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\title The Geometry of Vector Bundles and an Introduction to Gauge
Theory \\
Lecture 11
\endtitle
\author Professor Steven Bradlow \\ Class Notes From Math 433
\endauthor
\affil University of Illinois at Urbana-Champaign
\endaffil
\address 273 Altgeld Hall, 1409 W. Green Street, Urbana, IL 61801
\endaddress
\email bradlow\@math.uiuc.edu
\endemail
\date February 16, 1998
\enddate
\endtopmatter
\document
One of the central problems in vector bundles is the following classification
problem. Given a space $B$ and $n \in \Bbb N$, describe, up to isomorphism, all vector bundles of rank $n$ over $B$. This is denoted by $\text{Vect}_n(B)$.
\vskip.25in
\remark{Exercise 1} Using a two patch covering of $S^1$, examine the possibilities for the transition functions for a rank $n$ bundle over $S^1$. Make a conjecture for $\text{Vect}_n(S^1)$.
\endremark
\vskip.25in
The idea for the classification is to find a bundle, denoted
$\pi:EG \to BG$ and called the universal bundle for $B$, such that for any other bundle $E \to B$ of rank $n$, there is
a map $f:B \to BG$ such that $f^*(EG) \cong E$. By our earlier results, we already know that homotopic maps induced isomorphic bundles. Thus, if $[B,BG]$ denotes the homotopy classes of maps from $B$ to $BG$, then we have a well defined map,
$$\aligned
[B,BG] &\to \text{Vect}_n(B) \cr
[f] &\mapsto [f^*(EG)] \endaligned$$
Our goal is to construct an inverse to this map. First, we have to say
what $EG$ and $BG$ should be.
We will assume throughout that $B$ is compact, although the
following will hold for when $B$ is only paracompact.
\vskip.25in
\definition{Definition 1} A cover $\{U_\alpha\}$ of $B$ is called a
good cover if every non-empty intersection is diffeomorphic to $\R^d$, where $d$ is the dimension of $B$. [Note:
some texts call a cover good if every intersection is contractible.]
\enddefinition
\vskip.25in
Note that for any bundle over $B$, we may take the elements of the good
cover as our trivializing neighborhoods.
\vskip.25in
\proclaim{Proposition} For $\pi:E \to B$ a vector bundle there
exist finite many sections, $\{s_1,\dots,s_k\}$ such that for all $b \in B$,
the set $\{s_1(b),\dots,s_k(b)\}$ spans the fiber.
\endproclaim
{\bf Proof:} Consider first the local situation. Using the good cover, $\psi_i:E_{|U_i} \to U_i \times \Bbb R^n$, we claim that we have $n$ local sections which generate $E_b$ for all $b \in U_i$. If $\{e_1,\dots,e_n\}$ is
the standard basis, the local sections are $(x,e_a)$. Define $s_{i,a}=\psi_i^{-1}(b,e_a)$. This is a local frame over $U_i$. To patch
together to get a global set of sections, taker a partition of unity, $\{\rho_i\}$ subordinate to $\{U_i\}$. Extend $s_{i,a}(b)$ to $\rho_i(b)s_{i,a}(b)$. Then $$\bigcup_i \{\rho_i s_{i,a}\}_{a=1}^n$$ do the job. \qed
\vskip.25in
\remark{Exercise 2} Check that the collection $$\{\tilde s_i,\dots,\tilde s_k\}=\bigcup_i \{\rho_i s_{i,a}\}_{a=1}^n$$ globally generate the bundle $E$, i.e, $\{\tilde s_1(b),\dots,\tilde s_k(b)\}$ spans $E_b$ for all $b \in B$.
\endremark
\vskip.25in
Now that we have our global sections, how do we use them? Assume that
$s_1,\dots,s_k$ are the desired sections. Let $V=\Bbb R \langle s_1,\dots,s_k \rangle$ be the real vector space on the set of sections. Fix $b \in B$ and
define a map, $\text{ev}_b:V \to E_b$ by evaluation, $s_i \mapsto s_i(b)$. Then, the following properties hold:
\vskip.25in
\itemitem{(a)} The map is surjective since the $s_i(b)$ span the fiber.
\vskip.25in
\itemitem{(b)} The kernel of $\text{ev}_b$ is a codimension $n$ subspace of
$V$, so that $V/\text{ker}(\text{ev}_b) \cong E_b$
\vskip.25in
To complete our search for the universal bundle, we need to digress into
the realm of the Grassmanian, denoted by $G(k,n)$, which is the set of
all $k-$planes in $\Bbb R^n$. This is the subject of the next lecture.
\enddocument
\end