# Tuesday, September 21

:::{.remark}
Today: a short discussion on generalizations of $\B G$ to topological groups.
:::

:::{.definition title="Topological categories"}
A **topological category** is a category where the objects are topological spaces and morphisms form topological spaces in a coherent way, i.e. the following maps should be continuous:

- $\mathrm{source}, \mathrm{target}: \Mor_{\cat{C}} \to \Ob(\cat C)$, 
- $\id: \Ob(\cat C)\to \Mor_{\cat C}$,
- Composition: $\cat{C}(x, y) \times \cat{C}(y,z) \to \cat{C}(x, z)$. 

I.e. it is a category enriched over topological spaces (plus conditions).
:::

:::{.example title="?"}
If $G\in \Top\Grp$, then $\mcb G$ is a topological category since the morphism space $\Mor(\pt, \pt) = G$ has a topology.
Similarly $\mce G$ is a topological category.
:::

:::{.remark}
We can take nerves of topological categories; this just requires tracking the additional enrichment (i.e. the various topologies).
The same proof will yield a principal $G\dash$bundle $\nerve{\mce G} \mapsvia{\pi} \nerve{\mcb G}$, noting that $G$ again acts on $\nerve{\mce G}$.
:::

:::{.definition title="Absolute Neighborhood Retract"}
A space is called an **absolute neighborhood retract** (ANR) if for any $X\embeds Y$ (as a closed subspace) into a metric space, $X$ is a retract of a neighborhood in $Y$.
:::

:::{.example title="?"}
Every CW complex is an ANR.
This is also true if every point of $X$ has a contractible neighborhood.
:::

:::{.lemma title="?"}
If $G$ is ANR, then $EG = \nerve{\mce G}\to \nerve{\mcb G} = \B G \in \Prin\Bung$. 
:::

:::{.proof title="?"}
Note that $\B G$ is a $\Delta\dash$complex, so we'll try to build bundle charts by inducting over the skeleta.
Each graded piece of the complex is of the form $\Delta^i \times G\cartpower{i}$, so pick an interior point $((x_0, \cdots, x_i), (g_1,\cdots, g_i))$ so $x_i\neq 0$ for every $i$.
Define a map
\[
\Delta^i \times G\cartpower{i} \times G &\to E G \\
( (x_0,\cdots, x_i), (g_1, \cdots, g_i), g) &\mapsto (\id(\cdots), (g, gg_1, gg_1g_2, \cdots, gg_1\cdots g_i))
,\]
which corresponds to the sequence of composable morphisms
\[
(g \mapsvia{g_1} gg_1 \mapsvia{g_2} g g_1 g_2 \to \cdots \to gg_1\cdots g_i) 
.\]

:::{.exercise title="?"}
Show that this is not compatible with the gluing!
:::

Write $p: \Delta^i \times G\cartpower{i} \to \BG$ for the quotient attaching map, so we can write the $m\dash$skeleton as $\BG^{(m)} = \Union_{i\leq m} p(\Delta^i \times G\cartpower{i})$.
Now suppose $(U_m, \phi_m)$ is a chart for $\ro{\EG}{\BG^{(m)}} \to\BG^{(m)}$, we want to extend this to a chart or $\BG^{(m+1)}$.
We have a retraction $r: U_{m+1}\to U_m$ where $U_{m+1} \subseteq \BG^{(m+1)}$ is an open inclusion.
We construct a trivialization of $\pi\inv(U_{m+1}) \to U_{m+1}$:


\begin{tikzcd}
	{\pi\inv(p\inv(U_{m+1}))} && {\pi\inv(p\inv(U_m))} && {\pi\inv(U_m)} && {U_m\times G} \\
	\\
	{p\inv(U_{m+1})} && {p\inv(U_m)} && {U_m} \\
	{p\inv(U_{m+1})} &&&& {U_{m+1}}
	\arrow["{\phi_m}", from=1-5, to=1-7]
	\arrow["\pi"', from=1-5, to=3-5]
	\arrow["p", from=3-3, to=3-5]
	\arrow[from=1-3, to=1-5]
	\arrow[from=1-3, to=3-3]
	\arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=1-3, to=3-5]
	\arrow[hook, from=4-5, to=3-5]
	\arrow[hook, from=3-1, to=3-3]
	\arrow["r"', curve={height=18pt}, from=3-3, to=3-1]
	\arrow[from=1-1, to=3-1]
	\arrow["{\exists\tilde r}", dashed, from=1-1, to=1-3]
	\arrow["p", from=4-1, to=4-5]
	\arrow[Rightarrow, no head, from=3-1, to=4-1]
	\arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=1-1, to=3-3]
	\arrow["{\phi_{m+1}}", color={rgb,255:red,92;green,92;blue,214}, curve={height=-30pt}, from=1-1, to=1-7]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsOSxbNCwwLCJcXHBpXFxpbnYoVV9tKSJdLFs2LDAsIlVfbVxcdGltZXMgRyJdLFs0LDIsIlVfbSJdLFsyLDIsInBcXGludihVX20pIl0sWzIsMCwiXFxwaVxcaW52KHBcXGludihVX20pKSJdLFs0LDMsIlVfe20rMX0iXSxbMCwyLCJwXFxpbnYoVV97bSsxfSkiXSxbMCwwLCJcXHBpXFxpbnYocFxcaW52KFVfe20rMX0pKSJdLFswLDMsInBcXGludihVX3ttKzF9KSJdLFswLDEsIlxccGhpX20iXSxbMCwyLCJcXHBpIiwyXSxbMywyLCJwIl0sWzQsMF0sWzQsM10sWzQsMiwiIiwxLHsic3R5bGUiOnsibmFtZSI6ImNvcm5lciJ9fV0sWzUsMiwiIiwxLHsic3R5bGUiOnsidGFpbCI6eyJuYW1lIjoiaG9vayIsInNpZGUiOiJ0b3AifX19XSxbNiwzLCIiLDEseyJzdHlsZSI6eyJ0YWlsIjp7Im5hbWUiOiJob29rIiwic2lkZSI6InRvcCJ9fX1dLFszLDYsInIiLDIseyJjdXJ2ZSI6M31dLFs3LDZdLFs3LDQsIlxcZXhpc3RzXFx0aWxkZSByIiwwLHsic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiZGFzaGVkIn19fV0sWzgsNSwicCJdLFs2LDgsIiIsMCx7ImxldmVsIjoyLCJzdHlsZSI6eyJoZWFkIjp7Im5hbWUiOiJub25lIn19fV0sWzcsMywiIiwwLHsic3R5bGUiOnsibmFtZSI6ImNvcm5lciJ9fV0sWzcsMSwiXFxwaGlfe20rMX0iLDAseyJjdXJ2ZSI6LTUsImNvbG91ciI6WzI0MCw2MCw2MF19LFsyNDAsNjAsNjAsMV1dXQ==)

This extends the chart to $\BG^{(m+1)}$, noting that $p$ is a quotient map and thus preserves open sets.
:::

:::{.remark}
We can't necessarily extend over the entire $m+1$ skeleton! 
But here extending it over a retractable neighborhood was enough, so we needed $G$ to be an ANR in order for $\BG$ to be an ANR.
Why: consider 
\[
p\inv(U_m) \subseteq \Union_{i\leq m} \Delta^i \times G\cartpower{i} \subseteq \Union_{i\leq m-1} \Delta^i \times G\cartpower{i}
.\]
If $G$ is an ANR, use that $\Delta^i$ is an ANR and so their product will be, then pick a neighborhood and apply $p$ to get the required open.
:::

## Building $\BO_n$ and $\EO_n$

:::{.remark}
We'll assume all spaces paracompact from this point forward!
We have a correspondence
\[
\correspond{
  \text{$n\dash$dimensional CW complexes }
}
\mapstofrom
\correspond{
  \text{$n\dash$dimensional vector bundles }
  \\
  \text{with an $\Orth_n\dash$structure}
}
\mapstofrom
\Prin\Bun(\Orth_n)\slice X
\mapstofrom
[X, \B\Orth_n]
\]
Our next goal is to construct $\BO_n$ and $\EO_n$ as spaces.
Let $V_n(\RR^k) \da \ts{(\vector v_1, \cdots, \vector v_n) \text{orthonormal} }$.
Note that $\Orth_n\actson V_n(\RR^k)$ by
\[
\qty{\vector v_1, \cdots, \vector v_n} 
\cdot A = 
\qty{\sum_i a_{i, 1} \vector v_i,
\sum_i a_{i, 2} \vector v_i,
\cdots,
\sum_i a_{i, n} \vector v_i}
.\]

There is a projection

\begin{tikzcd}
	{F_{\vector v_1} = V_{n-1}(\RR^{k-1})} && {V_n(\RR^k)} & {(\vector v_1, \cdots, \vector v_1)} \\
	\\
	&& {V_1(\RR^k)} & {\vector v_1}
	\arrow[from=1-1, to=1-3]
	\arrow[from=1-3, to=3-3]
	\arrow[maps to, from=1-4, to=3-4]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNSxbMiwwLCJWX24oXFxSUl5rKSJdLFsyLDIsIlZfMShcXFJSXmspIl0sWzMsMCwiKFxcdmVjdG9yIHZfMSwgXFxjZG90cywgXFx2ZWN0b3Igdl8xKSJdLFszLDIsIlxcdmVjdG9yIHZfMSJdLFswLDAsIkZfe1xcdmVjdG9yIHZfMX0gPSBWX3tuLTF9KFxcUlJee2stMX0pIl0sWzQsMF0sWzAsMV0sWzIsMywiIiwwLHsic3R5bGUiOnsidGFpbCI6eyJuYW1lIjoibWFwcyB0byJ9fX1dXQ==)

We'll use the fact that $V_1(\RR^k)$ is $(k-2)\dash$connected, since it's homotopy equivalent to $S^{k-1}$. 

:::

:::{.lemma title="?"}
$V_n(\RR^{k})$ is $(k-n-1)\dash$connected.
:::

:::{.proof title="?"}
Induct on $n$ using the homotopy LES for the fiber bundle:


\begin{tikzcd}
	&& \cdots && {\pi_{i+1} V_{n-1} \RR^{k-1} \cong \pi_{i+1}(S^{k-1})} \\
	\\
	{\pi_i V_{n-1} \RR^{k-1} = 0} && \textcolor{rgb,255:red,92;green,92;blue,214}{\pi_i V_{n} \RR^{k}} && {\pi_i V_{1} \RR^{k} \cong \pi_iS^{k-2} = 0} \\
	{ \substack{(k-n-1)\dash\text{connected} \\ i\leq k-n-1} } && {\therefore\text{zero}} && {i\leq k-n-1 \implies i\leq k-3 }
	\arrow[from=1-5, to=3-1]
	\arrow[from=3-1, to=3-3]
	\arrow[from=3-3, to=3-5]
	\arrow[from=1-3, to=1-5]
\end{tikzcd}

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

:::

:::{.remark}
Using the inclusions $V_n(\RR^k) \injects V_n(\RR^{k+1})$, we can define $V_n(\RR^\infty) = \colim_k V_n(\RR^k) = \Union_{k\geq 0}V_n (\RR^k)$.
We equip it with the **weak topology**, i.e. $U \subseteq V_n(\RR^\infty)$ is open iff $U \intersect V_n(\RR^k)$ is open for all $k$.
:::

:::{.lemma title="?"}
\[
\pi_* V_n(\RR^\infty) = 0
.\]
:::

:::{.proof title="?"}
By compactness, any sphere $S^m$ maps to $V_n(\RR^k)$ for some large $k$, and using $V_n(\RR^k) \injects V_n(\RR^\ell)$ with $\ell-n-1 > m$ where $\pi_n V_n(\RR^\ell) = 0$ to make the map nullhomotopic.
:::

:::{.definition title="?"}
\[
V_n(\RR^\infty) / \Orth_n = \Gr_n(\RR^\infty)
.\]
:::

:::{.remark}
It will turn out that $\EO_n = V_n(\RR^\infty)$, sometimes referred to as the *Stiefel manifold* of $n\dash$frames in $\RR^\infty$.

:::