# Fiber Bundles with Structure and Principal $G\dash$ Bundles (Tuesday, August 24)

:::{.remark}
Setup:
 
- $B \in \Top$ is a space.
- $\pi:E\to B$ is a map of sets with fibers/preimages $F \da F_b \da \pi\inv(\ts{b})$.
- A *bundle atlas* for $\pi$ is $\phi$ where $\phi_U: \pi\inv(U) \to U \cross F$ is a bijection of sets

Then there is at most one topology on $E$ making $\pi:E\to B$ into a fiber bundle with the specified atlas.
:::

:::{.definition title="Dual of a vector bundle"}
Given a vector bundle $\pi:E\to B$, form the **dual bundle** $\pi\dual:E\dual\to B$ by setting

- $E\dual \da \disjoint_{b\in B} F_b\dual$
- Set $\pi\dual(F_b\dual) = \ts{b} \in B$.
- Given $\phi: \pi\inv(U)\to U\times \RR^n$, set
\[
\phi\dual: (\pi\dual)\inv(U) = \disjoint_{b\in U} F_b\dual \too U\cross (\RR^n)\dual \cong U\cross \RR^n
.\]

Here $A \subseteq \pi\inv(U)$ is open iff $\phi_U(A)$ is open in $B$.

:::

:::{.remark}
Consider what happens on overlapping charts -- looking at maps fiberwise yields:

\begin{tikzcd}
	{\pi\inv(U)} && {\pi\inv(U\intersect V)} && {\pi\inv(V)} \\
	&& {} \\
	{U\times F} && {(U\intersect V) \times F} && {V\times F}
	\arrow[hook', from=1-3, to=1-1]
	\arrow["{\varphi_U}", from=1-1, to=3-1]
	\arrow[hook', from=3-3, to=3-1]
	\arrow[hook, from=3-3, to=3-5]
	\arrow[hook, from=1-3, to=1-5]
	\arrow["{\varphi_V}"', from=1-5, to=3-5]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=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)

Starting at $(U \intersect V)\times F$ and running the diagram clockwise yields a map
\[
\phi_V \circ \phi_U \inv : (U \intersect V ) \times F &\to (U \intersect V ) \times F \\
(b, f) &\mapsto (b, \phi_{VU}(f) )
\]
where $\phi_{VU}$ is the following continuous map, defining a **transition function**:
\[
\phi_{VU}: U \intersect V &\to \Homeo(F)
,\]
where $\Homeo(F) \da \Hom_{\Top}(F, F)$ with the compact-open topology.
:::

:::{.definition title="The compact-open topology"}
Let $\Map(X, Y)\da \Hom_\Top(X, Y)$ be the set of continuous maps $X\to Y$, then a map $Z\to \Map(X, Y)$ is continuous iff the following map is continuous:
\[
Z\cross X &\to Y \\
(z, x) &\mapsto f(z)(x)
.\]

If $X$ is Hausdorff and locally compact then $\Map(X, Y)$ will have this property for all $Y$.
A subbasis for this topology will be given by taking $C \subseteq X$ compact, $U \subseteq Y$ open and taking the basic opens to be 
\[
S(C, U) \da \ts{f\in \Map(X, Y)\st f(X) \subseteq U}
.\]
If $Y$ has a metric, then this will coincide with the *compact convergence topology*, which has a basis
\[
\ts{ 
S(f, C, E) \st C \subseteq X \text{ compact}, \forall \eps>0,\,\forall f\in \Map(X, Y)
}, \\
S(f, C, E) \da \ts{ 
g\in \Map(X, Y) \st d( f(x), g(x) ) < \eps \,\,\forall x\in C 
} 
.\]
:::

:::{.definition title="Structure Groups"}
Let $G \subseteq \Homeo(F)$, then a **fiber bundle with structure group** $G$ is a fiber bundle $F\to E \mapsvia{\pi} B$ together with a bundle atlas such that $G \subseteq \Homeo(F)$.
:::

:::{.example title="?"}
An $\RR^n\dash$bundle is just a bundle where $F = \RR^n$ for all fibers, where we ignore the vector space structure and only take transition functions to be homeomorphisms.
An $\RR^n\dash$bundle with a $G\da \GL_n(\RR)$ is exactly a vector bundle, where we can use the structure group to put a vector space structure on the fibers.
We have charts $\phi_U: \pi\inv(U)\to U \cross \RR^n$, so for all $b\in U$, writing $F_b \da \pi\inv(\ts{b})$ and get $\phi_U(F_b) = \st{b} \cross \RR^n$.
We can then define addition and multiplication for $w_1, w_2 \in F_b$ as
\[
cw_1+ w_2 \da \phi_U\inv\qty{ c\phi_U(w_1) + \phi_U(w_2) }
.\]
This is well-defined because for any other chart containing $V\ni b$, we have $\phi_{VU}\in \GL_n(\RR)$.
This follows by just setting $A \da \phi_V \circ \phi_U \inv$ and writing
\[
\phi_V(w_1 + w_2) 
&= A \phi_U(w_1 + w_2) \\
&\da A\qty{\phi_U(w_1) + \phi_U(w_2) } \\
&= A\phi_U(w_1) + A\phi_U(w_2) \\
&= \phi_V(w_1) + \phi_V(w_2) \\
&\da \phi_V(w_1 + w_2)
.\]
:::

:::{.example title="Bundles with structure"}
An $\RR^n\dash$bundle with a $\GL_{n}^+(\RR)$ structure is an orientable vector bundle, where
\[
\GL_n^+(\RR) = \ts{ A \in \GL_n(\RR) \st \det(A) > 0 } 
.\]

A $G\da O_n(\RR)$ structure yields vector bundles with Riemannian metrics on fibers, where $O_n(\RR) \da \ts{ A\in \GL_n(\RR) \st AA^t = \id}$.
Here we use the fact that there is an equivalence between metrics (symmetric bilinear pairings) and choices of an orthonormal basis, e.g. using that if $\ts{e_1, \cdots, e_n}$, one can specify an inner product completely by writing
\[
v\da \sum v_i e_i,\quad w \da \sum w_i e_i &&\implies \inner{v}{w} = \sum v_i w_i
.\]
:::

:::{.definition title="Principal $G\dash$bundles"}
A **principal $G\dash$bundle** is a fiber bundle $\pi:P\to B$ with a right $G\dash$action $\psi: P\times G\to P$ such that

1. $\psi\qty{F_b} = F_b$, so the action preserves each fiber, and
2. $\psi$ is free and transitive.

:::