# Vector Bundle Classification Theorem (Tuesday, September 28)

> See homework posted on the website!
> Turn in 2 total problem sets, one by mid-October.

:::{.theorem title="?"}
\[
[X, \BO_n] \cong [X, \Gr_n(\RR^\infty)] &\iso \VectBundlerk{n}(\RR, X) \\
f & \mapsto f^* \gamma_n
.\]
:::

:::{.remark}
We proved surjectivity last time for $X \in \CW$ compact, using compactness to embed any bundle into a trivial bundle.
We proved injectivity in the form of $f^* \gamma_n \cong g^* \gamma_n \implies f\homotopic g$, again for $X\in \CW$ compact.
So we need to handle the case of $X$ not compact.
:::

:::{.lemma title="?"}
Let $\pi: E\to X$ be a vector bundle over $X$.
Suppose for $B\in \CW$ compact with $B \subseteq X$, we have $f: B\to \Gr_n(\RR^\infty)$ such that $f^* \gamma_n \cong \ro{E}{B}$.
Suppose also that there exists a $g: X\to \Gr_n(\RR^\infty)$ with $g^* \gamma_n \cong E$.
Then there exists an $h: X\to\Gr_n(\RR^\infty)$ such that $\ro{h}{B} = f$ and $h^* \gamma_n \cong E$.

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:::

:::{.remark}
Idea: write $X = \colim_n X^{(n)}$ as a limit of compact finite skeleta, define maps $f_n: X^{(n)} \to \BO_n$ and $f_{n+1}: X^{(n+1)}\to \BO_n$, then modify $\tilde f_{n+1} \homotopic f_{n+1}$ to extend $f_n$ in such a way that $f_n^* \gamma_i = \ro{E}{B_n}$.
:::

:::{.proof title="?"}
For $g^* \gamma_n \cong E$ with $(\ro{g}{B})^* \gamma_n \cong \ro{E}{B}$ and $f^* \gamma_n \cong \ro{E}{B}$, then $f\homotopic \ro{g}{B}$ by injectivity for compact $B$.
We can then extend the homotopy $H: I\cross X\to \BO_n$ where $H_0 = g$ and $h\da H_1$ with $\ro{h}{B} = f$.
:::

## Characteristic Classes

:::{.definition title="Characteristic classes and representability"}
Let $F, G$ be two contravariant functors with source $\cat{C}$.
A **characteristic class of $F$ valued in $\cat{G}$** is a natural transformation $c: F\to G$.
$F$ is **representable** if there exists an object $\B F$ such that $F(X) = [X, \B F]$ for every $X\in \Ob(\cat C)$.

> Note: we aren't requiring the target categories to coincide!

:::

:::{.example title="?"}
\envlist

- $F(X) \da \Prin\Bun(\Orth_n)\slice X = \VectBundlerk{n} {}\slice X = [X \BO_n]$ is a contravariant functor $\ho\CW\op\to \Set$, where contravariance is due to pullbacks.
- $G(X) \da H^j_\sing(X)$, which is representable: for any $X\in \CW$ and any $G\in \Ab\Grp$, we have $H^j(X; M) = [X, K(G, j)]$.
  This comes from taking the sphere that generates $\pi_j K(G, j) = \gens{\alpha}$ and pulling any $f: X\to K(G, j)$ back to $f^* \alpha$.

:::

:::{.lemma title="?"}
If $F = [\wait, \B F]$ is representable, then characteristic classes of $F$ valued in a functor $G$ biject with $G(B)$ 
:::

:::{.remark}
We can write $F(B) = [B, B] \ni \id_B$, and it turns out that the characteristic class is determined by where $\id_B$ is sent:

\begin{tikzcd}
	&&& \textcolor{rgb,255:red,92;green,92;blue,214}{\id_B\in F} && \textcolor{rgb,255:red,92;green,92;blue,214}{c(\id_B)} \\
	X &&& {F(B) = [B, B]} && {G(B)} \\
	\\
	B &&& {F(X) = [X, B]} && {G(X)} \\
	&&& \textcolor{rgb,255:red,92;green,92;blue,214}{\psi} && \textcolor{rgb,255:red,92;green,92;blue,214}{c(\psi) = G(\psi)(c(\id_B))}
	\arrow["c", from=2-4, to=2-6]
	\arrow["{\psi \in [X, B]}", from=2-1, to=4-1]
	\arrow["c", from=4-4, to=4-6]
	\arrow["{F(\psi)}", from=2-4, to=4-4]
	\arrow["{G(\psi)}", from=2-6, to=4-6]
	\arrow[color={rgb,255:red,92;green,92;blue,214}, dashed, from=1-4, to=1-6]
	\arrow[color={rgb,255:red,92;green,92;blue,214}, curve={height=18pt}, dashed, from=1-4, to=5-4]
	\arrow[color={rgb,255:red,92;green,92;blue,214}, dashed, from=5-4, to=5-6]
	\arrow[color={rgb,255:red,92;green,92;blue,214}, curve={height=24pt}, dashed, from=1-6, to=5-6]
\end{tikzcd}

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

For us, taking $B\da \Gr_n(\RR^\infty)$ and $G(B) = H^j(\Gr_n(\RR^\infty)) \ni \alpha = c(\id_{B})$, so we can pullback to define $c(f) = f^*\alpha \in H^j(X)$.
:::

:::{.example title="?"}
Take $F(X) = \Prin\Bun(U_n)\slice X = [X \BU_n]$ and $G(X) = H^j(X)$, then $\alpha\in H^j(\BU_n)$ maps to to $c_\alpha(E) = f^*(\alpha)$ for any $f\in [X, \BU_n]$.
:::


:::{.example title="?"}
Take $F(X) = H^n(X; M)$ and $G(X) = H^m(X; N)$ with $M, N \in \Ab\Grp$.
Then $F(X) = [X, K(M, n)]$, and taking $G(K(M, n)) = H^m(K(M, n), N) \ni \alpha$ yields a map 
\[
H^n(X; M) &\to H^m(X; N) \\
(f: X\to K(M, n)) &\mapsto f^* \alpha
,\]
i.e. a cohomology operation.
If for example
\[
\phi \in \Hom_\Grp(M, N) 
= \Hom_\Grp( H_n(K(M, n); \ZZ), N)
= H^n( K(M, n); N)
,\]
using that $H_n(K(M, n); \ZZ)\cong M$.
This yields a change of coefficient morphism
\[
H^n(X; M) \to H^n(X; N)
,\]
which turns out to be the same map as above.
So any element in $H^m(K(M, n), N)$ yields a map $H^n(X, M)\to H^m(X, N)$ by sending $f:X\to K(M, n)$ to $f^*\alpha$.
Taking $n=m$ yields $H^n(X; M)\to H^n(X; N)$.
:::