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\title The Geometry of Vector Bundles and an Introduction to Gauge
Theory \\
Lecture 13
\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 20, 1998
\enddate
\endtopmatter
\document
Last time we constructed a bundle, $\pi:Q \to G(N-n,N)$ with $Q_{|V} \cong \Bbb R^n/V$. We saw that for a rank $n$ bundle, $E \to B$ over a compact space with a finite good cover, that there is a collection of sections, $\{s_1,\dots,s_N\}$ which have the property that the evaluation map, $\text{ev}_b:\Bbb R^n \to E_b$ (via $e_i \mapsto s_i(b)$), generates the fiber. The kernel of $\text{ev}_b$ fits into an exact sequence, $$0 \to \text{ker ev}_b \to \Bbb R^N \to E_b \to 0$$ Hence, we can construct a map $f:B \to G(N-n,N)$ by $b \mapsto \text{ker ev}_b$.
\proclaim{Claim} $f^*(Q) \cong E$. \endproclaim
\vskip.25in
{\bf Proof:} $$F^*(Q)_{|b} \cong Q_{|(b)} \cong [\text{ker ev}_b] \cong \Bbb R^N/\text{ker ev}_b \cong E_b$$ \qed
\vskip.25in
Recall that this construction is meant to produce a map, $\text{Vect}_n(B) \to [B,BG]$ which is the inverse of the map, $[f] \mapsto [f^*(Q)]$. We thus need
to consider what happens if we pick a different set of sections, $\{s_1',\dots,s_n'\}$ and thus get a different map, $f':B \to G(N-n,N)$.
We would like to conclude that $f$ and $f'$ are homotopic, i.e, $$[f]=[f'] \quad \text{ in }[B,BG]$$
In order for this to be true, we need to embed $$G(N-n,N) \to G(N'-n,N')$$
for $N'>N$. This embedding is induced naturally by the embedding
$$\R^N \to \R^{N'}$$ in which $\R^N$ occupies the subspace spanned by
the first $N$ standard basis vectors in $\R^{N'}$. The extra room in $G(N'-n,N')$ is needed in order to construct the required homotopy between the
maps $f$ and $f'$.[ Details can be found in {\it Husemoller} or {\it Milnor} and {\it Stashoff}].
We can avoid questions about which $N'$ to use in $G(N'-n,N')$ by taking the
limit,
$$ \dots \to G(N-n,N) \to G(N+1-n,N+1) \to \dots$$ That is, by using
$$G_\infty(n)=\lim_{N \to \infty} G(N-n,N)$$
We can similarly take the limit,
$$\dots \to \gamma_n^k \to \gamma_{n+1}^k \to \dots $$
to produce a bundle, $\gamma^k \to G_\infty(n)$ with $BG=G_\infty$ and
$EG=Q$. We thus get:
\vskip.25in
\proclaim{Proposition} There is a bijective correspondance,
$$\text{Vect}_n(B) \leftrightarrow [B,BG]$$
\endproclaim
\vskip.25in
\remark{Remark} The above methods can be extended to the case where $B$ is only {\it paracompact}.
\endremark
\vskip.25in
\example{Example 1} We compute $\text{Vect}_n(S^1)$. We need to examine
$G(N-n,N)$ where $N$ is the number of elements in a good cover of $S^1$. It is
straightforward to show that any good cover of $S^1$ must involve at least
$3$ elements. Set $N=3n$. Now, $G(2,3) \approx G(1,3) \approx \Bbb RP^2$.
The work above actually shows that $\text{Vect}_n[S^1,\Bbb RP^2]=\Bbb Z_2$.
Note that $[,]$ refers to {\bf unbased} homotopies.
\endexample
\vskip.25in
For any principal $G-$bundle, there is a construction for a spaces $BG,EG$ and
a principal $G-$bundle, $EG \to BG$. If we define $\text{Prin}_G(B)$ to be
the set of isomorphism classes of principal $G-$bundles over $B$, then
we get $\text{Prin}_G(B) \cong [B,BG]$. $EG$ is obtained by something called
the Milnor construction (pg 54 of Husemoller). $EG$ is obtained by the infinite join of the group $G$. This is by definition
$$G \ast G \ast \dots = \{(t_1x_0,t_1x_1,\dots):t_i \in [0,1],x_i \in G,\text{ a finite number of the }t_1 \neq 0,\sum t_i=1\}$$
Now, let $G$ act on $EG$ by right multiplication. Let $BG$ be the orbit space
and $EG \to BG$ the natural map. It remains to be shown that this is
a principal $G-$bundle with the appropriate universal properties.
\vskip.25in
\remark{Note} $EG$, I believe, is the realization of the Bar construction for
the space $G$.
\endremark
\vskip.25in
\remark{Exercise 1} Take $G=\Bbb Z_2$. Show that $EG$ is infact $S^\infty$ (the limit of the system $\dots S^n \to S^{n+1} \to \dots$) and that
$BG=\rp^\infty$ (the limit of the system $\dots \to \rp^n \to \rp^{n+1} \to \dots $). If $G=S^1$, show that $EG=S^\infty$ and that
$BG=\cp^\infty$ (the limit of the system $\dots \to \cp \to \cp^{n+1} \to \dots $)
\endremark
\enddocument
\end