# Thursday, September 02

:::{.remark}
Recall that we have a correspondence
\[
\correspond{
  \text{Vector bundles }E
}
&\adjunction{\text{clutching}}{\text{mixing}}{}{}
\correspond{
  \text{Principal $\GL_n\dash$bundles $\Frame(E)$}
}
\]
We saw that $E \cong \Frame(E) \fiberprod{\GL_n(\RR)} \RR^n$.
If we take $\Frame(E)$, mix, and apply the clutching construction, is the result bundle-isomorphic to the frame bundle?
:::

:::{.remark}
Recall the clutching construction: we take a cover $\ts{U_i}_{i\in I}$ and $\phi_{ij}: U_i \intersect U_j \to G$ satisfying the cocycle condition $\phi_{ij}\phi_{jk} = \phi_{ik}$, then $G \subseteq \Homeo(F, F)$ and we construct a fiber bundle $\Union_{i\in I} U_i \times F / \sim$ where for $b\in (U_i \intersect U_j )$ and 
\[
(b, f) \in (U_i \intersect U_j )\cross F \subseteq U_i\cross F
,\] 
we send this to 
\[
(b, \phi_{ji}(b) f ) \in (U_i \intersect U_j)\times F \subseteq U_j \cross F
.\]
This will be a fiber bundle with fiber $F$ and structure group $G$.
Moreover, if $F=G$, this will be a principal $G\dash$bundle using right-multiplication.
:::

:::{.question}
How can we tell when two fiber bundles constructed via clutching are isomorphic?
:::

:::{.lemma title="when clutched bundles are trivial"}
The bundle formed by the clutching data $\ts{\phi_{ij}}$ is trivial (so isomorphic to the trivial bundle) iff there exist $\gamma_i: U_i\to G$ such that $\phi_{ji} = \gamma_j \circ \gamma_i\inv$.
:::

:::{.remark}
For principal bundles, these $\gamma_i$ will give sections assembling to a global section obtained from clutching data:

\begin{tikzcd}
	P && {U_i \times G} \\
	\\
	B && {U_i}
	\arrow[from=1-1, to=3-1]
	\arrow[from=1-3, to=3-3]
	\arrow["{s(b) = (b, \gamma_i(b))}"', curve={height=24pt}, from=3-3, to=1-3]
	\arrow["{\psi_B}"', from=3-1, to=3-3]
	\arrow["\psi"', from=1-1, to=1-3]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMiwwLCJVX2kgXFx0aW1lcyBHIl0sWzIsMiwiVV9pIl0sWzAsMiwiQiJdLFswLDAsIlAiXSxbMywyXSxbMCwxXSxbMSwwLCJzKGIpID0gKGIsIFxcZ2FtbWFfaShiKSkiLDIseyJjdXJ2ZSI6NH1dLFsyLDEsIlxccHNpX0IiLDJdLFszLDAsIlxccHNpIiwyXV0=)

The map on $U_i \to \Union_i U_i \times F$ will be $(b, \gamma_i(b))$, and we can use that
\[
(b, \gamma_i(b)) \sim (b, \phi_{ji}(b)\gamma_i(b) ) \sim (b, \gamma_j(b))
,\]
so these agree on overlaps.
:::

:::{.lemma title="?"}
If a principal bundle $P\to B$ has a global section, then $P$ is trivial, so $P\cong B\cross G$ as bundles.
The idea:

\begin{tikzcd}
	{(b, s(b)g)} &&&& {(b, g)} \\
	P &&&& {B\times G} \\
	\\
	&& B \\
	&& b
	\arrow["{p_1}"', from=2-5, to=4-3]
	\arrow["\pi", from=2-1, to=4-3]
	\arrow[dashed, from=2-1, to=2-5]
	\arrow["s", curve={height=-24pt}, from=4-3, to=2-1]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNixbMCwxLCJQIl0sWzIsMywiQiJdLFs0LDEsIkJcXHRpbWVzIEciXSxbMiw0LCJiIl0sWzAsMCwiKGIsIHMoYilnKSJdLFs0LDAsIihiLCBnKSJdLFsyLDEsInBfMSIsMl0sWzAsMSwiXFxwaSJdLFswLDIsIiIsMCx7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dLFsxLDAsInMiLDAseyJjdXJ2ZSI6LTR9XV0=)

:::

:::{.proof title="of lemma about when clutched bundles are trivial"}
$\implies$:

If $E$ is trivial, we have an isomorphism 


\begin{tikzcd}
E 
  \ar[rd, "f"]
  \ar[rr, "\pi"] 
& 
& 
P\times G
  \ar[ld, "p_1"] 
\\
& 
B 
& 
\end{tikzcd}

We have a $G\dash$isomorphism $E_1 \mapsvia{f} E_2$, and so a composition

\begin{tikzcd}
	{(U\intersect V) \times F} && {\pi\inv(U \intersect V)} && {\pi\inv(U \intersect V)} && {(U\intersect V) \times F}
	\arrow["{\phi_U}", from=1-3, to=1-1]
	\arrow["f"', from=1-3, to=1-5]
	\arrow["{\phi_V}"', from=1-5, to=1-7]
	\arrow["F", curve={height=-30pt}, from=1-1, to=1-7]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMCwwLCIoVVxcaW50ZXJzZWN0IFYpIFxcdGltZXMgRiJdLFsyLDAsIlxccGlcXGludihVIFxcaW50ZXJzZWN0IFYpIl0sWzQsMCwiXFxwaVxcaW52KFUgXFxpbnRlcnNlY3QgVikiXSxbNiwwLCIoVVxcaW50ZXJzZWN0IFYpIFxcdGltZXMgRiJdLFsxLDAsIlxccGhpX1UiXSxbMSwyLCJmIiwyXSxbMiwzLCJcXHBoaV9WIiwyXSxbMCwzLCJGIiwwLHsiY3VydmUiOi01fV1d)

Here we've used that $f$ commutes with the projection maps.
We want to show $\im(F) \subseteq G$.
We have a composite

\begin{tikzcd}
	{U\times F} && {\pi\inv(U_i)} && {U\times F}
	\arrow["{\phi_i}", from=1-3, to=1-1]
	\arrow["f"', from=1-3, to=1-5]
	\arrow["{\gamma_i = \phi_i\inv \circ f: U\to G}", curve={height=-30pt}, from=1-1, to=1-5]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsMyxbMCwwLCJVXFx0aW1lcyBGIl0sWzQsMCwiVVxcdGltZXMgRiJdLFsyLDAsIlxccGlcXGludihVX2kpIl0sWzIsMCwiXFxwaGlfaSJdLFsyLDEsImYiLDJdLFswLDEsIlxcZ2FtbWFfaSA9IFxccGhpX2lcXGludiBcXGNpcmMgZjogVVxcdG8gRyIsMCx7ImN1cnZlIjotNX1dXQ==)

We can fill this in to a commutative diagram:

\begin{tikzcd}
	{(U_i\intersect U_j)\times F} && {\phi\inv(U_i \intersect U_j)} && {(U_i\intersect U_j)\times F} \\
	\\
	{\phi\inv(U_i \intersect U_j)} && {\phi\inv(U_i \intersect U_j)} && {(U_i\intersect U_j)\times F}
	\arrow["f", from=1-3, to=1-5]
	\arrow["{\phi_i}"', from=1-3, to=1-1]
	\arrow["{\phi_j}"', from=3-3, to=3-1]
	\arrow["f", from=3-3, to=3-5]
	\arrow[Rightarrow, no head, from=1-5, to=3-5]
	\arrow[Rightarrow, no head, from=1-3, to=3-3]
	\arrow["{\gamma_i}", color={rgb,255:red,92;green,92;blue,214}, curve={height=-30pt}, from=1-1, to=1-5]
	\arrow["{\gamma_j}"', color={rgb,255:red,92;green,92;blue,214}, curve={height=30pt}, from=3-1, to=3-5]
	\arrow["{\therefore \phi_{ji}}", color={rgb,255:red,92;green,92;blue,214}, from=1-1, to=3-1]
\end{tikzcd}

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

The converse direction proceeds similarly!
:::

:::{.lemma title="?"}
A $G\dash$isomorphism between the bundles $E_1, E_2$ obtained from clutching data $\ts{\phi_{ij}}$ and $\ts{\psi_{ij}}$ respectively with the same cover $\ts{U_i}_{i\in I}$ give maps $\gamma_i: U_i\to G$ such that 
\[
\gamma_j \phi_{ji} \gamma_i\inv = \psi_{ji}
.\]
:::

:::{.proof title="?"}
We can form the composite

\begin{tikzcd}
	{U_i\times F} && {\pi_1\inv(U_i)} && {\pi_2\inv(U_i)} && {U_i \times F}
	\arrow["{\phi_i}", from=1-3, to=1-1]
	\arrow["f"', from=1-3, to=1-5]
	\arrow["{\psi_j}"', from=1-5, to=1-7]
	\arrow["{\gamma_i: U_i\to G}", curve={height=30pt}, from=1-1, to=1-7]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMCwwLCJVX2lcXHRpbWVzIEYiXSxbMiwwLCJcXHBpXzFcXGludihVX2kpIl0sWzQsMCwiXFxwaV8yXFxpbnYoVV9pKSJdLFs2LDAsIlVfaSBcXHRpbWVzIEYiXSxbMSwwLCJcXHBoaV9pIl0sWzEsMiwiZiIsMl0sWzIsMywiXFxwc2lfaiIsMl0sWzAsMywiXFxnYW1tYV9pOiBVX2lcXHRvIEciLDAseyJjdXJ2ZSI6NX1dXQ==)


And then assemble a commuting diagram:

\begin{tikzcd}
	{(U_i \intersect U_j)\times F} && {\pi_1\inv(U_i \intersect U_j)} && {\pi_2\inv(U_i \intersect U_j)} && {(U_i \intersect U_j)\times F} \\
	\\
	{(U_i \intersect U_j)\times F} && {\pi_1\inv(U_i \intersect U_j)} && {\pi_2\inv(U_i \intersect U_j)} && {\pi_2\inv(U_i \intersect U_j)}
	\arrow["{\phi_i}", from=1-3, to=1-1]
	\arrow["{\psi_j}", from=1-5, to=1-7]
	\arrow["f", from=1-3, to=1-5]
	\arrow["{\gamma_i}", curve={height=-30pt}, no head, from=1-1, to=1-7]
	\arrow["{\psi_{ji}}", from=1-7, to=3-7]
	\arrow["{\psi_j}"', from=3-5, to=3-7]
	\arrow[Rightarrow, no head, from=1-5, to=3-5]
	\arrow["f"', from=3-3, to=3-5]
	\arrow[Rightarrow, no head, from=1-3, to=3-3]
	\arrow["{\phi_i}"', from=3-3, to=3-1]
	\arrow["{\phi_{ji}}"', from=1-1, to=3-1]
	\arrow["{\gamma_j}"', curve={height=30pt}, from=3-1, to=3-7]
\end{tikzcd}

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


:::

## Nonabelian Čech Cohomology

:::{.definition title="Čech complex"}
Let $\mcu \da \ts{U_i}_{i\in I}\covers B$ an open cover, and define
\[
\Cc^0(\mcu; G) \da \ts{ \ts{\gamma_i: U_i\to G}_{i\in I} }
,\]
which is a group under pointwise multiplication.
Define
\[
\Cc^2(\mcu; G) &\da \ts{ \ts{\phi_{ij}: U_i \intersect U_j \to G}_{i, j\in I} }\\
\Cc^3(\mcu; G) &\da \ts{ \ts{\phi_{ijk}: U_i \intersect U_j \intersect U_k \to G}_{i,j,k \in I} }
,\]
and boundary maps
\[
\delta^{0}: \Cc^0(\mcu; G) &\to \Cc^1(\mcu; G)\\
\ts{\gamma_i: U_i\to G} &\mapsto \ts{\phi_{ji} \da \gamma_j\gamma_i\inv: U_i \intersect U_j \to G }
,\]

\[
\delta^{1}: \Cc^1(\mcu; G) &\to \Cc^2(\mcu; G)\\
\ts{\phi_{ij}: U_i \intersect U_j \to G} &\mapsto \ts{\eta{ijk} \da \phi_{ij} \phi_{jk} \phi_{ik}\inv: U_i \intersect U_j \intersect U_k \to G }
.\]
:::

:::{.remark}
One can check that $\delta^1 \circ \delta^0 = 0$ is trivial.
And 1-cocycle will yield a fiber bundle.
:::

:::{.lemma title="1"}
A bundle is trivial iff it is a 1-coboundary, where we take $Z^1(\mcu; G) \da \ker \delta^1$, $B^1(\mcu; G) \da \im \delta^0$.
:::

:::{.warnings}
We'd like to define homology as $Z/B$, but since these aren't abelian groups, the coboundaries $B$ may not be normal in $Z$ and the quotient may not yield a group.
:::

:::{.definition title="First Čech cohomology"}
There is an action of $\Cc^0(\mcu; G)\actson \Cc^1(\mcu; G)$ given by taking $\gamma \da \ts{\gamma_i}_{i\in I}$ and setting $(\gamma \phi)_{ij} = \gamma_i \phi_{ij} \gamma_j\inv$, which descends to an action on $Z^1$.
We can take the quotient by this action to define
\[
\Hc^1(\mcu; G) \da Z^1(\mcu; G) / \sim
.\]
:::

:::{.lemma title="2"}
Two bundles are isomorphic iff they yield the same element in $\Hc^1(\mcu; G)$.
:::

:::{.remark}
This works when bundles have the same open cover, and if not, we can take a common refinement.
:::