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\begin{document}

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\section*{direct sum of vector bundles}

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

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

[[!include bundles - contents]]

\hypertarget{linear_algebra}{}\paragraph*{{Linear algebra}}\label{linear_algebra}

[[!include homotopy - 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{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{whitney_summands_of_trivial_vector_bundles}{Whitney summands of trivial vector bundles}\dotfill \pageref*{whitney_summands_of_trivial_vector_bundles} \linebreak
\noindent\hyperlink{characteristic_classes_of_whitney_sums}{Characteristic classes of Whitney sums}\dotfill \pageref*{characteristic_classes_of_whitney_sums} \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}

For $E_1, E_2 \to X$ two [[vector bundles]], their [[direct sum]] over $X$, also called their \textbf{Whitney sum}, is the vector bundle $E_1 \oplus E_2 \to X$ whose [[fiber]] over any $x \in X$ is the [[direct sum]] of vector spaces of the fibers of $E_1$ and $E_2$.

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

\begin{defn}
\label{DirectSumOfTopologicalVectorBundlesViaTotalspaces}\hypertarget{DirectSumOfTopologicalVectorBundlesViaTotalspaces}{}
\textbf{([[direct sum]] of [[topological vector bundles]] via total spaces)}

Let

\begin{enumerate}%
\item $X$ be a [[topological space]],


\item $E_1 \overset{\pi_1}{\to} X$ and $E_2 \overset{\pi_2}{\to} X$ two [[topological vector bundles]] over $X$.



\end{enumerate}
Then the \emph{direct sum of vector bundles} $E_1 \oplus_X E_2 \to E$ is the [[topological vector bundle]] whose total space is the [[topological subspace]]

\begin{displaymath}
E_1 \oplus_X E_2
  \;\coloneqq\;
  \left\{
    (v_1, v_2)
    \in 
    E_1 \times E_2
    \,\vert\,
    \pi_1(v_1) = \pi_2(v_2)
  \right\}
  \;\subset\;
  E_1 \times E_2
\end{displaymath}
of the [[product topological space]] of the two total spaces, and whose projection map is

\begin{displaymath}
\itexarray{
    E_1 \oplus_X E_2 
      &\overset{\phantom{AA}\pi\phantom{AA}}{\longrightarrow}&
    X
    \\
    (v_1,v_2) &\overset{\phantom{AAA}}{\mapsto}& \pi_1(v_1) = \pi_2(v_2)
  }
  \,.
\end{displaymath}
For $x \in X$ the vector space structure on the [[fibers]]

\begin{displaymath}
(E_1 \oplus E_2)_x \simeq (E_1)_x \oplus (E_2)_x
\end{displaymath}
is the one on the [[direct sum]] of [[vector spaces]].

\end{defn}
\begin{defn}
\label{DirectSumOfTopologicalVectorBundlesViaTransitionFunctions}\hypertarget{DirectSumOfTopologicalVectorBundlesViaTransitionFunctions}{}
\textbf{([[direct sum]] of [[topological vector bundles]] via [[transition functions]])}

Let $X$ be a [[topological space]], and let $E_1 \to X$ and $E_2 \to X$ be two [[topological vector bundles]] over $X$.

Let $\{U_i \subset X\}_{i \in I}$ be an [[open cover]] with respect to which both vector bundles locally trivialize (this always exists: pick a [[local trivialization]] of either bundle and form the joint [[refinement]] of the respective [[open covers]] by [[intersection]] of their patches). Let

\begin{displaymath}
\left\{
    (g_1)_{i j}
    \colon U_i \cap U_j \to GL(n_1)
  \right\}
  \phantom{AAA}
  \text{and}  
  \phantom{AAA}
  \left\{
    (g_2)_{i j}
    \colon
     U_i \cap U_j \longrightarrow GL(n_2)
  \right\}
\end{displaymath}
be the [[transition functions]] of these two bundles with respect to this cover.

For $i, j \in I$ write

\begin{displaymath}
\itexarray{
    (g_1)_{i j} \oplus (g_2)_{i j}
    &\colon&
    U_i \cap U_j
      &\longrightarrow&
    GL(n_1 + n_2)
    \\
    &&
    x 
      &\overset{\phantom{AAA}}{\mapsto}&
    \left(
      \itexarray{
         (g_1)_{i j}(x) & 0
         \\
         0 & (g_2)_{i j}(x)
      }
    \right)
  }
\end{displaymath}
be the pointwise [[direct sum]] of these transition functions

Then the \emph{direct sum bundle} $E_1 \oplus E_2$ is the one glued from this direct sum of the transition functions (by \href{topological+vector+bundle#TopologicalVectorBundleFromCechCocycle}{this construction}):

\begin{displaymath}
E_1 \oplus E_2
  \;\coloneqq\;
  \left(
    \left( \underset{i}{\sqcup} U_i \right)
    \times 
    \left(
      \mathbb{R}^{n_1 + n_2}
    \right)
  \right)/ \left( \left\{ (g_1)_{i j} \oplus (g_2)_{i j} \right\}_{i,j \in I} \right)
  \,.
\end{displaymath}
\end{defn}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{example}
\label{DirectSumOnDisjointUnionSpace}\hypertarget{DirectSumOnDisjointUnionSpace}{}
Let $X$ and $Y$ be [[topological spaces]], and write $X \sqcup Y$ for their [[disjoint union space]].

Then every [[topological vector bundle]] on $X \sqcup Y$ is the direct sum of a vector bundle that has [[rank of a vector bundle|rank]] zero on $Y$ and one that has rank zero on $X$.

More explicitiy: let

\begin{displaymath}
i_X \colon Vect(X) \longrightarrow Vect(X \sqcup Y)
\end{displaymath}
and

\begin{displaymath}
i_Y \colon Vect(Y) \longrightarrow Vect(X \times Y)
\end{displaymath}
be the operations of extending a vector bundle on the other [[connected component]] by a rank-0 vector bundle, then

\begin{displaymath}
Vect(X) \times Vect(Y)
    \underoverset{\simeq}{ i_X \oplus_{(X \sqcup Y)} i_Y }{\longrightarrow}
  Vect(X \sqcup Y)
\end{displaymath}
is an [[isomorphism]] of [[isomorphism classes]] of vector bundles (and an [[equivalence of categories]] of [[Vect(X)|categories of vector bundles]] before passing to isomorphism classes).

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

\hypertarget{whitney_summands_of_trivial_vector_bundles}{}\subsubsection*{{Whitney summands of trivial vector bundles}}\label{whitney_summands_of_trivial_vector_bundles}

\begin{prop}
\label{TopologicalSubBundlesOverParacompactHausdorffSpacesAreDirectSummands}\hypertarget{TopologicalSubBundlesOverParacompactHausdorffSpacesAreDirectSummands}{}
\textbf{(sub-bundles over [[paracompact spaces]] are [[direct sum of vector bundles|direct summands]])}

Let

\begin{enumerate}%
\item $X$ be a [[paracompact Hausdorff space]],


\item $E \to X$ a [[topological vector bundle]].



\end{enumerate}
Then every vector subbundle $E_1 \hookrightarrow E$ is a direct vector bundle summand, in that there exists another vector subbundle $E_2 \hookrightarrow E$ such that their direct sum of vector bundles (def. \ref{DirectSumOfTopologicalVectorBundlesViaTotalspaces}) is $E$

\begin{displaymath}
E_1 \oplus E_2 \simeq E
  \,.
\end{displaymath}
\end{prop}
(\hyperlink{Hatcher}{e.g. Hatcher, prop. 1.3})

\begin{proof}
Since $X$ is assumed to be paracompact Hausdorff, there exists a [[inner product on vector bundles]]

\begin{displaymath}
\langle -,-\rangle
  \;\colon\;
  E \oplus_X E 
    \longrightarrow
  X \times \mathbb{R}
\end{displaymath}
(by \href{inner+product+of+vector+bundles#ExistenceOfInnerProductOfTopologicalVectorBundlesOverParacompactHausdorffSpaces}{this prop.}). This defines at each $x \in X$ the [[orthogonal complement]] $(E'_x)^\perp \subset E_x$ of $E'_x \hookrightarrow E$. The [[subspace]] of these orthogonal complements is readily checked to be a [[topological vector bundle]] $(E')^\perp \to X$. Hence by construction we have

\begin{displaymath}
E \;\simeq\; E' \oplus_X (E')^\perp
  \,.
\end{displaymath}
\end{proof}
\begin{prop}
\label{OverCompactHausdorffSpacesEveryTopologicalVectorBundleIsDirectSummandOfATrivialBundle}\hypertarget{OverCompactHausdorffSpacesEveryTopologicalVectorBundleIsDirectSummandOfATrivialBundle}{}
\textbf{(over [[compact Hausdorff spaces]] every vector bundle is [[direct sum of vector bundles|direct summand]] of a trivial bundle)}

Let

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


\item $E \to X$ a [[topological vector bundle]].



\end{enumerate}
Then there exists another topological vector bundle $\tilde E \to X$ such that the direct sum of vector bundles (def. \ref{DirectSumOfTopologicalVectorBundlesViaTotalspaces}) of the two is a trivial vector $X \times \mathbb{R}^n$:

\begin{displaymath}
E \oplus \tilde E
   \;\simeq\;
  X \times \mathbb{R}^n
  \,.
\end{displaymath}
\end{prop}
(e.g. \hyperlink{Hatcher}{Hatcher, prop. 1.4}, \hyperlink{Friedlander}{Friedlander, ptop. 3.1})

\begin{proof}
Let $\{U_i \subset X\}_{i \in I}$ be an [[open cover]] of $X$ over which $E \to X$ has a [[local trivialization]]

\begin{displaymath}
\left\{
    \phi_i
      \;\colon\;
     U_i \times \mathbb{R}^n
       \overset{\simeq}{\longrightarrow}
     E\vert_{U_i}
  \right\}_{i \in I}
  \,.
\end{displaymath}
By compactness of $X$, there is a [[finite cover|finite sub-cover]], hence a [[finite set]] $J \subset I$ such tat

\begin{displaymath}
\{U_i \subset X\}_{i \in J \subset I}
\end{displaymath}
is still an open cover over which $E$ trivializes.

Since [[paracompact Hausdorff spaces equivalently admit subordinate partitions of unity]] there exists a [[partition of unity]]

\begin{displaymath}
\left\{
    f_i \;\colon\; X \to [0,1]
  \right\}_{i \in J}
\end{displaymath}
with [[support]] $supp(f_i) \subset U_i$. Hence the functions

\begin{displaymath}
\itexarray{
    E\vert_{U_i} 
     &\overset{\phantom{AAAA}}{\longrightarrow}&
    U_i \times \mathbb{R}^n
    \\
    v 
       &\overset{\phantom{AAA}}{\mapsto}&
    f_i(x) \cdot \phi_i^{-1}(v)
  }
\end{displaymath}
extend by 0 to vector bundle homomorphism of the form

\begin{displaymath}
f_i \cdot \phi^{-1}_i
  \;\colon\;
  E
    \longrightarrow
  X \times \mathbb{R}^n
  \,.
\end{displaymath}
The finite pointwise [[direct sum]] of these yields a vector bundle homomorphism of the form

\begin{displaymath}
\underset{i \in J}{\oplus} f_i \cdot \phi_i
  \;\colon\;
  E
    \longrightarrow
  X \times \left( 
    \underset{i \in J}{\oplus}
       \mathbb{R}^n
  \right)
  \simeq
  X \times \mathbb{R}^{n \dot {\vert J\vert}}
  \,.
\end{displaymath}
Observe that, as opposed to the single $f_i \cdot \phi^{-1}_i$, this is a fiber-wise injective, because at each point at least one of the $f_i$ is non-vanishing. Hence this is an injection of $E$ into a trivial vector bundle.

With this the statement follows by prop. \ref{TopologicalSubBundlesOverParacompactHausdorffSpacesAreDirectSummands}.

\end{proof}
\begin{remark}
\label{}\hypertarget{}{}
Prop. \ref{OverCompactHausdorffSpacesEveryTopologicalVectorBundleIsDirectSummandOfATrivialBundle} is key for the construction of [[topological K-theory]] groups on compact Hausdorff spaces.

\end{remark}
Remark : Let $E_1\rightarrow M$ and $E_2\rightarrow M$ be vector bundles over $M$. This gives product map $E_1\times E_2\rightarrow M\times M$ which is still a vector bundle. Consider diagonal map $d:M\rightarrow M\times M$ given by $m\mapsto (m,m)$. The Whitney sum of $E_1\rightarrow M$ and $E_2\rightarrow M$ is the pull back of $E_1\times E_2\rightarrow M\times  M$ along the diagonal map $d:M\rightarrow M\times M$ which is denoted by $E_1\oplus E_2\rightarrow M$.

\hypertarget{characteristic_classes_of_whitney_sums}{}\subsubsection*{{Characteristic classes of Whitney sums}}\label{characteristic_classes_of_whitney_sums}

\begin{prop}
\label{EulerClassOfWhitneySumIsCupProductOfEulerClasses}\hypertarget{EulerClassOfWhitneySumIsCupProductOfEulerClasses}{}
\textbf{([[Euler class]] takes [[Whitney sum]] to [[cup product]])}

The Euler class of the [[Whitney sum]] of two [[orthogonal group|oriented]] [[real vector bundles]] to the [[cup product]] of the separate Euler classes:

\begin{displaymath}
\chi( E \oplus F )
  \;=\;
  \chi(E)  \smile \chi(F)
  \,.
\end{displaymath}
\end{prop}
For details see at [[Euler class]], \href{Euler+class#EulerClassOfWhitneySumIsCupProductOfEulerClasses}{this Prop.}.

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

\begin{itemize}%
\item [[inner product on vector bundles]]


\item [[tensor product of vector bundles]], [[external tensor product of vector bundles]], [[dual vector bundle]],


\item [[tensor category]]


\item [[Stiefel-Whitney class]]


\item [[virtual vector bundle]], [[topological K-theory]]


\item [[framed manifold]], [[Thom spectrum]]



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

Discussion with an eye towards [[topological K-theory]] is in

\begin{itemize}%
\item [[Max Karoubi]], \emph{K-theory. An introduction}, Grundlehren der Mathematischen Wissenschaften \textbf{226}, Springer 1978. xviii+308 pp.


\item [[Allen Hatcher]], section 1.1 of \emph{Vector bundles and K-Theory}, (partly finished book) \href{http://www.math.cornell.edu/~hatcher/VBKT/VBpage.html}{web}



\end{itemize}
and with an eye towards [[algebraic K-theory]] in

\begin{itemize}%
\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})

\end{itemize}
[[!redirects Whitney sum]]

[[!redirects direct sums of vector bundles]]



\end{document}