# Thursday, October 28

:::{.remark}
The Euler class is *natural* in the following sense: for $E\to X$ with $\dim E = n$ and $X\in \CW$, we can write

\begin{tikzcd}
	{E\cong f^* \gamma_n} && {\gamma_n} \\
	\\
	X && {\BO_n}
	\arrow[from=1-1, to=3-1]
	\arrow["f", from=3-1, to=3-3]
	\arrow[from=1-3, to=3-3]
	\arrow[from=1-1, to=1-3]
	\arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=1-1, to=3-3]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMCwwLCJFXFxjb25nIGZeKiBcXGdhbW1hX24iXSxbMCwyLCJYIl0sWzIsMiwiXFxCT19uIl0sWzIsMCwiXFxnYW1tYV9uIl0sWzAsMV0sWzEsMiwiZiJdLFszLDJdLFswLDNdLFswLDIsIiIsMSx7InN0eWxlIjp7Im5hbWUiOiJjb3JuZXIifX1dXQ==)

Then naturality is the following equality:
\[
e(E) = e(f^* \gamma_n) = f^*(e(\gamma_n))
.\]

Recall the Thom isomorphism theorem: for an oriented $E\in \VectBundle^n$, we have a disk bundle $\DD E$ and sphere bundle $\SS E$, and
\[
H^j(\DD E; \SS E) \cong
\begin{cases}
0 &  j< n
\\
\ZZ & j=n
\\
H^{j-n}(\DD E) & j> n
\end{cases}
,\]
noting that we can take $H^n(\DD E, \SS E) \to H^n (\DD E_x, \SS E_x) \iso H^n(D^n, S^{n-1})$, where the target has a canonical positive generator.
The preimage of this generator is $u_E$, the Thom class.
The isomorphisms in the range $j>n$ are given by $\wait \cupprod u_E$.

We had a claim:
\[
H^n(\DD E, \SS E) &\to H^n(\DD E) \cong H^n(X) \\
u_E &\mapsto e(E)
.\]

:::

:::{.remark}
On the **Gysin sequence**: use the bundle $S^{n-1} \injects \SS E \to X$ and the LES
\[
\cdots \to H^{j-1}(\SS E) \mapsvia{\delta} H^{j-n}(X) \mapsvia{(\wait) \cupprod e(E)} H^j(X) \to H^j(\SS E) \to \cdots
.\]
The connecting map $\delta$ comes from the Thom isomorphisms $H^j(\DD E, \SS E)\iso H^{j-n}(\DD E)$ and splicing the LES of the pair $(\DD E, \SS E)$.
:::

:::{.proposition title="?"}
If $E$ is odd dimensional, then $2 e(E) = 0$.
Note that $e(E_1 \oplus E_2) - e(E_1) \cupprod e(E_2)$, and if $E$ has a nonvanishing section then $e(E) = 0$.
:::

:::{.remark}
So $e(E)$ is the obstruction to finding a nonvanishing section over the $n\dash$skeleton, where $\dim E = n$.
Given a nonvanishing section over $X\skel{k-1}$, consider extending it over $X\skel{k}$.
We can use the cellular attaching maps to write

\begin{tikzcd}
	{\Delta^k \times S^{n-1}} && {i^* \SS E} && {\SS E} \\
	\\
	&& {\Delta^k} && X
	\arrow["i", from=3-3, to=3-5]
	\arrow[from=1-5, to=3-5]
	\arrow[from=1-3, to=1-5]
	\arrow[from=1-3, to=3-3]
	\arrow[from=1-1, to=1-3]
	\arrow[from=1-1, to=3-3]
	\arrow["{s: \bd \Delta^k \to S^{n-1} \in \pi_{k}^{S^{n-1}}}", curve={height=-18pt}, dashed, from=3-3, to=1-1]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNSxbMiwyLCJcXERlbHRhXmsiXSxbNCwyLCJYIl0sWzQsMCwiXFxTUyBFIl0sWzIsMCwiaV4qIFxcU1MgRSJdLFswLDAsIlxcRGVsdGFeayBcXHRpbWVzIFNee24tMX0iXSxbMCwxLCJpIl0sWzIsMV0sWzMsMl0sWzMsMF0sWzQsM10sWzQsMF0sWzAsNCwiczogXFxiZCBcXERlbHRhXmsgXFx0byBTXntuLTF9IFxcaW4gXFxwaV97a31ee1Nee24tMX19IiwwLHsiY3VydmUiOi0zLCJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19XV0=)

- If $s \in \pi_k(S^n)$ and $k<n$, it is nullhomotopic and can thus be extended from $\bd \Delta^k$ to $\Delta^k$.
- If $k=n$, we get $s_*: H_{n-1}(\bd \Delta^n) \to H_{n-1}(S^{n-1})$ where $a\mapsto na$ with $n\da \deg(s_*)$.
  This yields an cellular $k\dash$cochain $C^k \to \ZZ$ where $\Delta^n \mapsto \deg s_*$, and this represents the Euler class.

:::

:::{.remark}
For smooth manifolds, we have natural bundles $\T X$ and $\nu X$ to work with.
Then
\[
\inner{e(\T M)}{ [M] } = \chi(M)
,\]
using that $e(\T M) \in H^n(M) \cong \ZZ$.
In general, so $E\to M^n$ with a generic section $s$, writing $Z\da s\inv(0) \leq E$ a submanifold of dimension $n-k$ when $\dim Z = k$, we have $[Z] \in H_{n-k}(M)$ and $\PD[Z]= e(E)$.
:::

:::{.example title="?"}
Consider $N^n \embeds M^{2n}$, then $\nu N\to N \in \VectBundle^n(N)$.
To compute $e(\nu N)$, pick a generic section:

\begin{tikzpicture}
\fontsize{45pt}{1em} 
\node (node_one) at (0,0) { \import{/home/zack/SparkleShare/github.com/Notes/Class_Notes/2021/Fall/CharacteristicClasses/sections/figures}{2021-10-28_13-35.pdf_tex} };
\end{tikzpicture}

Then $e(\nu N) = N\cdot N \in H_n(M)$ is the self-intersection number.
:::

:::{.remark}
For $N^n \subseteq M^m$, we have 


\begin{tikzcd}
	\textcolor{rgb,255:red,92;green,92;blue,214}{u_{\nu N}} & {H^{m-n}(\DD \nu N, \SS \nu N)} && {H^{m-n}(M, M\sm N)} \\
	\\
	\textcolor{rgb,255:red,92;green,92;blue,214}{e(u_{\nu N})} & {H^{m-n}(N) \cong H^{m-n}(\DD \nu N)} && {H^{m-n}(M)} & \textcolor{rgb,255:red,92;green,92;blue,214}{\PD[N]}
	\arrow[from=1-4, to=3-4]
	\arrow["\res", from=3-4, to=3-2]
	\arrow["{\text{excision}}"', from=1-4, to=1-2]
	\arrow[from=1-2, to=3-2]
	\arrow[color={rgb,255:red,92;green,92;blue,214}, curve={height=-6pt}, dashed, from=1-1, to=3-5]
	\arrow[color={rgb,255:red,92;green,92;blue,214}, curve={height=-30pt}, dashed, from=3-5, to=3-1]
\end{tikzcd}

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

So we can compute $e(E)$ for $\T M$ and $\nu N$.
:::

:::{.remark}
Next topics: Chern and Stiefel-Whitney classes.
:::

:::{.theorem title="?"}
For $E\to X$ a real vector bundle, there are characteristic classes $w_i(E) \in H^i(X; C_2)$ and $w \da 1 + w_1(E) + w_2(E) + \cdots \in H^*(E)$ , the **Stiefel-Whitney classes**, satisfying the following properties:

1. Naturality: $w_i(f^*(E)) = f^*(w_i(E))$
2. $w(E_1 \bigoplus E_2) = w(E_1 )\cupprod w(E_2)$
3. $w_i(E) = 0$ if $i>\dim_\RR E$.
4. If $E\to \RP^\infty$ is the canonical line bundle, then $w_1(E) = \alpha$ for $\gens{\alpha} = H^1(\RP^\infty; C_2)$. 

Moreover, these properties characterize $w_i(E)$ uniquely.
For complex vector bundles, there are $c_i(E) \in H^{2i}(X; \ZZ)$ and $c_(E)$, the **Chern classes**, which satisfy the same properties with $\CC$ instead of $\RR$ and $\gens{\alpha} = H^2(\CP^\infty; \ZZ)$.
:::


:::{.remark}
Next time: existence and uniqueness.
:::