# Universal Bundles (Thursday, September 16)

:::{.definition title="Universal $G\dash$bundles"}
A **universal $G\dash$bundle** is a principal $G\dash$bundle $EG \mapsvia{\pi} \B G$ such that $EG$ is weakly contractible, i.e. $\pi_*(EG) = 0$.
:::

:::{.remark}
We looked at a theorem stating the correspondence
\[
\Prin\Bung \slice X \mapstofrom [X, \B G]
.\]
:::

:::{.proof title="of surjectivity in theorem, continued"}
We showed surjectivity of the following map:
\[
[X, \B G] &\surjects \Prin\Bung\slice X \\
(f \in [X, \B G]) &\mapsto f^* EG
.\]

Given a principal $G\dash$bundle $P \mapsvia{\pi} B$, the mixing construction used an action $G\actson F$ to construct the fiber bundle $P\mix{G} F \mapsvia{\pi} B$.
Then the data of an equivariant map $f: P\to F$, so $f(pg) = f(p)\cdot g \da g\inv f(p)$ is equivalent to a section $s: B\to P\mix{G} F$.
Note that fixing the first coordinate $p$ in $[p, y]$ also fixes the second coordinate.
For $b\in B$, we can set $s(b) = [p, y] \sim [pg, g\inv y]$ (noting that these are equivalent in the mixed space), and we can define $f(p) = y$ to get an equivariant map since $f(pg) = g\inv y = g\inv f(p)$

So send $P \mapsvia{\pi} X \in \Prin\Bung\slice X$ to $P\mix{G} EG \to X$.
We proved that this has a section $s: X \to P\mix{G} EG$ and any two sections are homotopic, so from this we extract a $G\dash$equivariant map $f:P\to EG$.
Modding out the $G$ action yields $h: P/G\to EG/G$.
But $P/G\cong X$ and $EG/G\cong \B G$, so $h: X\to \B G$, and moreover $h^* EG \cong P$.
:::

:::{.proof title="of injectivity in theorem"}
Suppose $h_0, h_1\in [X, \B G]$, then $h_0^* EG \mapsvia{\phi, \cong} h_1^* EG$.
We can construct the following section $s_0$:

\begin{tikzcd}
	{(x, y)} && {y \da f_0([x, y])} \\
	{h_0^* EG} && EG &&& {h_0^* EG \mix{G} EG} & {[(x, y), y]} \\
	&&& \leadsto \\
	X && {\B G} &&& X & x
	\arrow["{f_0}", dashed, from=2-1, to=2-3]
	\arrow[from=2-3, to=4-3]
	\arrow["{h_0}"', from=4-1, to=4-3]
	\arrow[from=2-1, to=4-1]
	\arrow[from=2-6, to=4-6]
	\arrow[maps to, from=1-1, to=1-3]
	\arrow["{s_0}"', curve={height=30pt}, dashed, from=4-6, to=2-6]
	\arrow[maps to, from=4-7, to=2-7]
\end{tikzcd}

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


We can build another section $s_1$ in the following way:
use the isomorphism $\phi: h_0^* EG \to h_1^* EG$ to construct the composite

\begin{tikzcd}
	{(x, y)} && {(x, \phi(y))} && {\phi(y)} \\
	\\
	{h_0^* EG} && {h_1^* EG} && EG \\
	\\
	&& X && {\B G}
	\arrow["{f_1}", from=3-3, to=3-5]
	\arrow["{f_1\circ \phi}", curve={height=-30pt}, from=3-1, to=3-5]
	\arrow["\phi", from=3-1, to=3-3]
	\arrow[from=3-5, to=5-5]
	\arrow[from=3-3, to=5-3]
	\arrow[from=5-3, to=5-5]
	\arrow[from=3-1, to=5-3]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsOCxbMCwyLCJoXzBeKiBFRyJdLFsyLDIsImhfMV4qIEVHIl0sWzQsMiwiRUciXSxbMCwwLCIoeCwgeSkiXSxbMiwwLCIoeCwgXFxwaGkoeSkpIl0sWzQsMCwiXFxwaGkoeSkiXSxbMiw0LCJYIl0sWzQsNCwiXFxCIEciXSxbMSwyLCJmXzEiXSxbMCwyLCJmXzFcXGNpcmMgXFxwaGkiLDAseyJjdXJ2ZSI6LTV9XSxbMCwxLCJcXHBoaSJdLFsyLDddLFsxLDZdLFs2LDddLFswLDZdXQ==)

So we have
\[
s_0(x) &\da [x,y,y] \\
s_1(x) &\da [x,y,\phi(y)] 
.\]


By the lemma, $s_0\homotopic s_1$ through sections, so there is a homotopy
\[
s: I\times X &\to I \times h_0^* EG\mix{G} EG \\
(t, x) &\mapsto (t, x, y, z)
.\]
But this is a section of a principle $EG\dash$bundle over a CW complex, which yields a $G\dash$equivariant map 
\[
f: I \times h_0^* EG &\to EG \\
(t, x, y) &\mapsto z
.\]

Then

- At $t=0$, we have $(0, x)\mapsto (0, x,y,y)$, so $f(0, x, y) = y$,
- At $t=1$, we have $(1, x)\mapsto (1, x,y,\phi(y))$, so $f(1,x,y) = \phi(y)$.

Since $f$ is $G\dash$equivariant, we can quotient both sides by $G$ to get a map
\[
h: I\times X &\to \B G \\
(0, x) &\mapsto h_0(x) \\
(1, x) &\mapsto h_1(x) 
.\]
:::

:::{.exercise title="?"}
Try this proof yourself!
:::

:::{.remark}
The same proof shows the following:
:::

:::{.lemma title="?"}
If $F\to E \mapsvia{\pi} B$ is a fiber bundle and $B\in \CW$, if $\pi_{0\leq i\leq n}(F) = 0$ then we can inductively build sections over skeleta $B_{(k)}$ fir $k\leq n$ to construct a section over $B_{(n+1)}$.
Moreover, any two sections over the $n\dash$skeleton are homotopic.
:::

:::{.proposition title="?"}
If $P \mapsvia{\pi} B \in \Prin\Bung\slice X$ and $\pi_{0\leq i \leq n} P = 0$ (so $B$ is a "weak universal bundle") then $[X, B] \surjects \Prin\Bung\slice X$ for any $X\in \CW$ with $\dim(X) \leq n+1$, and it is bijective if $\dim X \leq n$.
Here the map is again $h \mapsto h^* P$.
:::

## Existence of Universal Bundles


:::{.theorem title="Milnor, 1966"}
For any group $G \in \Top\Grp$, there is a universal bundle $EG\to \B G$.
:::


:::{.remark}
Uniqueness up to homotopy: 

\begin{tikzcd}
	{h_*E'} & E && {E'} & {h'_*E} \\
	\\
	& B && {B'}
	\arrow["f", from=1-2, to=1-4]
	\arrow["h"', from=3-2, to=3-4]
	\arrow[from=1-2, to=3-2]
	\arrow[from=1-4, to=3-4]
	\arrow["{f'}"', curve={height=30pt}, from=1-4, to=1-2]
	\arrow["{h'}", curve={height=-30pt}, from=3-4, to=3-2]
	\arrow["\sim"', from=1-1, to=1-2]
	\arrow["\sim"', from=1-5, to=1-4]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNixbMSwwLCJFIl0sWzEsMiwiQiJdLFszLDAsIkUnIl0sWzMsMiwiQiciXSxbMCwwLCJoXypFJyJdLFs0LDAsImgnXypFIl0sWzAsMiwiZiJdLFsxLDMsImgiLDJdLFswLDFdLFsyLDNdLFsyLDAsImYnIiwyLHsiY3VydmUiOjV9XSxbMywxLCJoJyIsMCx7ImN1cnZlIjotNX1dLFs0LDAsIlxcc2ltIiwyXSxbNSwyLCJcXHNpbSIsMl1d)

Then since $(h'h\inv)^* E \cong E$, $h' h\inv \homotopic \id$ and $h (h')\inv \homotopic \id$, so we get a homotopy equivalence $B \homotopic B'$.
:::


:::{.exercise title="?"}
Show $E \homotopic E'$.
:::





:::{.remark}
We'll prove this theorem using Segal's construction.
For discrete groups $G$, the construction is covered in Hatcher 1B.
Hatcher constructs $K(G, 1)$, a space with $\pi_1 = G$ and contractible universal cover.
Then the universal cover $\hat X\to X$ is a principle $G\dash$bundle, and since $\hat X$ is contractible, it is universal:

\begin{tikzcd}
	&& {\tilde X\homotopic \pt} \\
	&&& {\in \Prin\Bung\slice X \text{ universal}} \\
	{K(G, 1)} && X
	\arrow["\pi", from=1-3, to=3-3]
	\arrow[from=3-1, to=3-3]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMCwyLCJLKEcsIDEpIl0sWzIsMiwiWCJdLFsyLDAsIlxcdGlsZGUgWFxcaG9tb3RvcGljIFxccHQiXSxbMywxLCJcXGluIFxcUHJpblxcQnVuZ1xcc2xpY2UgWCBcXHRleHR7IHVuaXZlcnNhbH0iXSxbMiwxLCJcXHBpIl0sWzAsMV1d)

:::


:::{.definition title="Nerve of a category"}
Given a category $\cat C$, the **nerve** $\nerve{\cat{C}}$ is the following $\Delta\dash$complex:

- $0\dash$simplices are objects of $\cat{C}$
- $n\dash$simplices for $n\geq 1$ are sequences of composable morphisms 
\[
x_0 \mapsvia{f_0} x_1 \mapsvia{f_1} \cdots \mapsvia{f_{n-1}} x_n
.\]
- Gluing data for 1-simplices: for $x_0 \mapsvia{f} x_1$, set $\bd_1(f) = x_1, \bd_0(f) = x_0$.
- Gluing data for $n\dash$simplices: the $i$th boundary maps are given by dropping vertex $i$:
\[
\bd_i(f_1, \cdots, f_n)
=
\begin{cases}
(f_2, f_3, \cdots, f_n) & i=0 \\
(f_1, f_2, \cdots, f_{i+1} \circ f_{i}, \cdots, f_n) & i=0 \\
(f_1, f_2, \cdots, f_{n-1}) & i=n.
\end{cases}
\]

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