# Principal $G\dash$bundles (Thursday, August 26)

:::{.remark}
Today: relating $\Prin\Bun\slice{G}$ to fiber bundles with a $G\dash$structure.
Recall that a principal $G\dash$bundle is a fiber bundle $\pi:P\to B$ with a fiberwise $G\dash$action $P\times G\to P$ which induces a free and transitive action on each fiber.
Note that we assume $G\in\Top\Grp$.
Any bundle in $\Prin\Bun\slice{G}$ is a fiber bundle with fibers $F$ homeomorphic to $G$ and admits a $G\dash$structure:
\[
G &\injects \Homeo(G) \\
g &\mapsto (h\mapsto gh)
.\]
Using that $F\cong G$, taking charts $(U, \varphi), (V, \psi)$ for $\pi:P\to B$, we can identify

\begin{tikzcd}
	{\pi\inv(U \intersect V)} && {(U \intersect V) \times G} && {(U \intersect V) \times G} && {\pi\inv(U \intersect V)} \\
	&& {(b, 1)} && {(b, g)} \\
	&& {(b, h)} && {(b, gh)}
	\arrow["{\phi_V \circ \phi_U\inv}", from=1-3, to=1-5]
	\arrow["{\phi_U, \cong}", from=1-1, to=1-3]
	\arrow["{\phi_V, \cong}"', from=1-7, to=1-5]
	\arrow[maps to, from=2-3, to=2-5]
	\arrow[maps to, from=3-3, to=3-5]
\end{tikzcd}

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

So every transition function is given by left-multiplication by some element in $G$, as opposed to arbitrary homeomorphisms of $G$ as a topological group.
:::

:::{.example title="of principal bundles"}
\envlist

- Trivial actions: $B \cross G \mapsvia{p_1} B$.
- Regular covering spaces $\pi:\tilde X\to X$, then $G = \Deck(\tilde X/X)$ with the discrete topology.
- Given an $n\dash$dimensional vector bundle $\pi: E\to B$, take 
\[
\Frame(F_b) \da \ts{(e_1, \cdots, e_n) \in F_b} \subseteq F_b^{\times n}
,\]
the collection of all ordered bases of $F_b$.
  Then set 
  \[
\Frame_n \da \disjoint_{b\in B} \Frame(F_b) \to B
  \]
  to get a principal $G\dash$bundle for $G = \GL_n(F_b)$ under the following action: picking a framing $(e_1, \cdots, e_n)$ in $F_b$,
  then for $A\in \GL_n(F_b)$ regarded as a linear map, define
  \[
  (\vector e_1, \cdots, \vector e_n) \cdot A \da \qty{\sum_i a_{i,1} \vector e_i, \sum_i a_{i, 2} \vector e_i, \cdots, \sum_i a_{i, n} \vector e_i }
  .\]

- Given an *oriented* $n\dash$dimensional vector bundle $\pi:E\to B$, one gets a $G\da \GL_n^+(\FF_b)$ by taking positively oriented frames.
- Given a vector bundle with a Riemannian metric, we get a principal $\OO_n(\RR)\dash$bundle by taking orthonormal frames.

:::

:::{.definition title="?"}
Given two principal $G\dash$bundles $\pi: P\to B$ and $\pi': Q\to B$, an **isomorphism of principal bundles** is a $G\dash$equivariant map $P \mapsvia{f} Q$ commuting over $B$:


\begin{tikzcd}
	P && Q \\
	\\
	& B
	\arrow["\pi", from=1-1, to=3-2]
	\arrow["{\pi'}"', from=1-3, to=3-2]
	\arrow["f"', from=1-1, to=1-3]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsMyxbMCwwLCJQIl0sWzIsMCwiUSJdLFsxLDIsIkIiXSxbMCwyLCJcXHBpIl0sWzEsMiwiXFxwaSciLDJdLFswLDEsImYiLDJdXQ==)

Here *equivariant* means commuting with the $G\dash$action, in the following precise sense:
let $(U, \varphi)$ and $(V, \psi)$ be charts for $\pi, \pi'$, then consider the composition
\[
F: \qty{ (U \intersect V) \times F \mapsvia{\phi\inv} \pi\inv(U \intersect V) \mapsvia{f} (\pi')\inv (U \intersect V) \mapsvia{\psi} (U \intersect V) \times F}
.\]
Note that this fixes every point $b\in U \intersect V$, so we can regard $F: U \intersect V \to \Homeo(F)$, using that $f$ commutes with the projection maps:
\[
(b, ?) \mapsto \pi\inv(b) \mapsto (f\circ \pi\inv)(b) = (\pi')\inv b \mapsto b
.\]
We say $f$ is a $G\dash$isomorphism iff $F$ sends everything to $G$.
:::

## Sending Fiber Bundles to Principal $G\dash$bundles

:::{.remark}
Given a principal $G\dash$bundle $\pi:P\to B$ and a $F\in \Top$ with a left $G\dash$action.
Then define 
\[
P \fiberprod{G} F / (pg, f)\sim (p, gf)
\]
as a fiber bundle over $B$ using $\pi$ as the projection.
Note that this looks like a tensor product, and this works in general for any space $P$ with a right $G\dash$action and $F$ with a left $G\dash$action.
This will be a fiber bundle with fiber $F$ and structure group $G \leq \Homeo(F)$.

Locally there is a homeomorphism:
\[
(U\times G) \mix{G} F &\mapsvia{\sim} U\times F \\
(p, g, f) &\mapsto (p, gf)
.\]
This is well defined since $(p, gh, f)$ and $(p, g, hf)$ map to $(p, ghf)$.
The inverse is $(p, f) \mapsto (p, 1, gf)$.
:::

:::{.exercise title="?"}
Check that this is a fiber bundle with $G\dash$structure.
:::