 \longrightarrow K(\mathbb{Z},2)
      \\
      \text{up to homotopy}
    }
  \right\}
  \;\simeq\;
  \pi_2(K(\mathbb{Z},2))
  \;\simeq\;
  \mathbb{Z}
  \,.
\end{displaymath}
The ([[isomorphism class]]) of the complex line bundle which corresponds to $+1 \in \mathbb{Z}$ under this sequence of [[isomorphisms]] is called the \emph{basic complex line bundle on the 2-sphere}.

Hence the basic complex line bundle on the 2-sphere is [[generalized the|the]] [[pullback bundle]] of the [[universal complex line bundle]] on $B U(1)$ along the map $S^2 \to B U(1)$ which represents the element $1 \in \mathbb{Z} \simeq \pi_2(B U(1))$. If the [[classifying space]] $B U(1)$ is represented by the infintie [[complex projective space]] $\mathbb{C}P^\infty$ with its canonical [[CW-complex]] structure (\href{complex+projective+space#CellComplexStructureOnComplexProjectiveSpace}{this prop.}), then this map is represented by the canonical cell incusion $S^2 \hookrightarrow\mathbb{C}P^\infty$.

Notice that there is a non-trivial [[automorphism]] of $\mathbb{Z}$ as an [[abelian group]] given by $n \mapsto -n$. This means that there is an ambiguity in the definition of the basic line bundle on the 2-sphere.

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

\begin{lemma}
\label{ClutchingConstructionOfBasicLineBundle}\hypertarget{ClutchingConstructionOfBasicLineBundle}{}
\textbf{([[clutching construction]] of the basic line bundle)}

Under the [[clutching construction]] of [[vector bundles]] on the [[2-sphere]], the basic complex line bundle on the 2-sphere is given by the [[transition function]]

\begin{displaymath}
\mathbb{C} \supset \, S^1 \longrightarrow GL(1,\mathbb{C}) \, \subset \mathbb{C}
\end{displaymath}
from the [[Euclidean space|Euclidean]] [[circle]] $S^1 \subset \mathbb{R}^2 \simeq \mathbb{C}$ to the complex [[general linear group]] in 1-dimension, which is $GL(1,\mathbb{C}) \simeq \mathbb{C} \setminus \{0\}$ given simply by

\begin{displaymath}
z \mapsto z
  \,,
\end{displaymath}
Alternatively, due to the sign ambiguity in the definition of the basic bundle, its clutching transition function is given by

\begin{displaymath}
z \mapsto - z
  \,.
\end{displaymath}
\end{lemma}
\begin{proof}
Under the [[clutching construction]] the [[isomorphism class]] of a complex line bundle corresponds to the [[homotopy class]] of its clutching transition function

\begin{displaymath}
S^1 \to GL(1, \mathbb{C}) \simeq \mathbb{C} \setminus \{0\}
\end{displaymath}
hence to an element of the [[fundamental group]] $\pi_1(\mathbb{C} \setminus \{0\}) \simeq \mathbb{Z}$. Hence by definition, the basic bundle has clutching transition function corresponding to $\pm 1  \in \mathbb{Z} \simeq [S^1, GL(1,\mathbb{Z})]$ and this element is represented by the function $z \mapsto \pm z$.

\end{proof}
\begin{prop}
\label{TensorRelationForBasicLineBundleOn2Sphere}\hypertarget{TensorRelationForBasicLineBundleOn2Sphere}{}
\textbf{(fundamental tensor/sum relation of the basic complex line bundle)}

Under [[direct sum of vector bundles]] $\oplus_{S^2}$ and [[tensor product of vector bundles]] $\otimes_{S^2}$, the basic line bundle on the 2-sphere $H \to S^2$ satisfies the following relation

\begin{displaymath}
H \oplus_{S^2} H
    \;\simeq\;
  \left(
    H \otimes_{S^2} H
  \right)
    \oplus_{S^2}
  1_{S^2}
\end{displaymath}
(where $1_{S^2}$ denotes the [[trivial vector bundle]] [[complex line bundle]] on the 2-sphere).

\end{prop}
(e.g (\hyperlink{Hatcher}{Hatcher, Example 1.13}))

\begin{proof}
Via the [[clutching construction]] there is a single [[transition function]] of the form

\begin{displaymath}
S^1 \longrightarrow GL(n,\mathbb{C})
\end{displaymath}
that characterizes all the bundles involved. With $S^1 \hookrightarrow \mathbb{C}$ identified with the [[topological subspace]] of [[complex numbers]] of unit [[absolute value]], the standard choice for these functions is

\begin{itemize}%
\item for the [[trivial vector bundle|trivial]] [[line bundle]] $1_{S^2}$ we may choose

$f_1 \colon z \mapsto \left( 1 \right)$;


\item for the basic line bundle we may choose (by lemma \ref{ClutchingConstructionOfBasicLineBundle})

$f_H \colon z \mapsto \left( z\right)$



\end{itemize}
This yields

\begin{itemize}%
\item for $H \otimes H \oplus 1_{S^2}$ the clutching function

$z \mapsto \left( \itexarray{ z^2 & 0 \\ 0 & 1 }\right)$


\item for $H \oplus H$ the clutching function

$z \mapsto \left( \itexarray{ z & 0 \\ 0 & z }  \right)$.



\end{itemize}
Since the complex [[general linear group]] $Gl(n,\mathbb{C})$ is [[path-connected topological space|path-connected]] (by \href{general+linear+group#ConnectednessOfGeneralLinearGroup}{this prop.}), there exists a [[continuous function]]

\begin{displaymath}
\gamma \colon [0,1] \longrightarrow GL(2,\mathbb{C})
\end{displaymath}
connecting the identity matrix on $\mathbb{C}^2$ with the one that swaps the two entries, i.e.

with $\gamma(0) = \left( \itexarray{ 1 & 0 \\ 0 & 1 }  \right)$

and $\gamma(1) = \left(  \itexarray{ 0 & 1 \\ 1 & 0 } \right)$

Therefore the function

\begin{displaymath}
\itexarray{
    S^1 \times [0,1] &\overset{}{\longrightarrow}& GL(2,\mathbb{C})
    \\
    (z,t) 
      &\overset{\phantom{AA}}{\longrightarrow}& 
    f_{H \oplus 1}(z) \cdot \gamma(t) \cdot f_{1 \oplus H}(z) \cdot \gamma(t)
  }
\end{displaymath}
(with [[matrix multiplication]] on the right) is a [[left homotopy]] from $f_{H \oplus H}$ to $f_{H \otimes H \oplus 1}$.

\end{proof}
\begin{remark}
\label{}\hypertarget{}{}
\textbf{([[fundamental product theorem in topological K-theory]])}

Under the map

\begin{displaymath}
Vect(S^2)_{/\sim} \longrightarrow K(X)
\end{displaymath}
that sends [[complex vector bundles]] to their class in the [[topological K-theory]] ring $K(X)$, the fundamental tensor/sum relation of prop. \ref{TensorRelationForBasicLineBundleOn2Sphere} says that the K-theory class $H$ of the basic line bundle in $K(X)$ satisfies the relation

\begin{displaymath}
\begin{aligned}
    (H - 1)^2 
    & = 
    H^2  + 1 - \underset{= H^2 + 1}{\underbrace{2 H}}
    \\
    = &   0
  \end{aligned}
\end{displaymath}
in $K(X)$.

(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.}).)

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.

It turns out that this homomorphism is in fact an [[isomorphism]], hence that the relation $(H-1)^2 = 0$ from prop. \ref{TensorRelationForBasicLineBundleOn2Sphere} is the \emph{only} relation satisfied by the basic complex line bundle in topological K-theory.

More generally, for $X$ a [[topological space]], then there is a composite ring homomorphism

\begin{displaymath}
\itexarray{
    K(X) \otimes \mathbb{Z}[h]/((h-1)^2)
      & \longrightarrow & 
    K(X) \times K(S^2)
      & \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 is the [[external tensor product]] of vector bundles.

This composite is an [[isomorphism]] if $X$ is a [[compact Hausdorff space]] (for $X = \ast$ the [[point space]] this reduces to the previous statement).

This is called the \emph{[[fundamental product theorem in topological K-theory]]}. It is the main ingredient in the [[proof]] of [[Bott periodicity]] in complex topological K-theory.

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

\begin{itemize}%
\item [[universal complex line bundle]]

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

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


\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})



\end{itemize}
[[!redirects basic line bundles on the 2-sphere]]



\end{document}