# Bordism (Tuesday, November 30)

:::{.definition title="Bordism"}
Let $M_1, M_2\in \smooth\Mfd^n$ be closed, then $M_1$ is **bordant** to $M_2$ iff $\exists W\in \smooth\Mfd^{n+1}$ with $\bd W = M_1 \disjoint M_2$.
:::

:::{.remark}
This defines an equivalence relation on $\smooth\Mfd^n$, and yields a ring $(\prod_n \mcn^n, \disjoint, \cross)$. 
We'll add extra structure to get refinements of this ring, for which we'll need a bit about stable normal bundles and the Whitney embedding theorem.
:::

:::{.theorem title="Whitney Embedding"}
Any $M\in \smooth\Mfd^n$ (possibly with boundary) embeds in $\RR^{2n+1}$, and any two such embeddings into $\RR^{2n+3}$ are isotopic.
:::

:::{.remark}
Moreover if $i, i': M\embeds \RR^n$ with $n\geq 2n+3$, then the normal bundles $\nu M, (\nu M)'$ are bundle isomorphic.
:::

:::{.corollary title="?"}
Every such $M$ has a stable normal bundle $\nu_M$ which is well-defined up to adding trivial line bundles.
:::

:::{.remark}
Note that adding trivial line bundles yields a tower $\BO_1 \to \BO_{2}\to \cdots$, where the classifying maps of $E$ and $E \oplus L$ are classified by maps to $\BO_k$ and $\BO_{k+1}$ respectively.
We can define lifts of extra structures by looking for commutative towers over $X_k$ for e.g. $X = \BO, \Spin$, etc:

\begin{tikzcd}
	& \cdots && {X_k} && {X_{k+1}} && \cdots \\
	\\
	& \cdots && {\BO_k} && {\BO_{k+1}} && \cdots \\
	M
	\arrow[from=1-2, to=1-4]
	\arrow[from=1-4, to=1-6]
	\arrow[from=1-6, to=1-8]
	\arrow[from=3-2, to=3-4]
	\arrow[from=3-4, to=3-6]
	\arrow[from=3-6, to=3-8]
	\arrow[from=1-4, to=3-4]
	\arrow[from=1-6, to=3-6]
	\arrow["{\nu_M}"{description}, curve={height=6pt}, from=4-1, to=3-4]
	\arrow[curve={height=24pt}, from=4-1, to=3-6]
	\arrow[curve={height=12pt}, dashed, from=4-1, to=1-4]
	\arrow[curve={height=12pt}, dashed, from=4-1, to=1-6]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsOSxbMSwwLCJcXGNkb3RzIl0sWzMsMCwiWF9rIl0sWzUsMCwiWF97aysxfSJdLFs3LDAsIlxcY2RvdHMiXSxbMywyLCJcXEJPX2siXSxbNSwyLCJcXEJPX3trKzF9Il0sWzcsMiwiXFxjZG90cyJdLFsxLDIsIlxcY2RvdHMiXSxbMCwzLCJNIl0sWzAsMV0sWzEsMl0sWzIsM10sWzcsNF0sWzQsNV0sWzUsNl0sWzEsNF0sWzIsNV0sWzgsNCwiXFxudV9NIiwxLHsiY3VydmUiOjF9XSxbOCw1LCIiLDEseyJjdXJ2ZSI6NH1dLFs4LDEsIiIsMSx7ImN1cnZlIjoyLCJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19XSxbOCwyLCIiLDEseyJjdXJ2ZSI6Miwic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiZGFzaGVkIn19fV1d)

:::

:::{.definition title="Structures"}
An $X\dash$structure on $M$ is a family of lifts for large $k$:

\begin{tikzcd}
	&& {X_k} \\
	\\
	M && {\BO_k}
	\arrow[from=1-3, to=3-3]
	\arrow["{\nu_M^k}"', from=3-1, to=3-3]
	\arrow[dashed, from=3-1, to=1-3]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsMyxbMiwwLCJYX2siXSxbMCwyLCJNIl0sWzIsMiwiXFxCT19rIl0sWzAsMl0sWzEsMiwiXFxudV9NXmsiLDJdLFsxLDAsIiIsMCx7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dXQ==)

We also require these to be compatible with the morphism $\BO_k\to \BO_{k+1}$ and $X_k\to X_{k+1}$.
:::

:::{.example title="?"}
Some examples:

- $\BSO_k$ for orientation,
- $V_k$ (the Stiefel manifold) for framings.

:::

:::{.definition title="$X\dash$bordism"}
Given $M_i$ with $X\dash$structures, we say $N$ is an **$X\dash$bordism** if

- $\bd N = M_1 \disjoint M_2$
- There is an $X\dash$structure on $N$ that restricts to the $X\dash$structures on $M_1$ and $M_2$. 

:::

:::{.remark}
For orientations, we get a bordism ring $\Omega_n$, and for framings we get $\Omega_n^{\Fr}$.
:::

:::{.theorem title="Identification of framed bordism group"}
\[
\Omega_n^{\Fr} \cong \pi_n^s
,\]
the stable homotopy groups of spheres $\colim_k \pi_{n+k} S^n$ (which stabilize for $k>n+1$).
:::

:::{.definition title="Stiefel-Whitney numbers"}
Given $M\in \smooth\Mfd^n$ and $I \da \ts{r_1,\cdots, r_n}$ such that $\sum_{1\leq k \leq n} kr_k = n$, consider $W_n \da \prod_{1\leq k \leq n} w_k(\T M)^{r_k} \in H^n(M; C_2)$.
We can evaluate this to get
\[
w_I(M) \da \inner{W_n}{[M]} \in C_2
.\]
Note that the condition on $I$ guarantees that $W_n\in H^n$.
:::

:::{.remark}
For $n=3$, the only possibilities for $I$ are 

- $(3,0,0) \leadsto W_n = w_1^3(\T M)$
- $(1, 1, 0 \leadsto W_n = w_1(\T M)w_2(\T M) )$
- $(0,0,1) \leadsto W_n = w_3(\T M)$

:::

:::{.definition title="Pontryagin number"}
Given an *oriented* $M \in \smooth\Mfd^n$, the **Pontryagin number** of $M$ is given by taking $I \da\ts{r_1,\cdots, r_{n\over 4}}$ such that $4r_1 + 8r_2 + \cdots + nr_{n\over 4} = n$, setting $P_n \da \prod_k p_k(\T M)^{r_k}$, and evaluating
\[
p_I(M) \da \inner{P_n}{[M]}
.\]
:::

:::{.corollary title="?"}
If any Pontryagin number is nonzero, then $M$ can not admit an orientation reversing diffeomorphism.
:::

:::{.proof title="?"}
Use that 
\[
P_I(-M) = \inner{\prod p_1 \TM }{[-M]} = -P_I(M)
,\]
but also
\[
P_I(M)
&= \inner{\prod p_i \T M}{[M]} \\
&= \inner{f^* \qty{ \prod p_i \T M} }{[M]} \\
&= \inner{\prod p_i \T M}{[f_* M]} \\
&= \inner{\prod p_i \T M}{[- M]} \\
&= P_I(-M)
.\]
:::

:::{.example title="?"}
Some examples:

- $P_I(\CP^{2n}) \neq 0$ by a computation.
- $P_I(\CP^{2n+1}) = 0$ since conjugating the complex structure $J\mapsto -J$ is an orientation reversing diffeomorphism.

:::

:::{.theorem title="Pontryagin, Thom"}
$M=0$ in $\mcn_n$ iff $w_I(M) = 0$ for all $I$.
:::

:::{.proof title="?"}
$\implies$: Easy, a simple algebraic topology calculation.

$\impliedby$:
Difficult and omitted!

:::

:::{.corollary title="?"}
There is an injective group morphism
\[
\mcn_n (\mcn_{I_1}, \cdots, \mcn_{I_k}) \to C_2\cartpower{k}
.\]
:::

:::{.theorem title="Pontryagin-Thom"}
If $M=0$ in $\Omega_n$, then $P_I(M) = 0$ for all $I$.
Conversely, if $P_I(M)=0$ for all $I$ then $M$ is torsion in $\Omega_n$, so there is some $k$ such that $M\disjointpower{k} = \bd W$ for some $W$.
:::

:::{.remark}
Milnor and Wall: the only torsion in $\Omega_n$ is order 2, and an oriented manifold $M$ is 0 in $\Omega_n \iff w_I(M), P_I(M) = 0$ for all $I$.
For a proof (for at least the first statement), see Milnor-Stasheff.
:::

:::{.remark}
We can kill torsion to get an injective map
\[
\Omega_n \tensor \QQ \mapsvia{(P_{I_1}, \cdots, P_{i_k} ) } \ZZ\cartpower{k}
.\]

:::

:::{.theorem title="Pontryagin-Thom"}
The ring $\mcn_n$ is a polynomial ring over $\FF_2$ with one generator in each dimension:
\[
\mcn_n \cong \FF_2[\ts{v_i \st i\neq 2^i-1}]
.\]
:::

:::{.theorem title="Pontryagin-Thom"}
The ring $\Omega_* \tensor \QQ$ is polynomial ring over $\QQ$ generated by $\CP^n$ for $n=1,2,\cdots$:
\[
\Omega_* \cong \cong \QQ[v_1,v_2,\cdots] && \abs{v_i} = i
.\]
:::