# Tuesday, October 26

:::{.remark}
For $F\to E \mapsvia{\pi} B$ a fiber bundle with a $G\dash$structure, we have maps:

\begin{tikzcd}
	&& G \\
	\\
	{\phi_{ij}(U_i \intersect U_j)} && {\Homeo(F, F)}
	\arrow[from=1-3, to=3-3]
	\arrow[from=3-1, to=3-3]
	\arrow[from=3-1, to=1-3]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsMyxbMiwwLCJHIl0sWzIsMiwiXFxIb21lbyhGLCBGKSJdLFswLDIsIlxccGhpX3tpan0oVV9pIFxcaW50ZXJzZWN0IFVfaikiXSxbMCwxXSxbMiwxXSxbMiwwXV0=)

A vector bundle was an $\RR^n$ bundle with a $\GL_n\dash$structure, having a Riemannian metric meant having an $\Orth_n$ structure, and orientability meant having a $\GL^+_n$ structure.
A **principal** $G\dash$bundle was $P\to B$ where $P$ has a $G\dash$action acting freely and transitively on each fiber.
Taking the frame bundle sent vector bundles to principal $\GL_n\dash$bundles.

We had a construction sending fiber bundles to principal $G\dash$bundles, namely the **clutching** construction:
given $\mcu \covers X$ and $\phi_{ij}: U_i \intersect U_j \to G$ satisfying the cocycle condition $\phi_{ij} \phi_{jk} = \phi_{ik}$, we get a fiber bundle with fiber $F$ and transition functions for any $F\in\GSpaces{G}$.
:::

:::{.remark}
On universal bundles and classifying spaces: for $X\in \CW$, 
\[
\Prin\Bung(X) \iso [X, \B G] \\
f^* EG &\mapsfrom f
.\]

We noted

- $G$ discrete implies $\B G \homotopic K(G, 1)$
- $K(C_2, 1) = \RP^{\infty}$ with $EC_2 = S^{\infty}$
- $\B U_1 = \CP^{\infty}$ with $EU_1 = S^{\infty}$
- $\BO_n = \Gr_n(\RR^{\infty})$, with $\EO_n = V_n(\RR^{\infty})$ the Stiefel manifold of $n\dash$dimensional frames.
- $\B \SO_n$ are oriented $n\dash$planes in $\RR^\infty$.
- For $H\leq G$, $EH = EG$ and $\B H = EG/H$.

We had a canonical bundle $\gamma_n \to \Gr_n(\RR^\infty)$ whose fiber above $W\leq \RR^{\infty}$ was exactly $W$, so $\gamma_n = \ts{(W, w\in W) \subseteq \Gr_n(\RR^\infty) \times \RR^{\infty}}$.
Every vector bundle $E\to X$ is of the form $f^* \gamma_n\to X$ for some $f\in [X, \Gr_n(\RR^\infty)]$, and similarly $\Orth_n$ bundles are pullbacks of $V_n(\RR^\infty)\to \Gr_n(\RR^\infty)$.
:::

:::{.example title="?"}
A useful application: characteristic classes.
For any $c\in H^d(\BO_n)$ can be pulled back:
\[
H^d(X) &\to \Vect_n(X) \\
f^*c &\mapsfrom f\in [X, \BO_n]
.\]

Noting that $\BU_1 =\CP^\infty$ and $H^2(\CP^\infty) = \ZZ\gens{c_1}$, so any line bundle $L\to X$ $f:X\to \BU_1$ yields $c_1(L) \da f^* c_1\in H^2(X)$, the **first Chern class**.
Noting $H^2(X; \ZZ) \cong [X, K(\ZZ, 2)] \cong [X, \CP^\infty]$.
:::

:::{.remark}
Note: why $c_1=0$ in symplectic settings, related to Maslov index and ensures that the dimension of the relevant moduli space is zero.
:::