# Kirsten Wickelgren, Talk 1 (Wednesday, July 14)


## Intro

**Abstract**:

> Morel and Voevodsky's $\AA^1$ homotopy theory imports tools from algebraic topology into the study of schemes, or in other words, into the study of the solutions to polynomial equations. 
  This theory produces greater understanding of arithmetic and geometric aspects these solutions. 
  We will introduce some of this theory using as a guide questions such as "How many lines meet 4 lines in 3-space?"  


**References/Background**:

- Some algebraic topology or algebraic geometry, for example as described in Hatcher's and Hartshorne's books
- [Lecture Notes 1](https://www.ias.edu/sites/default/files/L1%20-%20Intro%20to%20A1-homotopy%20theory%20using%20enumerative%20examples%20-%20Jul%2013%2C%202021_1.pdf)
- [Lecture Notes 2](https://www.ias.edu/sites/default/files/L2%20-%20Intro%20to%20A1-homotopy%20using%20enumerative%20examples%20-%20Jul%2015%2C%202021.pdf)
- [Exercises](https://www.ias.edu/sites/default/files/Pauli-Exercises_PCMI.pdf)

:::{.remark}
Enumerative geometry counts algebro-geometric objects over $\CC$.
Example: how many lines meet 4 generic lines in $\PP^3$?
The answer is 2, and our goal is to record this kind of arithmetic information about geometric objects over a field $k$ whose intersections are fixed over $\bar k$ but not necessarily $k$ itself.
Our main tool will be $\AA^1\dash$homotopy theory, due to Morel-Voevodsky.
:::

## Classical Theory

:::{.remark}
First some classical homotopy theory.
The sphere can be defined as 
\[
S^n \da \ts{\elts{x}{n} \st \sum x_i^2 = 1} \homotopic \PP^n(\RR) / \PP^{n-1}(\RR)
,\]
and we have a degree map $[S^n, S^n] \to \ZZ$.
Given any $f\in \Top(S^n, S^n)$ and $p\in S^n$, we can write $f\inv(p) = \ts{\elts{q}{N}}$ and compute $\deg f = \sum_{i=1}^N \deg_{q_i} f$ in terms of **local degrees**.
Letting $V$ be a ball containing $p$, we have $F\inv (V) \contains U \ni q_i$ another ball such that $U \intersect f\inv(p) = q_i$.
Then $U/\bd U \homotopic S^n \homotopic V/\bd V$, so we can define a map
\[
\bar f: {U \over U\smts{q_i}} \to {V \over V\smts{p}}
\]
and define $\deg_{q_i} f \da \deg \bar{f}$.

Letting $\tv{\elts{x}{n}}$ be oriented coordinates about $q_i$ and $\tv{\elts{y}{n}}$ about $p$, then $f = \tv{\elts{f}{n}}: \RR^n\to \RR^n$ and we can consider $J_f \da \det \qty{\del f_i \over \del x_i}$.
There is then a formula
\[
\deg_{q_i}(f) = 
\begin{cases}
+1 & J_f(q_i) > 0 
\\
-1 & J_f(q_i) < 0.
\end{cases}
,\]
and for all $q_i$ we have $\deg f = \# f\inv(\pt)$, i.e. the number of solutions to the polynomial system $\ts{f_1 = f_2 = \cdots f_n = 0}$.
:::

:::{.example title="?"}
If $f\in \CC[x]$ of degree $n$, we can regard $f$ as a function $f: \PP^1(\CC)\to \PP^1(\CC)$ and 
by the fundamental theorem of algebra,
\[
\deg f = n = \# \ts{f=0}
.\]
:::

:::{.remark}
We can similarly count solutions to $f=0$ when $f$ is a section of a rank $n$ vector bundle 

\begin{tikzcd}
V
\ar[d, "p"] 
\\
X 
\ar[u, bend left, dotted, "f"]
\end{tikzcd}
This count can be computed using the Euler class:
\[
e(V) = e(V, f) = \sum_{q_i\in \ts{f=0}} \deg q_i f
.\]
:::

:::{.example title="?"}
Let $X \da \Gr(1, 3)_{/\CC}$, the Grassmannian parameterizing dimension 2 subspaces $W \subseteq \CC^4$, or equivalently lines in $\PP W \subseteq \PP(\CC^4) \cong \PP^3(\CC)$, where $\PP W$ is defined as $W\smts{0}$ where $\lambda w\sim w$ for any $\lambda \in \CC\units$.
The tautological is a rank 2 bundle:

\begin{tikzcd}
S_{[\PP W]} = W
  \ar[r] 
& 
S
  \ar[d] 
\\
& 
\Gr(1, 3)\slice{\CC}
\end{tikzcd}

Let $L_1, \cdots, L_4$ be four lines in $\PP^3$, then $\ts{\text{lines intersecting all } L_i} = \ts{f=0}$ where $f$ is a section (depending on the $L_i$) of  the bundle 

\begin{tikzcd}
\mce\da \qty{\Extalg^2 S^*}^{\oplus 4}\
\ar[d] 
\\
\Gr(1, 3)
\end{tikzcd}

and the Euler number of this bundle counts the number of such intersections.
In particular, $e(\mce)$ is independent of the choice of lines and section, provided they're sufficiently generic (so the $L_i$ do not pairwise intersect).
Using the splitting principle and knowledge of $H^*(\Gr)$, one can compute $e(\mce) = 2$.
:::

## Over arbitrary fields: Grothendieck-Witt

:::{.remark}
We'd like to do this over arbitrary fields $k$.
Lannes and Morel defined degrees for rational maps $f:\PP^1 \to \PP^1$.
Above we only remembered the sign of $J_f$, and here we'll allow remembering more: $\deg f$ will be valued in $\GW(k)$.
We can realize $\GW(k)$ as the group completion of the semiring of nondegenerate symmetric bilinear forms under $\perp, \tensor_k$, where we complete with respect to $\perp$.
It is related to the Witt group by
\[
W(K) \iso {\GW(k) \over \ZZ\gens{q_{ \hyp} } } \da {\GW(k) \over \ZZ\adjoin{\gens{1} + \gens{-1}}}
.\]
There is a **rank map**
\[
\rank: \GW(k) &\to \ZZ \\
q: (V\tensorpower{k}{2} \to K) &\mapsto \dim_k V
,\]
which can be realized by a pullback

\begin{tikzcd}
	{\GW(k)} && {W(k)} & {} \\
	\\
	\ZZ && {\ZZ/2}
	\arrow["\rank", from=1-3, to=3-3]
	\arrow["\rank", from=1-1, to=3-1]
	\arrow[from=3-1, 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=WzAsNSxbMCwwLCJcXEdXKGspIl0sWzMsMF0sWzIsMCwiVyhrKSJdLFswLDIsIlxcWloiXSxbMiwyLCJcXFpaLzIiXSxbMiw0LCJcXHJhbmsiXSxbMCwzLCJcXHJhbmsiXSxbMyw0XSxbMCwyXSxbMCw0LCIiLDEseyJzdHlsZSI6eyJuYW1lIjoiY29ybmVyIn19XV0=)

We can also write $\GW(k)$ in terms of generators $\gens{a}$ where $a\in k\units/ (k\units)^{\times 2}$, where $\gens{a}$ is associated to a bilinear form
\[
\gens{a}: k^{\times 2} &\to k \\
(x, y) &\mapsto axy
,\]
subject to relations

- $\gens{a}\gens{b} = \gens{ab}$
- $\gens{u} + \gens{v} = \gens{uv(u+v)} + \gens{u+v}$
- $\gens{u} + \gens{-u} = \gens{1} + \gens{-1} = q_\hyp$, which is the matrix
\[
q_\hyp = \matt{0}{1}{1}{0}
.\]

:::

:::{.example title="of known $\GW$ groups"}
The **signature** is the difference between the numbers of positive and negative ones in the associated matrix, and one can show
\[
\rank: \GW(\CC) &\iso \ZZ \\ 
(\rank, \sig): \GW(\RR) &\iso \ZZ^{\times 2} \\
(\rank, \disc): \GW(\FF_q) &\iso \ZZ \cross \FF_q\units/ (\FF_q\units)^{\times 2}
\]
where the last is a situation where we can compute étale cohomology.

:::

:::{.example title="Springer's theorem"}
Let $k\in \Field$ be complete and discretely valued with residue field $\kappa$, e.g. $k= \QQpadic$ or $\FF_p\laurent{t}$ with $\kappa = \FF_p$. 
Then if $\ch k \neq 2$,
\[
GW(k) \iso W(k)\sumpower{2}
.\]
:::

:::{.remark}
For $E\slice k$ a finite separable field extension, we'll have transfers
\[
\Tr_{E\slice k}: \GW(E) &\to \GW(k) \\
(V\tensorpower{k}{2} \mapsvia{\beta} E) &\mapsto (V\tensorpower{k}{2} \mapsvia{\beta} E \mapsvia{\Tr_{E_{/k}}} k)
,\]
which coincide with classical transfers for field extensions.
:::

:::{.remark}
For Lannes/Morel's formula, given $\PP^1\slice{k} \mapsvia{f} \PP^1\slice{k}$ and $p\in \PP^1\slice{k}$, we can write $f\inv(p) = \ts{\elts{q}{N}}$ and suppose $J(q_i) = f'(q_i) \neq 0$ for all $i$.
Then we remember the entire Jacobian and set
\[
\deg f \da \sum_{i=1}^N \Tr_{k(q_i)\slice k} \gens{J(q_i)}
,\]
which in fact doesn't depend on $p$.
Morel defines an $\AA^1\dash$degree
\[
\deg^{\AA^1}: [\PP^n/\PP^{n+1}, \PP^n/\PP^{n+1}]^{\AA^1} \to \GW(k)
,\]
where we are taking unstable $\AA^1\dash$homotopy classes of maps.
Noting that an element of $\GW(\RR)$ was determined by its rank and signature, we get a commutative diagram showing that $\deg^{\AA^1}$ is compatible with rank, signature, and the classical algebraic topological degree.
There are other ways of computing this degree besides taking the above sum: Cazanave, [Brazelton-McKean-Pauli](https://arxiv.org/abs/2103.16614) give formulas in terms of Bézoutians.
:::

## Homotopy

:::{.remark}
Recall that 
\[
X\smashprod Y \da { X\times Y \over (X\cross \pt) \union (\pt \cross Y)} && \in \Top_*
,\]
and $S^n \smashprod S^m \mapsvia{\sim} S^{n+m}$ and $(S^1)\smashpower{n} \mapsvia{\sim}  S^n$, so we define 
\[
\Suspend_{S^1} X \da S^1 \smashprod X
.\]
In $\AA^1$ homotopy theory we declare $\AA^1 \homotopic \pt$.
:::

:::{.example title="?"}
We can take a pushout of the following form:

\begin{tikzcd}
	& z && z \\
	z & {\GG_m} && {\AA^1\homotopic \pt} \\
	\\
	{1/z} & {\pt \homotopic \AA^1} && { \therefore \PP^1 \homotopic \Suspend_{S^1} \GG_m}
	\arrow[maps to, from=2-1, to=4-1]
	\arrow[from=2-2, to=4-2]
	\arrow[from=2-2, to=2-4]
	\arrow[maps to, from=1-2, to=1-4]
	\arrow[from=4-2, to=4-4]
	\arrow[from=2-4, to=4-4]
	\arrow["\lrcorner"{anchor=center, pos=0.125, rotate=180}, draw=none, from=4-4, to=2-2]
\end{tikzcd}


> [Link to Diagram](https://q.uiver.app/?q=WzAsOCxbMSwxLCJcXEdHX20iXSxbMSwzLCJcXHB0IFxcaG9tb3RvcGljIFxcQUFeMSJdLFswLDEsInoiXSxbMCwzLCIxL3oiXSxbMywxLCJcXEFBXjFcXGhvbW90b3BpYyBcXHB0Il0sWzEsMCwieiJdLFszLDAsInoiXSxbMywzLCJcXFBQXjEgXFxob21vdG9waWMgXFxTdXNwZW5kX3tTXjF9IFxcR0dfbSJdLFsyLDMsIiIsMCx7InN0eWxlIjp7InRhaWwiOnsibmFtZSI6Im1hcHMgdG8ifX19XSxbMCwxXSxbMCw0XSxbNSw2LCIiLDIseyJzdHlsZSI6eyJ0YWlsIjp7Im5hbWUiOiJtYXBzIHRvIn19fV0sWzEsN10sWzQsN10sWzcsMCwiIiwxLHsic3R5bGUiOnsibmFtZSI6ImNvcm5lciJ9fV1d)

Here the formalism of homotopy pushouts allows us to conclude that in an appropriate $\AA^1\dash$homotopy category, 
\[
\Suspend_{S^1} \GG_m \da S^1\smashprod \GG_m \homotopic \PP^1
.\]

:::

:::{.remark title="on motivic spheres"}
We have 
\[
\GG_m \da \spec k[z, 1/z] = \AA^1\smts{\pt}
.\]
By taking pushouts inductively we can realize 
\[
\AA^n\smts{\pt} \homotopic \Suspend_{S^1} (\AA^1\smts{\pt}) \smashprod (\AA^n\smts{\pt}) \homotopic (S^1)\smashpower{n-1} \smashprod (\GG_m)\smashpower{n}
.\]
:::

:::{.remark}
We can use this to write
\[
\PP^n/\PP^{n-1} 
&\homotopic {\PP^n \over \PP^n\smts{\pt} }\\
&\homotopic {\AA^n \over \AA^n\smts{\pt} }\\
&\homotopic {\pt \over \AA^n\smts{\pt} }\\
&\homotopic \Suspend_{S^1} \qty{ \AA^n\smts{\pt} } \\
&\homotopic (S^1)\smashpower{n} \smashprod (\GG_m)\smashpower{n}
.\]
:::

:::{.remark}
Stable homotopy shows that inverting $\Suspend$ is useful, which we also do in the $\AA^1\dash$setting by inverting $\Suspend_{\PP^1} (\wait) \da \PP^1 \smashprod (\wait)$ to obtain a stable homotopy category $\SH(k)$.
:::

:::{.theorem title="Morel, Hopkins-Morel"}
For $k\in \Field$, stably we have
\[
[S^0, S^0] \mapsvia{\sim} [\PP^n/\PP^{n-1}, \PP^n/\PP^{n-1}] \mapsvia{\sim} \GW(k)
.\]
Moreover, there is a ring structure on homotopy classes which yields an isomorphism of rings into **Milnor-Witt $K\dash$theory**,
\[
\bigoplus_{n\in \ZZ} [ S^0, \GG_m\smashpower{n}] \mapsvia{\sim} \KMW_*(k)
.\]

:::

:::{.remark}
$\KMW_*$ is a graded associative algebra with generators $[u] \in \KMW_1(k)$ for $u\in k\units$ and $\eta \in \KMW_{-1}(k)$, with relations

- $[u][1-u] = 0$, the *Steinberg relations*,
- $[ab] = [a] + [b] + \eta[a][b]$,
- $[a]\eta = \eta[a]$,
- $\eta q_\hyp = 0$ for $q_\hyp \da \eta[-1] + 2$

:::

:::{.remark}
There is an isomorphism
\[
\GW(k) &\mapsvia{\sim} \KMW_0(k) \\
\gens{a} &\mapstofrom 1 + \eta[a] \\
q_\hyp \da \gens{1} + \gens{-1} &\mapstofrom 1 + 1 + \eta[-1]
.\]
:::

:::{.remark title="on the proof"}
$[a]$ yields a map
\[
[a]: S^0 = (\spec k)^{\coprod 2} &\to \GG_m \\
p &\mapsto a
,\]
where $p$ is the non-basepoint,
and 
\[
\eta: \AA^2\smts{\pt} &\to \PP^1 \\
(x, y) &\mapsto [x: y]
.\]
On $\CC\dash$points, $\CC^2\smz \homotopic S^3$ maps to $\CP^1\homotopic S^2$ by the Hopf map, but on $\RR\dash$points we get $S^1 \mapsvia{\deg = -2} S^1$ implying that $\eta$ is not nilpotent, which is a new fact.
:::

:::{.remark}
We can define 
\[
X\wedgeprod Y = {X \cross Y \over \pt_X \sim \pt_Y}
\]
and get maps 
\[
X\wedgeprod Y \to X\cross Y\to X\smashprod Y
.\]
which yields
\[
\Suspend(X\cross Y) \mapsvia{\sim } \Suspend X \wedgeprod \Suspend Y \wedgeprod \Suspend(X\smashprod Y)
.\]

:::

:::{.lemma title="?"}
In $\SH(k)$,we get

\begin{tikzcd}
	{\GG_m\prodpower{2} } && {\GG_m} \\
	\\
	{ {\GG_m\smashpower{2}} \wedgeprod {\GG_m\smashpower{2}} } && {\GG_m}
	\arrow["\mult", from=1-1, to=1-3]
	\arrow["{(1, 1, \eta)}", from=3-1, to=3-3]
	\arrow["\simeq", from=1-1, to=3-1]
	\arrow[equal, from=1-3, to=3-3]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMCwwLCJcXEdHX21ee1xcdGltZXMgMn0iXSxbMCwyLCJcXEdHX21ee1xcd2VkZ2Vwcm9kIDJ9IFxcd2VkZ2Vwcm9kIFxcR0dfbV57XFxzbWFzaHByb2QgMn0iXSxbMiwyLCJcXEdHX20iXSxbMiwwLCJcXEdHX20iXSxbMCwzLCJcXG11bHQiXSxbMSwyLCIoMSwgMSwgXFxldGEpIl0sWzAsMSwiXFxzaW1lcSIsMV0sWzMsMiwiXFxzaW1lcSIsMV1d)

:::

:::{.lemma title="?"}
The map 
\[
f: \PP^1 &\to \PP^1 \\
z &\mapsto az
\]
is equal to $1 + \eta[a]$ in $\SH(k)$, since $f = \Suspend g$ where 
\[
g: \GG_m &\to \GG_m \\ 
z &\mapsto az
,\]
which is equal to 
\[
\Suspend(\GG_m \cross k \mapsvia{1\times a} \GG_m^{\times 2} \mapsvia{\mult} \GG_m)
.\]
:::

:::{.remark}
The lemma implies the relation $[ab] = [a] + [b] + \eta[a][b]$, and it turns out there's an isomorphism to motivic homotopy groups of spheres:
\[
\KMW_*(k) \mapsvia{\sim} \bigoplus _{n\in \ZZ} [S^0, \GG_m\smashpower{n}]
.\]
:::

## Big Problems

:::{.remark}
Notation: we'll write 
\[ 
\bigoplus_{n\in \ZZ} \pi_{n, n} \SS \da \bigoplus _{n\in \ZZ} [S^0, \GG_m\smashpower{n} ]
\] 
to be the zero line of the homotopy groups of spheres, and generally \( \bigoplus _n \pi_{n+r, n} \SS \) for the $r\dash$line.
Classical homotopy groups of spheres encode interesting geometric information, and we're finding that the corresponding motivic homotopy groups of spheres do as well.
[Röndigs-Spitzweck-Østvær](https://arxiv.org/abs/1604.00365) compute the 1-line for $\ch k\neq 2$ in a 2019 Annals paper, and we have some information about the 2-line.
:::

:::{.question}
What is $[\PP^n/\PP^{n-1}, \PP^n/\PP^{n-1}]$ for more general rings?
[Bachmann-Østvær (2021)](https://arxiv.org/abs/2102.01618) do this over $\ZZ\invert{2}$ and show \[
\pi_{0,0}\SS \tensor \ZZlocal{2} \mapsvia{\sim} \GW(\ZZ\invert{2})\tensor \ZZlocal{2}
.\]
:::

:::{.question}
What is $\pi_{*, *}\SS$ in general?
:::

:::{.question}
Is there a Freudenthal suspension theorem? 
I.e. which stable elements of $\pi_{*, *} \SS$ correspond to unstable groups?
:::

## Counting Things

:::{.remark}
Many people have used the $\AA^1\dash$Euler class for interesting things!
Let $X\in \smooth\Sch\slice{k}$ with $\dim X = d$ and let $V\to X \in \VectBun\slice{X}$ be a vector bundle.
:::

:::{.definition title="Orientation of bundles"}
A bundle $V\to X$ is **oriented** by the following data: $(L, \phi)$ where $L\to X$ is a line bundle and $\phi$ is a trivialization 
\[
\phi: \det V \iso L^{\tensor 2}
.\]
It is **relatively oriented** when $\Hom(\det TX, \det V)$ is oriented, where $\det (\wait) = \Extalg^{\text{top}}(\wait)$.
:::

:::{.example title="?"}
For $X = \PP^n$ or $\Gr(m, n)$ (parameterizing copies of $\PP^m$ in $\PP^n$), then 
\[
\omega_x = \det T^n X = \OO(-n-1)
,\]
and $X$ is orientable iff $n$ is odd.
For $\PP^1$, $\OO(n)$ is relatively orientable iff $n$ is even.
:::

## Euler Numbers

:::{.definition title="Euler Number in $\GW(k)$"}
Suppose $X\in \smsch\slice{k}$ is proper with $\dim X = d$ and consider a vector bundle
\begin{tikzcd}
\GL_d \actson F
  \ar[r] 
& 
V
  \ar[d] 
\\
& 
X 
\ar[u, "f", bend left, dotted]
\end{tikzcd}
Suppose 

  - $V$ is relatively oriented by $(L, \phi)$, and $f$ is a section with isolated zeros, so $\ts{f=0}$ consists of zeros of multiplicity 1, or equivalently
  - For all $x\in \ts{f = 0}$, the composite map
\[
Tf \da \qty{T_x X \to T_{f(x)} V \mapsvia{\sim} T_x X \oplus V_x \mapsvia{p_2} V_x} && f(x) \da (x, 0)
\]
  has nonvanishing determinant.

Then the **Euler number of $(V, \phi)$** with respect to $f$ is defined as 
\[
n(V, \phi, f) \da \sum_{x \in \ts{f = 0 } \subseteq X} \deg_x f
.\]
where $\deg_x f$ can be computed by 

- Choosing local Nisnevich coordinates on $X$ ,
- Choosing local trivializations of $V$ which are "compatible" with $\phi$,

Then locally writing
\[
f: \AA^d \to \AA^d \implies J_f \da \det \qty{\del f_i \over \del x_j}
,\]
one has
\[
\text{For } J_f(x) \neq 0 \in \kappa(x), \quad
\deg_x f \da \Tr_{\kappa(x) \slice k} \gens{J_f(x)}
.\]
:::

:::{.remark}
Equivalently, $T_x f\in \Hom(T_x X, V_x)$ and we can define 
\[
J_f(x) \da \det T_x f \in \Hom(\det T_x X, \det V_x) \iso L_x\tensorpower{k}{2}
,\]
where the orientation provides the isomorphism. 
Picking any basis for $L_x\tensorpower{k}{2}$ yields a number which is well-defined in $\kappa(x) / (\kappa(x)\units)^2$ by choosing a trivialization of $L_x$.
:::

:::{.question}
What happens if the zeros of $f$ have multiplicity $m_i > 1$?
In the classical setting, we didn't say what happens when $J_f(x) = 0$.
We'll answer this next time.
:::

:::{.question}
Why is the Euler number $n(V, f)$ independent of the section $f$?
Analogously, why is the number of intersections in the original problem 2, not depending on which specific lines were chosen?
:::

:::{.answer}
Sections with isolated zeros are often connected by 1-parameter $\AA^1\dash$families of such sections, and $\GW(k[x]) \mapsvia{\sim} \GW(k)$, although this is hard to show.

Alternatively, the Euler number is a pushforward of an Euler class taking values in interesting cohomology theories, so $n(V, f) = \pi_* e(V, f)$.
:::