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\begin{document}

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\section*{tautological line bundle}

\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{AsAtopologicalLieBundle}{As a topological line bundle}\dotfill \pageref*{AsAtopologicalLieBundle} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

The canonical [[line bundle]] over a [[projective space]] is sometimes called its ``tautological line bundle''. For more see at \emph{[[classifying space]]}.

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

\hypertarget{AsAtopologicalLieBundle}{}\subsubsection*{{As a topological line bundle}}\label{AsAtopologicalLieBundle}

We discuss the tautological line bundle as a [[topological vector bundle]]. Hence let $k$ be the [[topological field]] either

\begin{itemize}%
\item $k = \mathbb{R}$ the [[real numbers]]


\item or $k = \mathbb{C}$ the [[complex numbers]]



\end{itemize}
equipped with their [[Euclidean space|Euclidean]] [[metric topology]].

\begin{defn}
\label{ToplogicalProjectiveSpace}\hypertarget{ToplogicalProjectiveSpace}{}
\textbf{(topological [[projective space]])}

Let $n \in \mathbb{N}$. Consider the [[Euclidean space]] $k^{n+1}$ equipped with its [[metric topology]], let $k^{n+1} \setminus \{0\} \subset k^{n+1}$ be the [[topological subspace]] which is the [[complement]] of the origin, and consider on its underlying set the [[equivalence relation]] which identifies two points if they differ by [[multiplication]] with some $c \in k$ (necessarily non-zero):

\begin{displaymath}
(\vec x_1 \sim \vec x_2)
   \;\Leftrightarrow\;
  \left(
    \underset{c \in k}{\exists}
    ( \vec x_2 = c \vec x_1 )
  \right)
  \,.
\end{displaymath}
The [[equivalence class]] $[\vec x]$ is traditionally denoted

\begin{displaymath}
[x_1 : x_2 : \cdots : x_{n+1}]
  \,.
\end{displaymath}
Then the \emph{[[projective space]]} $k P^n$ is the corresponding [[quotient topological space]]

\begin{displaymath}
k P^n \;\coloneqq\; \left(k^{n+1} \setminus \{0\}\right) / \sim
  \,.
\end{displaymath}
\end{defn}
\begin{defn}
\label{TopologicalProjectiveSpaceStandardOpenCover}\hypertarget{TopologicalProjectiveSpaceStandardOpenCover}{}
\textbf{(standard open cover of topological projective space)}

For $n \in \mathbb{N}$ the \emph{standard open cover} of the projective space $k P^n$ (def. \ref{ToplogicalProjectiveSpace}) is

\begin{displaymath}
\left\{
    U_i \subset k P^n
  \right\}_{i \in \{1, \cdots, n+1\}}
\end{displaymath}
with

\begin{displaymath}
U_i 
    \coloneqq 
  \left\{ 
    [x_1 : \cdots : x_{n+1}] \in k P^n
    \;\vert\;
    x_i \neq 0
  \right\}
  \,.
\end{displaymath}
To see that this is an open cover:

\begin{enumerate}%
\item This is a cover because with the orgin removed in $k^n \setminus \{0\}$ at every point $[x_1: \cdots : x_{n+1}]$ at least one of the $x_i$ has to be non-vanishing.


\item These subsets are open in the [[quotient topology]] $k P^n = (k^n \setminus \{0\})/\sim$, since their [[pre-image]] under the quotient co-projection $k^{n+1} \setminus \{0\} \to k P^n$ coincides with the pre-image $(pr_i\circ\iota)^{-1}( k \setminus \{0\} )$ under the [[projection]] onto the $i$th coordinate in the [[product topological space]] $k^{n+1} = \underset{i \in \{1,\cdots, n\}}{\prod} k$ (where we write $k^n \setminus \{0\} \overset{\iota}{\hookrightarrow} k^n \overset{pr_i}{\to} k$).



\end{enumerate}
\end{defn}
\begin{defn}
\label{TautologicalTopologicalLineBundle}\hypertarget{TautologicalTopologicalLineBundle}{}
\textbf{(tautological topological line bundle)}

For $k$ a [[topological field]] and $n \in \mathbb{N}$, the \emph{tautological line bundle} over the [[projective space]] $k P^n$ is topological $k$-[[line bundle]] whose total space is the following [[subspace]] of the [[product space]] of the [[projective space]] $k P^n$ with $k^n$:

\begin{displaymath}
T
   \coloneqq
  \left\{
    ( [x_1: \cdots : x_{n+1}], \vec v)
    \in
    k P^n \times k^{n+1}
    \;\vert\;
    \vec v \in \langle \vec x\rangle_k
  \right\}
  \,,
\end{displaymath}
where $\langle \vec x\rangle_k \subset k^{n+1}$ is the $k$-[[linear span]] of $\vec x$.

(The space $T$ is the space of pairs consisting of the ``name'' of a $k$-line in $k^{n+1}$ together with an element of that $k$-line)

This is a bundle over [[projective space]] by the projection function

\begin{displaymath}
\itexarray{
    T &\overset{\pi}{\longrightarrow}& k P^n
    \\
    ([x_1: \cdots : x_{n+1}], \vec v) &\mapsto& [x_1: \cdots : x_{n+1}]
  }
  \,.
\end{displaymath}
\end{defn}
\begin{prop}
\label{WellDefinedTautologicalTopologicalLineBundle}\hypertarget{WellDefinedTautologicalTopologicalLineBundle}{}
\textbf{(tautological topological line bundle is well defined)}

The tautological line bundle in def. \ref{TautologicalTopologicalLineBundle} is well defined in that it indeed admits a [[local trivialization]].

\end{prop}
\begin{proof}
We claim that there is a local trivialization over the canonical cover of def. \ref{TopologicalProjectiveSpaceStandardOpenCover}. This is given for $i \in \{1, \cdots, n\}$ by

\begin{displaymath}
\itexarray{
    U_i \times k &\overset{}{\longrightarrow}& T\vert_{U_i}
    \\
    ( [x_1 : \cdots x_{i-1}: 1 : x_{i+1} : \cdots : x_{n+1}] , c )
      &\mapsto&
    ( [x_1 : \cdots x_{i-1} : 1 : x_{i+1} : \cdots : x_{n+1} ], (c x_1, c x_2, \cdots , c x_{n+1}) )
  }
  \,.
\end{displaymath}
This is clearly a [[bijection]] of underlying sets.

To see that this function and its inverse function are continuous, hence that this is a [[homeomorphism]] notice that this map is the [[extension]] to the [[quotient topological space]] of the analogous map

\begin{displaymath}
\itexarray{
    ( (x_1, \cdots, x_{i-1}, x_{i+1}, \cdots, x_{n+1}) , c)
     &\mapsto&
    ( (x_1, \cdots, x_{i-1}, x_{i+1}, \cdots, x_{n+1}) , (c x_1, \cdots c x_{i-1}, c, c x_{i+1}, \cdots, c x_{n+1}) )
  }
  \,.
\end{displaymath}
This is a [[polynomial]] function on [[Euclidean space]] and since [[polynomials are continuous]], this is continuous. Similarly the [[inverse function]] lifts to a [[rational function]] on a subspace of Euclidean space, and since [[rational functions are continuous]] on their domain of definition, also this lift is continuous.

Therefore by the [[universal property]] of the [[quotient topology]], also the original functions are continuous.

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

\begin{itemize}%
\item The [[basic complex line bundle on the 2-sphere]] is the tautological [[complex line bundle]] over the [[complex projective space]] $\mathbb{C}P^1 \simeq S^2$ (the [[Riemann sphere]])

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

Lecture notes include

\begin{itemize}%
\item [[Klaus Wirthmüller]], section 2 of \emph{Vector bundles and K-theory}, 2012 (\href{ftp://www.mathematik.uni-kl.de/pub/scripts/wirthm/Top/vbkt_skript.pdf}{pdf})

\end{itemize}
See also

\begin{itemize}%
\item Wikipedia, \emph{\href{https://en.wikipedia.org/wiki/Tautological_bundle}{Tautological bundle}}

\end{itemize}
[[!redirects tautological line bundles]]



\end{document}