Assume that $u\neq v\in k^{\times}$. By proposition 118 this implies
$\pi_{\mathrm{N}}^{*}([X/u])\neq\pi_{\mathrm{N}}^{*}([X/v])$. By proposition
99 we have $\pi_{\mathrm{N}}^{*}([X/u])=[g_{u,1}]\oplus\pi$ and
$\pi_{\mathrm{N}}^{*}([X/v])=[g_{v,1}]\oplus\pi$. Since
$[\mathcal{J},\mathbb{P}^{1}]^{\mathrm{N}}$ is a group, this implies
$[g_{u,1}]\neq[g_{v,1}]$. ∎

######  Proposition 120.

For every $u\in k^{\times}$ we have

$\left(\deg^{\mathbb{A}^{1}}(\xi[\pi_{\mathrm{N}}^{*}(X/u)]),\mathrm{res}(\xi[\pi_{\mathrm{N}}^{*}(X/u)])\right)=\left(\langle
u\rangle,u\right)~{}\text{in}~{}\mathrm{GW}(k)\times_{k^{\times}/k^{\times
2}}k^{\times}.$

In particular, for $\pi_{\mathrm{N}}^{*}(X/1)=\pi$, we get

$\left(\deg^{\mathbb{A}^{1}}(\xi[\pi]),\mathrm{res}(\xi[\pi])\right)=\left(\langle
1\rangle,1\right)~{}\text{in}~{}\mathrm{GW}(k)\times_{k^{\times}/k^{\times
2}}k^{\times}.$

###### Proof.

By lemma 109 we know
$\xi[\pi_{\mathrm{N}}^{*}(X/u)]=\nu_{\mathbb{P}^{1}}([X/u])$. In [13, 3.4]
Cazanave shows that the image of $\nu_{\mathbb{P}^{1}}([X/u])$ in
$\mathrm{GW}(k)\times_{k^{\times}/k^{\times 2}}k^{\times}$ is $\left(\langle
u\rangle,u\right)$ by assigning it to the rank $1$ symmetric matrix $[u]$,
which has determinant $u$ and corresponds to the quadratic form $\langle
u\rangle$. ∎

In light of question 91 we would like to show that $\rho(\xi[\tilde{\pi}])$ is
the class $(-\langle 1\rangle,1)$ in
$\mathrm{GW}(k)\times_{k^{\times}/k^{\times 2}}k^{\times}$. We are not able to
confirm this yet, since we do not know how to compute the resultant of
$\xi[\tilde{\pi}]$. We can, however, make the following observation based on
computations of topological degrees in appendix C. We thank Kirsten Wickelgren
for mentioning to us the idea to use the arguments of [9] and [10] to reduce
the computation to the Grothendieck–Witt group of the integers.

######  Theorem 121.

For every field $k$, we have

$\displaystyle\deg^{\mathbb{A}^{1}}\left(\xi[\tilde{\pi}]\right)=-\langle
1\rangle~{}\text{in}~{}\mathrm{GW}(k).$

###### Proof.

First we assume $k=\mathbb{F}_{2}$. By [13, Lemma 3.13],
$[\mathbb{P}^{1},\mathbb{P}^{1}]^{\mathrm{N}}$ is generated by elements in
degree $1$, i.e., the class of $X/1$ generates
$[\mathbb{P}^{1},\mathbb{P}^{1}]^{\mathrm{N}}$. Since
$\nu_{\mathbb{P}^{1}}\colon[\mathbb{P}^{1},\mathbb{P}^{1}]^{\mathrm{N}}\to[\mathbb{P}^{1},\mathbb{P}^{1}]^{\mathbb{A}^{1}}$
is a group completion, this implies that
$\deg_{\mathbb{F}_{2}}^{\mathbb{A}^{1}}\colon[\mathbb{P}^{1},\mathbb{P}^{1}]^{\mathbb{A}^{1}}\xrightarrow{\cong}\mathrm{GW}(\mathbb{F}_{2})=\mathbb{Z}$
is an isomorphism, where we refer to [20] for the Grothendieck group
$\mathrm{GW}(\mathbb{F}_{2})$ of symmetric bilinear forms over
$\mathbb{F}_{2}$ and the isomorphism $\mathrm{GW}(\mathbb{F}_{2})=\mathbb{Z}$
(see [10, Lemma B.5], [20, III Remark (3.4)]). Since the right-hand side of
diagram (117) commutes, the fact that we have
$\deg\left([\tilde{\pi}]\right)=-1$ implies
$\deg^{\mathbb{A}^{1}}\left(\xi[\tilde{\pi}]\right)=-\langle 1\rangle$ in
$\mathrm{GW}(k)$.

Next we let $k$ be a field of characteristic $2$. Then the canonical morphism
$\operatorname{Spec}{k}\to\operatorname{Spec}{\mathbb{F}_{2}}$ induces a
commutative diagram of group homomorphisms

Since $\tilde{\pi}$ is defined over $\mathbb{F}_{2}$ and
$\deg_{\mathbb{F}_{2}}^{\mathbb{A}^{1}}(\xi[\tilde{\pi}])=-\langle 1\rangle$
by the first case, this implies
$\deg_{k}^{\mathbb{A}^{1}}(\xi[\tilde{\pi}])=-\langle 1\rangle$ in
$\mathrm{GW}(k)$.

Now we assume that $k$ is a field of characteristic $\neq 2$. The proof for
this case is also based on the fact that $\tilde{\pi}$ is already defined over
$\mathbb{Z}$ and not just $k$. To make the argument work, however, requires a
bit more effort. For a ring $A$, let $\mathcal{SH}(A)$ denote the stable
motivic homotopy category over $\operatorname{Spec}{A}$. Let
$KO_{k}\in\mathcal{SH}(k)$ denote the motivic spectrum over
$\operatorname{Spec}{k}$ which represents Hermitian K-theory. It is equipped
with a unit morphism $\varepsilon_{k}\colon\mathbbm{1}_{k}\to KO_{k}$ in
$\mathcal{SH}(k)$. Let
$\mathfrak{s}\colon[\mathbb{P}^{1},\mathbb{P}^{1}]^{\mathbb{A}^{1}}\to\mathbbm{1}_{k}^{0,0}(\operatorname{Spec}{k})$
denote the homomorphism defined by stabilisation and note that there is a
canonical isomorphism
$KO_{k}^{0,0}(\operatorname{Spec}{k})\cong\mathrm{GW}(k)$. We then define the
homomorphism $\delta$ as the following composition.

$\displaystyle\delta\colon[\mathbb{P}^{1},\mathbb{P}^{1}]^{\mathbb{A}^{1}}\xrightarrow{\mathfrak{s}}\mathbbm{1}_{k}^{0,0}(\operatorname{Spec}{k})\xrightarrow{\varepsilon_{k}}KO_{k}^{0,0}(\operatorname{Spec}{k})\cong\mathrm{GW}(k)$

We claim that the homomorphism $\delta$ can be identified with the motivic
Brouwer degree $\deg_{k}^{\mathbb{A}^{1}}$ over $k$. To prove the claim we
follow the argument of Levine and Raksit in [19, proof of Theorem 8.6, page
1845]. By Morel’s computation [22, Theorem 6.40] the isomorphism
$\mathrm{GW}(k)\cong\mathbbm{1}_{k}^{0,0}(\operatorname{Spec}{k})$ sends
$\langle u\rangle\in\mathrm{GW}(k)$, for $u\in k^{\times}$, to
$\mathfrak{s}(\nu_{\mathbb{P}^{1}}[X/u])$, the image of the class of
$X/u\colon\mathbb{P}^{1}_{k}\to\mathbb{P}^{1}_{k}$,
$[x_{0}:x_{1}]\mapsto[x_{0}:ux_{1}]$, in
$\mathbbm{1}_{k}^{0,0}(\operatorname{Spec}{k})$. Hence the classes
$\mathfrak{s}(\nu_{\mathbb{P}^{1}}[X/u])$ for all $u\in k^{\times}$ generate
$\mathbbm{1}_{k}^{0,0}(\operatorname{Spec}{k})$. Thus, in order to prove the
claim it suffices to show that $\delta(\nu_{\mathbb{P}^{1}}[X/u])=\langle
u\rangle$ in $\mathrm{GW}(k)$, since $\deg_{k}^{\mathbb{A}^{1}}([X/u])=\langle
u\rangle\in\mathrm{GW}(k)$. That is we need to show
$\varepsilon_{k}(\mathfrak{s}(\nu_{\mathbb{P}^{1}}[X/u]))=\langle u\rangle$.
This follows from [1, Corollary 6.2] which proves the claim.

In [9, §3.8.3] Bachmann and Hopkins construct a motivic spectrum
$KO^{\prime}_{\mathbb{Z}}\in\mathcal{SH}(\mathbb{Z})$ with a unit morphism
$\varepsilon^{\prime}_{\mathbb{Z}}\colon\mathbbm{1}_{\mathbb{Z}}\to
KO^{\prime}_{\mathbb{Z}}$, and write $KO^{\prime}_{k}$ and
$\varepsilon^{\prime}_{k}$ for the pullback of $KO^{\prime}_{\mathbb{Z}}$ and
$\varepsilon^{\prime}_{\mathbb{Z}}$ to $\mathcal{SH}(k)$ along the canonical
morphism $\operatorname{Spec}{k}\to\operatorname{Spec}{\mathbb{Z}}$. Since the
characteristic of $k$ is not $2$, there is an equivalence of ring spectra
$KO^{\prime}_{k}\simeq KO_{k}$ by [9, Lemma 3.38 (3)], which induces an
isomorphism $(KO^{\prime}_{k})^{0,0}(\operatorname{Spec}{k})\cong
KO^{0,0}_{k}(\operatorname{Spec}{k})$. Thus, there is an isomorphism
$(KO^{\prime}_{k})^{0,0}(\operatorname{Spec}{k})\cong
KO_{k}^{0,0}(\operatorname{Spec}{k})$ which fits into the following
commutative diagram.

By the above, we may therefore identify $\deg_{k}^{\mathbb{A}^{1}}$ over $k$
with the composed homomorphism

$\displaystyle[\mathbb{P}^{1},\mathbb{P}^{1}]^{\mathbb{A}^{1}}\xrightarrow{\mathfrak{s}}\mathbbm{1}_{k}^{0,0}(\operatorname{Spec}{k})\xrightarrow{\varepsilon^{\prime}_{k}}(KO^{\prime}_{k})^{0,0}(\operatorname{Spec}{k})\cong
KO_{k}^{0,0}(\operatorname{Spec}{k})\cong\mathrm{GW}(k).$

Furthermore, by [9, Lemma 3.38 (2)], there is an isomorphism
$\pi_{0,0}(KO^{\prime}_{\mathbb{Z}})\cong\mathrm{GW}(\mathbb{Z})$, where
$\mathrm{GW}(\mathbb{Z})$ denotes the Grothendieck–Witt group over
$\mathbb{Z}$ defined in [20, Chapter II]. Let
$[\mathbb{P}^{1},\mathbb{P}^{1}]^{\mathbb{A}^{1}}_{\mathbb{Z}}$ denote the set
of endomorphisms of $\mathbb{P}^{1}$ in the pointed unstable
$\mathbb{A}^{1}$-homotopy category over $\operatorname{Spec}{\mathbb{Z}}$. We
now define the homomorphism
$\deg^{\mathbb{A}^{1}}_{\mathbb{Z}}\colon[\mathbb{P}^{1},\mathbb{P}^{1}]^{\mathbb{A}^{1}}_{\mathbb{Z}}\to\mathrm{GW}(\mathbb{Z})$
as the composition

$\displaystyle\deg^{\mathbb{A}^{1}}_{\mathbb{Z}}\colon[\mathbb{P}^{1},\mathbb{P}^{1}]^{\mathbb{A}^{1}}_{\mathbb{Z}}\xrightarrow{\mathfrak{s}_{\mathbb{Z}}}\mathbbm{1}_{\mathbb{Z}}^{0,0}(\operatorname{Spec}{\mathbb{Z}})\xrightarrow{\varepsilon^{\prime}_{\mathbb{Z}}}(KO^{\prime}_{\mathbb{Z}})^{0,0}(\operatorname{Spec}{\mathbb{Z}})\cong\mathrm{GW}(\mathbb{Z}).$

The canonical homomorphism $\mathbb{Z}\to k$ then induces the following
commutative square

where $\mathfrak{b}_{k}\colon\mathrm{GW}(\mathbb{Z})\to\mathrm{GW}(k)$ denotes
the change of coefficients homomorphism. As a consequence we see that if
$[\alpha]\in[\mathbb{P}^{1},\mathbb{P}^{1}]^{\mathbb{A}^{1}}_{k}$ is in the
image of the homomorphism
$[\mathbb{P}^{1},\mathbb{P}^{1}]^{\mathbb{A}^{1}}_{\mathbb{Z}}\to[\mathbb{P}^{1},\mathbb{P}^{1}]^{\mathbb{A}^{1}}_{k}$,
then

(122)
$\displaystyle\deg^{\mathbb{A}^{1}}_{k}([\alpha])=\mathfrak{b}_{k}(\deg^{\mathbb{A}^{1}}_{\mathbb{Z}}([\alpha])).$

By [10, Lemma 5.6] (see also [20, Theorem II.4.3]), $\mathrm{GW}(\mathbb{Z})$
is generated over $\Z$ by the classes $\langle 1\rangle$ and
$\langle-1\rangle$. For a class $q\in\mathrm{GW}(\mathbb{Z})$, let
$q_{\mathbb{C}}$ and $q_{\mathbb{R}}$ denote the images of $q$ in
$\mathrm{GW}(\mathbb{C})$ and $\mathrm{GW}(\mathbb{R})$, respectively. Then
$q\in\mathrm{GW}(\mathbb{Z})$ is uniquely determined by the integers
$r(q):=\mathrm{rank}(q_{\mathbb{C}})$ and
$s(q):=\mathrm{sgn}(q_{\mathbb{R}})$, given by the rank and signature of
$q_{\mathbb{C}}$ and $q_{\mathbb{R}}$, respectively, via the formula

(123) $\displaystyle q=\frac{r(q)+s(q)}{2}\langle
1\rangle+\frac{r(q)-s(q)}{2}\langle-1\rangle\in\mathrm{GW}(\Z).$

Since $\mathcal{J}$ and both morphisms $\pi$ and $\tilde{\pi}$ are defined
over $\operatorname{Spec}{\mathbb{Z}}$, we can now apply the above
observations to prove the assertion of the proposition. Since $\pi$ is an
$\mathbb{A}^{1}$-weak equivalence over $\operatorname{Spec}{\mathbb{Z}}$ as
well, we can form the pointed $\mathbb{A}^{1}$-homotopy class
$\xi_{\mathbb{Z}}\left([\tilde{\pi}]\right):=[\tilde{\pi}\circ\pi^{-1}]^{\mathbb{A}^{1}}\in[\mathbb{P}^{1},\mathbb{P}^{1}]^{\mathbb{A}^{1}}_{\mathbb{Z}}$
defined by the zig-zag
$\mathbb{P}^{1}_{\mathbb{Z}}\xleftarrow{\pi}\mathcal{J}_{\mathbb{Z}}\xrightarrow{\tilde{\pi}}\mathbb{P}^{1}_{\mathbb{Z}}$.
The class $\xi_{\mathbb{Z}}\left([\tilde{\pi}]\right)$ is sent to
$\xi\left([\tilde{\pi}]\right)$ under base change. Thus, by the above
arguments, to determine
$\deg^{\mathbb{A}^{1}}_{k}\left(\xi[\tilde{\pi}]\right)$ in $\mathrm{GW}(k)$
it suffices to compute the rank and signature of
$\deg^{\mathbb{A}^{1}}_{\mathbb{Z}}\left(\xi_{\mathbb{Z}}[\tilde{\pi}]\right)$
after base change to $\mathbb{C}$ and $\mathbb{R}$, respectively. Since the
right-hand side of diagram (117) commutes, the fact that we have
$\deg\left([\tilde{\pi}]\right)=-1$ implies
$\mathrm{rank}(\deg_{\mathbb{C}}^{\mathbb{A}^{1}}\left(\xi[\tilde{\pi}]\right))=-1$.
In appendix C and example 168 we show that the signature of
$\deg^{\mathbb{A}^{1}}_{\mathbb{R}}\left(\xi[\tilde{\pi}]\right)$ over
$\mathbb{R}$ is $-1$. Thus, by formula (123), we get
$\deg^{\mathbb{A}^{1}}_{\mathbb{Z}}\left(\xi_{\mathbb{Z}}[\tilde{\pi}]\right)=-\langle
1\rangle$ in $\mathrm{GW}(\mathbb{Z})$. By equation (122) we can therefore
conclude that $\deg_{k}^{\mathbb{A}^{1}}\left(\xi[\tilde{\pi}]\right)=-\langle
1\rangle$ in $\mathrm{GW}(k)$. ∎

###  6.2 Group completion of naive homotopy classes

We will now describe the homomorphism
$\psi\colon\left([\mathbb{P}^{1},\mathbb{P}^{1}]^{\mathbb{A}^{1}},\oplus^{\mathbb{A}^{1}}\right)\to\left([\mathcal{J},\mathbb{P}^{1}]^{\mathrm{N}},\oplus\right)$
induced by the universal property of the group completion
$\nu_{\mathbb{P}^{1}}$ in more detail. By [13, Lemma 3.13],
$[\mathbb{P}^{1},\mathbb{P}^{1}]^{\mathrm{N}}$ is generated by elements in
degree $1$, i.e., it is generated by the set of classes $[X/u]$ for all $u\in
k^{\times}$. Hence, since $\nu_{\mathbb{P}^{1}}$ is a group completion, every
element in $[\mathbb{P}^{1},\mathbb{P}^{1}]^{\mathbb{A}^{1}}$ of degree $0$
can be written as a sum of the differences
$\gamma_{u,v}:=\nu_{\mathbb{P}^{1}}([X/u])-\nu_{\mathbb{P}^{1}}([X/v])$ for
suitable $u,v\in k^{\times}$. Thus the set of classes $\gamma_{u,v}$ for all
$u,v\in k^{\times}$ generates the subgroup
$[\mathbb{P}^{1},\mathbb{P}^{1}]_{0}^{\mathbb{A}^{1}}$ of degree $0$ elements.
Because of this we would like to understand the image of the $\gamma_{u,v}$
under $\psi$. Since $\psi$ is a group homomorphism, we know
$\psi(\gamma_{u,v})\oplus\psi(\nu_{\mathbb{P}^{1}}([X/v]))=\psi(\nu_{\mathbb{P}^{1}}([X/u]))$.
Since $\psi\circ\nu_{\mathbb{P}^{1}}=\pi_{\mathrm{N}}^{*}$, this implies
$\psi(\gamma_{u,v})\oplus\pi_{\mathrm{N}}^{*}([X/v])=\pi_{\mathrm{N}}^{*}([X/u])$.
By lemma 98, the map $g_{u,v}$ satisfies
$[g_{u,v}]\oplus\pi_{\mathrm{N}}^{*}([X/v]))=\pi_{\mathrm{N}}^{*}([X/u])$.
Hence, since $[\mathcal{J},\mathbb{P}^{1}]^{\mathrm{N}}$ is a group, we get

$\psi(\gamma_{u,v})=[g_{u,v}]~{}\text{in}~{}[\mathcal{J},\mathbb{P}^{1}]^{\mathrm{N}}.$

This motivates the following definition of the subgroup
$\mathbf{G}\subseteq[\mathcal{J},\mathbb{P}^{1}]^{\mathrm{N}}$.

######  Definition 124.

Let $\mathbf{G}_{0}:=\langle[g_{u,v}]\,|\,u,v\in
k^{\times}\rangle\subseteq[\mathcal{J},\mathbb{A}^{2}\setminus\\{0\\}]^{\mathrm{N}}$
denote the subgroup generated by the homotopy classes of the maps $g_{u,v}$.
Let $\mathbf{G}\subseteq[\mathcal{J},\mathbb{P}^{1}]^{\mathrm{N}}$ be the
subgroup generated by $\mathbf{G}_{0}$ and $[\pi]$.

We note that by lemma 94, we have $-[g_{u,v}]=[g_{v,u}]$, while $[g_{u,u}]$ is
the neutral element, and we therefore have $-([g_{u,v}]\oplus\pm
n[\pi])=[g_{v,u}]\oplus\mp n[\pi]$ in $\mathbf{G}$.

######  Remark 125.

Let $\mathbf{M}_{0}$ denote the submonoid of
$[\mathcal{J},\mathbb{A}^{2}\setminus\\{0\\}]^{\mathrm{N}}$ generated by the
set of homotopy classes $[g_{u,1}]$ for all $u\in k^{\times}$. By lemma 94, we
have $[g_{u,v}]\oplus[g_{v,1}]=[g_{u,1}]$ for all $u,v\in k^{\times}$. Thus,
every element in $\mathbf{G}_{0}$ is a sum of differences of elements in
$\mathbf{M}_{0}$. The implies that the inclusion
$\mathbf{M}_{0}\subset\mathbf{G}_{0}$ is a group completion.

######  Lemma 126.

The morphism of monoids
$\pi_{\mathrm{N}}^{*}\colon[\mathbb{P}^{1},\mathbb{P}^{1}]^{\mathrm{N}}\to[\mathcal{J},\mathbb{P}^{1}]^{\mathrm{N}}$
has image in $\mathbf{G}$.

###### Proof.

By [13, Lemma 3.13], $[\mathbb{P}^{1},\mathbb{P}^{1}]^{\mathrm{N}}$ is
generated by elements in degree $1$, i.e., it is generated by the set of
classes $[X/u]$ for all $u\in k^{\times}$. Hence it suffices to show that
$\pi_{\mathrm{N}}^{*}([X/u])$ is contained in $\mathbf{G}$. This follows from
lemma 97. ∎

######  Proposition 127.

The morphism of monoids
$\pi_{\mathrm{N}}^{*}\colon[\mathbb{P}^{1},\mathbb{P}^{1}]^{\mathrm{N}}\to\mathbf{G}$
is a group completion.

###### Proof.

Let $H$ be an abelian group and
$\mu\colon[\mathbb{P}^{1},\mathbb{P}^{1}]^{\mathrm{N}}\to H$ be a morphism of
monoids. We will show that there is a unique homomorphism of groups
$\widetilde{\mu}\colon\mathbf{G}\to H$ such that
$\widetilde{\mu}\circ\pi_{\mathrm{N}}^{*}=\mu$.

We set $\widetilde{\mu}([\pi]):=\mu([X/1])$. By lemma 98, we have
$[g_{u,v}]\oplus\pi_{\mathrm{N}}^{*}([X/v])=\pi_{\mathrm{N}}^{*}([X/u])$ in
$\mathbf{G}\subseteq[\mathcal{J},\mathbb{P}^{1}]^{\mathrm{N}}$. Hence
compatibility with $\mu$ forces the definition

$\displaystyle\widetilde{\mu}([g_{u,v}]):=\mu([X/u])-\mu([X/v]).$

By definition of $\mathbf{G}$ this induces a unique group homomorphism
$\widetilde{\mu}$, once we have shown that it is well-defined.

Now we show that $\widetilde{\mu}$ is well-defined. Because
$\mathbf{G}\cong\mathbf{G}_{0}\oplus\Z$, all relations in $\mathbf{G}$ amongst
the generators arise from relations of the classes $[g_{u,v}]$. Consider a
relation of the form

(128) $[g_{u_{1},v_{1}}]\oplus\ldots\oplus[g_{u_{s},v_{s}}]=0.$

We must then show that
$\sum_{i}\widetilde{\mu}\left([g_{u_{i},v_{i}}]\right)=0$ in $H$. Since
$\mathbf{G}$ is a group and by lemma 98, we have

$\sum_{i}g_{u_{i},v_{i}}\oplus\pi_{\mathrm{N}}^{*}([X/v_{i}])=\sum_{i}\pi_{\mathrm{N}}^{*}([X/u_{i}]).$

Since $\mathbf{G}$ is abelian, this implies

$\sum_{i}[g_{u_{i},v_{i}}]=\sum_{i}\pi_{\mathrm{N}}^{*}([X/u_{i}])-\pi_{\mathrm{N}}^{*}([X/v_{i}])=\sum_{i}\pi_{\mathrm{N}}^{*}([X/u_{i}])-\sum_{i}\pi_{\mathrm{N}}^{*}([X/v_{i}])=0.$

Hence

$\sum_{i}\pi_{\mathrm{N}}^{*}([X/u_{i}])=\sum_{i}\pi_{\mathrm{N}}^{*}([X/v_{i}])$

in $[\mathbb{P}^{1},\mathbb{P}^{1}]^{\mathrm{N}}$. Since
$\pi_{\mathrm{N}}^{*}$ is an injective monoid morphism, in
$[\mathbb{P}^{1},\mathbb{P}^{1}]^{\mathrm{N}}$ we have the equation

$\sum_{i}[X/u_{i}]=\sum_{i}[X/v_{i}].$

It thus follows that

$\mu\left(\sum_{i}[X/u_{i}]\right)=\mu\left(\sum_{i}[X/v_{i}]\right)~{}\text{in}~{}H.$

We calculate

$\widetilde{\mu}\left(\sum_{i}[g_{u_{i},v_{i}}]\right)=\sum_{i}\mu([X/u_{i}])-\mu([X/v_{i}])=0,$

as desired. This shows that $\widetilde{\mu}$ is well-defined.

It remains to show $\widetilde{\mu}\circ\pi_{\mathrm{N}}^{*}=\mu$. By [13,
Lemma 3.13], $[\mathbb{P}^{1},\mathbb{P}^{1}]^{\mathrm{N}}$ is generated by
the set of classes $[X/u]$ for all $u\in k^{\times}$. Hence $\mu$ is
completely determined by the images of $[X/u]$ for all $u\in k^{\times}$.
Thus, in order to show $\widetilde{\mu}\circ\pi_{\mathrm{N}}^{*}=\mu$, it
suffices to show
$\widetilde{\mu}\left(\pi_{\mathrm{N}}^{*}([X/u])\right)=\mu([X/u])$ for every
$u\in k^{\times}$. This is now immediate from the definition of
$\widetilde{\mu}$ and lemma 98:

$\displaystyle\widetilde{\mu}\left(\pi_{\mathrm{N}}^{*}([X/u])\right)$
$\displaystyle=\widetilde{\mu}\left([g_{u,1}]\oplus\pi_{\mathrm{N}}^{*}([X/1])\right)$
$\displaystyle=\widetilde{\mu}\left([g_{u,1}]\right)+\widetilde{\mu}\left([\pi]\right)$
$\displaystyle=\mu\left([X/u]\right)-\mu\left([X/1]\right)+\mu\left([X/1]\right)$
$\displaystyle=\mu\left([X/u]\right).$

This shows that
$\pi_{\mathrm{N}}^{*}\colon[\mathbb{P}^{1},\mathbb{P}^{1}]^{\mathrm{N}}\to\mathbf{G}$
has the universal property of a group completion and finishes the proof. ∎

######  Theorem 129.

There is a unique isomorphism of groups

$\chi\colon\mathbf{G}\to[\mathbb{P}^{1},\mathbb{P}^{1}]^{\mathbb{A}^{1}}$

such that $\chi\circ\pi_{\mathrm{N}}^{*}=\nu_{\mathbb{P}^{1}}$.

The homomorphism $\chi$ sends $[g_{u,v}]$ to the unique element $\gamma_{u,v}$
that satisfies
$\nu_{\mathbb{P}^{1}}^{*}([X/u])=\gamma_{u,v}\oplus^{\mathbb{A}^{1}}\nu_{\mathbb{P}^{1}}([X/v])$
and $[\pi]$ to $[\operatorname{id}]$. Moreover, $\chi$ and the homomorphism
$\psi\colon[\mathbb{P}^{1},\mathbb{P}^{1}]^{\mathbb{A}^{1}}\to\mathbf{G}\subseteq[\mathcal{J},\mathbb{P}^{1}]^{\mathrm{N}}$
are mutual inverses to each other.

###### Proof.

The existence of $\chi$ and its definition is a consequence of proposition 127
and its proof. The assertion that $\chi$ is the inverse of $\psi$ follows from
the fact that $\nu_{\mathbb{P}^{1}}$ is a group completion proven by Cazanave
in [13, Theorem 3.22] and the universal property of group completion. ∎

As a particular consequence of theorem 129 we get the following result.

######  Proposition 130.

The restriction $\chi_{0}$ of the homomorphism $\chi$ to $\mathbf{G}_{0}$
defines an isomorphism of groups

$\chi_{0}\colon\mathbf{G}_{0}\xrightarrow{\cong}[\mathbb{P}^{1},\mathbb{A}^{2}\setminus\\{0\\}]^{\mathbb{A}^{1}}.$

###### Proof.

The elements $\gamma_{u,v}$ are of degree $0$, and hence they lie in the
subgroup $[\mathbb{P}^{1},\mathbb{A}^{2}\setminus\\{0\\}]^{\mathbb{A}^{1}}$.
As a consequence, the homomorphism $\chi$ and its restriction $\chi_{0}$ fit
in the following commutative diagram of abelian groups.

Since the middle and right-most maps are isomorphisms, the assertion follows
from the five-lemma. ∎

The existence of the isomorphisms
$\mathbf{G}\xrightarrow{\chi}[\mathbb{P}^{1},\mathbb{P}^{1}]^{\mathbb{A}^{1}}\xleftarrow{\phi}[\mathcal{J},\mathbb{P}^{1}]^{\mathrm{N}}$
does not imply that $\mathbf{G}$ equals
$[\mathcal{J},\mathbb{P}^{1}]^{\mathrm{N}}$. However, we make the following
conjecture on the a prioiri subgroups $\mathbf{G}_{0}$ and $\mathbf{G}$.

######  Conjecture 131.

The inclusions
$\mathbf{G}_{0}\subseteq[\mathcal{J},\mathbb{A}^{2}\setminus\\{0\\}]^{\mathrm{N}}$
and $\mathbf{G}\subseteq[\mathcal{J},\mathbb{P}^{1}]^{\mathrm{N}}$ are
equalities.

We will show in theorem 138 in section 6.3 that conjecture 131 is true
whenever $k=\mathbb{F}_{q}$ is a finite field. This follows from an explicit
computation of $\mathbf{G}_{0}$ and $K_{1}^{MW}(\mathbb{F}_{q})$, the first
Milnor–Witt K-theory of $\mathbb{F}_{q}$.

######  Remark 132.

It follows from the structure of the group $\mathbf{G}$ as a product of
$\mathbf{G}_{0}$ and $\\{n[\pi]\,|\,n\in\mathbb{Z}\\}$ that in order to prove
conjecture 131 it suffices to show that the inclusion
$\mathbf{G}_{0}\subseteq[\mathcal{J},\mathbb{A}^{2}\setminus\\{0\\}]^{\mathrm{N}}$
is an equality, i.e., that the set of homotopy classes $[g_{u,v}]$ for all
$u,v\in k^{\times}$ generates the group
$[\mathcal{J},\mathbb{A}^{2}\setminus\\{0\\}]^{\mathrm{N}}$.

######  Remark 133.

If conjecture 131 is true, then the group homomorphism
$\psi\colon[\mathbb{P}^{1},\mathbb{P}^{1}]^{\mathbb{A}^{1}}\to[\mathcal{J},\mathbb{P}^{1}]^{\mathrm{N}}$,
induced by the fact that $\nu_{\mathbb{P}^{1}}$ is a group completion, is an
isomorphism and it agrees with $\phi^{-1}$, the inverse of the isomorphism of
theorem 84. We note, however, that this does not yet imply that the bijection
$\xi$ is a group homomorphism.

###  6.3 Milnor–Witt K-theory and morphisms in degree 0

Our final goal is to prove conjecture 131 for finite fields. For the proof we
use the Milnor–Witt K-theory of a field which we now recall from [22,
Definition 3.1].

######  Definition 134.

The Milnor–Witt K-theory of the field $k$, denoted $K_{*}^{MW}(k)$, is the
graded associative ring generated by symbols $[u]$ in degree $1$ for $u\in
k^{\times}$ and the symbol $\eta$ in degree $-1$ subject to the following
relations:

  1. 1.

For each $u\in k^{\times}\setminus\\{1\\}$, $[u].[1-u]=0$.

  2. 2.

For each pair $u,v\in(k^{\times})^{2}$, $[uv]=[u]+[v]+\eta.[u].[v]$.

  3. 3.

For each $u\in k^{\times}$, $\eta.[u]=[u].\eta$.

  4. 4.

Let $h:=\eta.[-1]+2$. Then $\eta.h=0$.

######  Remark 135.

It follows directly from the defining relations for $K_{*}^{MW}(k)$ that
$[1]=0$ and $\eta.[u^{2}]=0$ for each $u\in k^{\times}$. See [22, §3.1] for a
proof and other basic properties of Milnor–Witt K-theory.

Recall that $\mathbf{G}_{0}:=\langle[g_{u,v}]\,|\,u,v\in
k^{\times}\rangle\subseteq[\mathcal{J},\mathbb{A}^{2}\setminus\\{0\\}]^{\mathrm{N}}$
denotes the subgroup generated by the homotopy classes of the maps $g_{u,v}$.
In this subsection we write $\mathbf{G}_{0}(k)$ and $\mathbf{G}(k)$ for the
groups $\mathbf{G}_{0}$ and $\mathbf{G}$, respectively, to emphasize the
dependency of the base field $k$.

######  Proposition 136.

For every field $k$, there is an isomorphism $\mathbf{G}_{0}(k)\cong
K_{1}^{MW}(k)$.

###### Proof.

By proposition 130, we have an isomorphism
$\chi_{0}\colon\mathbf{G}_{0}(k)\xrightarrow{\cong}[\mathbb{P}^{1},\mathbb{A}^{2}\setminus\\{0\\}]^{\mathbb{A}^{1}}$.
As recalled in remark 64, the work of Morel in [22, §7.3] implies that there
is an isomorphism of groups
$[\mathbb{P}^{1},\mathbb{A}^{2}\setminus\\{0\\}]^{\mathbb{A}^{1}}\cong
K_{1}^{MW}(k)$. The composition of these isomorphisms yields the assertion. ∎

######  Proposition 137.

Let $k=\mathbb{F}_{q}$ be a finite field. Then $K_{1}^{MW}(\mathbb{F}_{q})$ is
a finite cyclic group of order $q-1$.

###### Proof.

First we assume that $q$ is even. Then the squaring homomorphism is
surjective, and hence every unit is a square. Fix $u$ to be a multiplicative
generator of $\mathbb{F}_{q}^{\times}$. It follows from [22, Lemma 3.6 (1)]
that $K_{1}^{MW}(\mathbb{F}_{q})$ is generated by the elements $[v]$ for $v\in
k^{\times}$, which are subject to the relation
$[vv^{\prime}]=[v]+[v^{\prime}]$ for all
$v,v^{\prime}\in\mathbb{F}_{q}^{\times}$. The fact that $u^{q-1}=1$ yields the
result that $K_{1}^{MW}(\mathbb{F}_{q})$ is cyclic of order $q-1$ generated by
the symbol $[u]$.

Now we assume that $q$ is odd. Then the kernel of the squaring homomorphism
has two elements, $-1$ and $1$, i.e.,
$\mathbb{F}_{q}^{\times}/\mathbb{F}_{q}^{\times 2}\cong\Z/2\Z$. Because $1$ is
a square, $\mathbb{F}_{q}\setminus\\{0,1\\}$ contains more non-squares than
squares. Construct pairs $(s,1-s)$ from elements
$s\in\mathbb{F}_{q}\setminus\\{0,1\\}$, and observe that there must exist at
least one non-square $s$ such that $1-s$ is also a non-square. For the rest of
the proof we fix $s$ to be one such non-square and pick a multiplicative
generator $u$ of $\mathbb{F}_{q}^{\times}$.

By relation (1) in $K_{1}^{MW}(\mathbb{F}_{q})$, we have $[s].[1-s]=0$. Let
$v_{1},v_{2}$ be non-squares in $\mathbb{F}_{q}^{\times}$. Then there exist
units $c_{1}$ and $c_{2}$ such that $c_{1}^{2}s=v_{1}$ and
$c_{2}^{2}(1-s)=v_{2}$ since $\mathbb{F}_{q}^{\times}/\mathbb{F}_{q}^{\times
2}\cong\Z/2\Z$. We can then compute

$\displaystyle[v_{1}v_{2}]$
$\displaystyle=[v_{1}]+[v_{2}]+\eta.[v_{1}].[v_{2}]$
$\displaystyle=[v_{1}]+[v_{2}]+\eta.[c_{1}^{2}s].[c_{2}^{2}(1-s)]$
$\displaystyle=[v_{1}]+[v_{2}]+\eta.([c_{1}^{2}]+[s]).([c_{2}^{2}]+[1-s])$
$\displaystyle=[v_{1}]+[v_{2}],$

where the last equality follows from the fact $\eta.[c_{i}^{2}]=0$ of remark
135 and the relation $[s].[1-s]=0$. Additionally, for every non-square $v$ and
every square $c^{2}$, we get $[vc_{2}^{2}]=[v]+[c^{2}]$. Since
$\mathbb{F}_{q}^{\times}$ is cyclic and $[u^{n}]=n[u]$, this shows that
$K_{1}^{MW}(\mathbb{F}_{q})$ is cyclic of order $q-1$ generated by the symbol
$[u]$. ∎

######  Theorem 138.

Let $k=\mathbb{F}_{q}$ be a finite field. Then conjecture 131 is true, i.e.,
the inclusions
$\mathbf{G}_{0}(\mathbb{F}_{q})\subseteq[\mathcal{J},\mathbb{A}^{2}\setminus\\{0\\}]^{\mathrm{N}}$
and
$\mathbf{G}(\mathbb{F}_{q})\subseteq[\mathcal{J},\mathbb{P}^{1}]^{\mathrm{N}}$
are equalities.

###### Proof.

By remark 132 it suffices to prove the assertion for
$\mathbf{G}_{0}(\mathbb{F}_{q})$. By propositions 136 and 137, both
$\mathbf{G}_{0}(\mathbb{F}_{q})$ and $K_{1}^{MW}(\mathbb{F}_{q})$ are finite
groups of the same cardinality. Since
$[\mathbb{P}^{1},\mathbb{A}^{2}\setminus\\{0\\}]^{\mathbb{A}^{1}}$ and
$K_{1}^{MW}(\mathbb{F}_{q})$ are isomorphic and since
$\xi_{0}\colon[\mathcal{J},\mathbb{A}^{2}\setminus\\{0\\}]^{\mathrm{N}}\to[\mathbb{P}^{1},\mathbb{A}^{2}\setminus\\{0\\}]^{\mathbb{A}^{1}}$
is an isomorphism, $[\mathcal{J},\mathbb{A}^{2}\setminus\\{0\\}]^{\mathrm{N}}$
is a finite group of the same cardinality as $\mathbf{G}_{0}(\mathbb{F}_{q})$
as well. Hence
$\mathbf{G}_{0}(\mathbb{F}_{q})\subseteq[\mathcal{J},\mathbb{A}^{2}\setminus\\{0\\}]^{\mathrm{N}}$
is an inclusion of finite groups of the same cardinality. This implies that
the inclusion
$\mathbf{G}_{0}(\mathbb{F}_{q})\subseteq[\mathcal{J},\mathbb{A}^{2}\setminus\\{0\\}]^{\mathrm{N}}$
is an equality. ∎

We conclude this section with the following observation. While proposition 136
shows that there is an isomorphism between $\mathbf{G}_{0}(k)$ and
$K_{1}^{MW}(k)$, the proof of proposition 137 suggests that the following map
provides a concrete isomorphism. We consider this an interesting observation
about $K_{1}^{MW}(k)$ that arises from our work on maps
$\mathcal{J}\to\mathbb{A}^{2}\setminus\\{0\\}$.

######  Proposition 139.

Let $k$ be one of the following fields: a field in which every unit is a
square, a finite field, or $\mathbb{R}$. Then the assignment
$[u]\mapsto[g_{u,1}]$ defines an isomorphism $\kappa\colon
K_{1}^{MW}(k)\to\mathbf{G}_{0}(k)$.

###### Proof.

We will show that $K_{1}^{MW}(k)$ and $\mathbf{G}_{0}(k)$ are generated by the
classes $[u]$ and $[g_{u,1}]$, respectively, and that these generators satisfy
exactly the same type of relations. This implies that $\kappa$ is both a well-
defined homomorphism and an isomorphism. We will prove the claim by looking at
each type of field separately.

First we assume that $k$ is a field where every unit is a square. The group
$K_{1}^{MW}(k)$ is generated by the elements $[u]$ and the relation
$[uv]=[u]+[v]$ for all $u,v\in k^{\times}$. To prove that $\kappa$ is an
isomorphism we need to show that $\mathbf{G}_{0}(k)$ is generated by the
classes $[g_{u,1}]$ subject to the relation
$[g_{uv,1}]=[g_{u,1}]\oplus[g_{v,1}]$. Since every unit in $k$ is a square, we
get $[g_{u,v}]=[g_{u/v,1}]$ by lemma 95. Hence $\mathbf{G}_{0}(k)$ is
generated by elements $[g_{u,1}]$. By proposition 119 we know that
$[g_{u,1}]\neq[g_{v,1}]$ for $u\neq v\in k^{\times}$. By lemma 96 we get the
relation $[g_{u,1}]\oplus[g_{v,1}]=[g_{uv,1}]$. Thus, the map sending $[u]$ to
$[g_{u,1}]$ induces a homomorphism which is surjective and injective. Hence
$\kappa$ is an isomorphism.

For $k=\mathbb{F}_{q}$, proposition 137 shows that
$K_{1}^{MW}(\mathbb{F}_{q})$ is generated by the symbol $[u]$ for a
multiplicative generator $u\in\mathbb{F}_{q}^{\times}$. When $q$ is even,
every unit is a square, and in this case $\kappa$ is an isomorphism. So we
assume that $q$ is odd, and will now show that every element in
$\mathbf{G}_{0}(\mathbb{F}_{q})$ can be written in the form
$m[g_{u,1}]\oplus[g_{u^{2m^{\prime}},1}]$ for some
$m,m^{\prime}\in\mathbb{Z}$. We will use that every square in $\mathbb{F}_{q}$
is equal to an even power of the generator $u\in\mathbb{F}_{q}^{\times}$, and
distinguish three cases: Assume first $v_{1},v_{2}\in\mathbb{F}_{q}^{\times}$
are squares. By lemma 95 we then have
$[g_{v_{1},v_{2}}]=[g_{v_{1}/v_{2},1}]=[g_{u^{2m}},1]$ for some $m$. Second,
if $v_{1}$ is not a square and $v_{2}$ is a square, then
$v_{1}/v_{2}=u^{2m+1}$ for some $m\in\Z$. Then by lemma 95 and 96 we know
$[g_{v_{1},v_{2}}]=[g_{u^{2m+1},1}]=[g_{u,1}]\oplus[g_{u^{2m},1}]$. Note that
$[g_{v_{2},v_{1}}]=-[g_{v_{1},v_{2}}]=-[g_{u,1}]\oplus[g_{u^{-2m},1}]$. Third,
assume that both $v_{1}$ and $v_{2}$ are non-squares in
$\mathbb{F}_{q}^{\times}$. Since
$\mathbb{F}_{q}^{\times}/\mathbb{F}_{q}^{\times 2}\cong\Z/2\Z$, we can find an
$m\in\mathbb{Z}$ such that $v_{1}/v_{2}=u^{2m}$. We have
$[g_{v_{1},v_{2}}]=[g_{u\cdot u^{2m},u}]$ by lemma 95 and scaling by the
square $u/v_{2}$. We can now apply lemma 94 and then lemma 96 to get

$[g_{u\cdot u^{2m},u}]=[g_{u\cdot
u^{2m},1}]\oplus[g_{1,u}]=[g_{u,1}]\oplus[g_{u^{2m},1}]\oplus[g_{1,u}]=[g_{u^{2m},1}].$

To conclude the argument we note that, for
$v_{1},v_{2}\in\mathbb{F}_{q}^{\times}$ with $v_{1}+v_{2}\neq 1$, there is the
relation $\langle v_{1}\rangle+\langle v_{2}\rangle=\langle
v_{1}+v_{2}\rangle+\langle(v_{1}+v_{2})v_{1}v_{2}\rangle$ in
$\mathrm{GW}(\mathbb{F}_{q})$. For $s$ and $1-s$ in $\mathbb{F}_{q}^{\times}$,
this gives $\langle s\rangle+\langle 1-s\rangle=\langle 1\rangle+\langle
s(1-s)\rangle=\langle 1,1\rangle$. In particular, since $u,s,$ and $1-s$ all
differ by squares and hence $\langle u\rangle=\langle s\rangle=\langle
1-s\rangle$ in $\mathrm{GW}(\mathbb{F}_{q})$, we have $\langle
u\rangle+\langle u\rangle=\langle 1,1\rangle=\langle u^{2},1\rangle$ in
$\mathrm{GW}(\mathbb{F}_{q})$. By [13, Corollary 3.10] this relation implies
$[X/u]\oplus^{\mathrm{N}}[X/u]=[X/u^{2}]\oplus^{\mathrm{N}}[X/1]$ in
$[\mathbb{P}^{1},\mathbb{P}^{1}]^{\mathrm{N}}$. By proposition 102 and lemma
98, this implies the equality $[g_{u,1}]\oplus[g_{u,1}]=[g_{u^{2},1}]$ in
$\mathbf{G}_{0}(\mathbb{F}_{q})$. Iterating this argument, we get
$(q-1)[g_{u,1}]=[g_{u^{q-1},1}]=[g_{1,1}]$. Since
$[g_{v_{1}},1]\neq[g_{v_{2}},1]$ for $v_{1}\neq
v_{2}\in\mathbb{F}_{q}^{\times}$ by proposition 119, this implies that
$\mathbf{G}_{0}(\mathbb{F}_{q})$ is cyclic of order $q-1$ generated by
$[g_{u,1}]$. Hence the map $[u]\mapsto[g_{u,1}]$ is a well-defined
homomorphism which is surjective and injective. Thus $\kappa$ is an
isomorphism in this case as well.

Finally, we assume $k=\mathbb{R}$. First we determine the generators and
relations for $K_{1}^{MW}(\mathbb{R})$. For $u>0$, we have
$[-u]=[-1]+[u]+\eta.[-1].[u]=[-1]+[u]$ and $-[u]=[1/u]$. Thus every element in
$K_{1}^{MW}(\mathbb{R})$ can be written as $n[-1]+[u]$ with $u>0$ for some
$n\in\mathbb{Z}$ subject to the relation
$(n[-1]+[u])+(m[-1]+[v])=(n+m)[-1]+[uv]$. Next we show that
$\mathbf{G}_{0}(\mathbb{R})$ has analogous generators and relations. Assume
$u,v>0$. Since $v$ is a square, lemmas 95 and 96 imply the identities

$[g_{u,v}]=[g_{u/v,1}],~{}\text{and}~{}[g_{-u,v}]=[g_{-u/v,1}]=[g_{-1,1}]\oplus[g_{u/v,1}].$

Using that $v$ is a square, lemma 95 yields the following identity

$[g_{u,-v}]=[g_{u/v,-1}]=[g_{u/v,1}]\oplus[g_{1,-1}]=-[g_{-1,1}]\oplus[g_{u/v,1}]$

where we have $[g_{1,-1}]=-[g_{-1,1}]$ by lemma 94. Finally, using lemma 95
and 96 we get

$[g_{-u,-v}]=[g_{-u/v,-1}]=[g_{-u/v,1}]\oplus[g_{1,-1}]=[g_{-1,1}]\oplus[g_{u/v,1}]\oplus[g_{1,-1}]=[g_{u/v,1}].$

This implies every element of $\mathbf{G}_{0}(\mathbb{R})$ can be written as a
sum $n[g_{-1,1}]\oplus[g_{u,1}]$ with $u>0$ and $n\in\mathbb{Z}$. By
proposition 119 we know that $[g_{u,1}]\neq[g_{v,1}]$ for $u\neq
v\in\mathbb{R}^{\times}$. By lemma 96 we get the relation
$\left(n[g_{-1,1}]\oplus[g_{u,1}]\right)\oplus\left(m[g_{-1,1}]\oplus[g_{v,1}]\right)=(n+m)[g_{-1,1}]\oplus[g_{uv,1}]$
when $u,v>0$. Hence the map $[u]\mapsto[g_{u,1}]$ is a well-defined
homomorphism which is surjective and injective. Thus $\kappa$ is an
isomorphism in this case. This finishes the proof. ∎

##  Appendix A Affine representability for pointed spaces and homotopies

In this section, we discuss the results of Asok, Hoyois, and Wendt in [5],
[6], and how we apply them. While the definition of our proposed group
operation on $[\mathcal{J},\mathbb{P}^{1}]^{\mathrm{N}}$ in definition 81 is
independent of motivic homotopy theory and the results of [6], we use the
affine representability results of [6] to compare our group operation with the
conventional group structure on
$[\mathbb{P}^{1},\mathbb{P}^{1}]^{\mathbb{A}^{1}}$. A minor technical point to
overcome is that Asok, Hoyois, and Wendt work in the unpointed motivic
homotopy category, whereas we need the analogous results in the pointed
setting. The purpose of this appendix is to explain how the pointed analogs
can be deduced. To keep the presentation brief, we use the conventions and
notation of the papers [5] and [6]. We thank Marc Hoyois for helpful comments.

Let $\mathrm{Spc}(k)$ denote the category of simplicial presheaves on
$\mathrm{Sm}_{k}$. Let $\mathrm{Spc}_{*}(k)$ denote the category of pointed
simplicial presheaves on $\mathrm{Sm}_{k}$. We refer to an object in
$\mathrm{Spc}(k)$ (respectively in $\mathrm{Spc}_{*}(k)$) as a (pointed)
motivic space. For a motivic space $\mathcal{Y}$, let
$\mathrm{Sing}^{\mathbb{A}^{1}}\mathcal{Y}$ denote the singular functor
defined in [5, §4.1], see also [23, page 88]. If $\mathcal{Y}$ is pointed by a
morphism $y\colon\operatorname{Spec}{k}\to\mathcal{Y}$, then the pointed
singular functor $\mathrm{Sing}_{*}^{\mathbb{A}^{1}}\mathcal{Y}$ is defined as
the fiber over $y$. More precisely, let $X$ be a pointed smooth $k$-scheme
pointed by the morphism $x\colon\operatorname{Spec}{k}\to X$, then
$\mathrm{Sing}_{*}^{\mathbb{A}^{1}}\mathcal{Y}$ is determined by the pullback
square of simplicial sets

(140)
$\textstyle{(\mathrm{Sing}_{*}^{\mathbb{A}^{1}}\mathcal{Y})(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\mathrm{Sing}^{\mathbb{A}^{1}}\mathcal{Y})(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{x^{*}}$$\textstyle{\mathrm{point}=(\mathrm{Sing}^{\mathbb{A}^{1}}\operatorname{Spec}{k})(\operatorname{Spec}{k})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{y_{*}}$$\textstyle{(\mathrm{Sing}^{\mathbb{A}^{1}}\mathcal{Y})(\operatorname{Spec}{k}).}$

We note that, on $0$-simplices, $x$ induces induces a map of sets
$x^{*}\colon\mathrm{Spc}(k)(X,\mathcal{Y})\to\mathrm{Spc}(k)(\operatorname{Spec}{k},\mathcal{Y})$.
Hence, $(\mathrm{Sing}_{*}^{\mathbb{A}^{1}}\mathcal{Y})(X)_{0}$ is the set
$\mathrm{Spc}_{*}(k)(X,\mathcal{Y})$ of pointed morphisms $X\to\mathcal{Y}$.
On $1$-simplices, $x$ induces a map of sets
$x^{*}\colon\mathrm{Spc}(k)(X\times\mathbb{A}^{1},\mathcal{Y})\to\mathrm{Spc}(k)(\mathbb{A}^{1},\mathcal{Y})$.
Hence, $(\mathrm{Sing}_{*}^{\mathbb{A}^{1}}\mathcal{Y})(X)_{1}$ is the set of
pointed naive $\mathbb{A}^{1}$-homotopies of pointed morphisms
$X\to\mathcal{Y}$.

######  Remark 141.

In particular, if $\mathcal{Y}=Y$ is represented by a pointed smooth
$k$-scheme $Y$, the set $\pi_{0}((\mathrm{Sing}_{*}^{\mathbb{A}^{1}}Y)(X))$ is
the set of pointed naive homotopy classes of pointed morphisms $X\to Y$
described in section 2.4, that is,

$\displaystyle\pi_{0}((\mathrm{Sing}_{*}^{\mathbb{A}^{1}}Y)(X))=[X,Y]^{\mathrm{N}}.$

We recall the following definition from [6]:

######  Definition 142.

[6, Definition 2.1.1] Let $\mathcal{F}\in\mathrm{Spc}(k)$ and let
$\mathcal{F}\to\widetilde{\mathcal{F}}$ be a fibrant replacement in the
$\mathbb{A}^{1}$-model structure on $\mathrm{Spc}(k)$. There is a canonical
map $\mathrm{Sing}^{\mathbb{A}^{1}}\mathcal{F}\to\widetilde{\mathcal{F}}$ that
is well-defined up to simplicial homotopy equivalence. Then
$\mathcal{F}\in\mathrm{Spc}(k)$ is called _$\mathbb{A}^{1}$ -naive_ if the map
$(\mathrm{Sing}^{\mathbb{A}^{1}}\mathcal{F})(X)\to\widetilde{\mathcal{F}}(X)$
is a weak equivalence of simplicial sets for every affine smooth $k$-scheme
$X$.

We will now show how the unpointed notion of $\mathbb{A}^{1}$-naivity of
definition 142 translates to the pointed setting.

######  Proposition 143.

Let $\mathcal{Y}\in\mathrm{Spc}_{*}(k)$ be a pointed motivic space. Assume
that the underlying unpointed motivic space $\mathcal{Y}$ is
$\mathbb{A}^{1}$-naive. Then, for every affine pointed smooth $k$-scheme $X$,
the canonical map
$\pi_{0}((\mathrm{Sing}_{*}^{\mathbb{A}^{1}}\mathcal{Y})(X))\xrightarrow{\cong}[X,\mathcal{Y}]^{\mathbb{A}^{1}}$
is a bijection.

###### Proof.

Let $(X,x)$ be a pointed smooth $k$-scheme, and let $p\colon
X\to\operatorname{Spec}{k}$ denote the canonical morphism. Then $p$ induces a
map
$p^{*}\colon(\mathrm{Sing}^{\mathbb{A}^{1}}\mathcal{Y})(\operatorname{Spec}{k})\to(\mathrm{Sing}^{\mathbb{A}^{1}}\mathcal{Y})(X)$
of simplicial sets such that $x^{*}\circ p^{*}$ is the identity on
$(\mathrm{Sing}^{\mathbb{A}^{1}}\mathcal{Y})(\operatorname{Spec}{k})$. This
shows that the map $x^{*}$ is a Kan fibration. Since the Kan–Quillen model
structure on simplicial sets is right proper, this implies that
$(\mathrm{Sing}_{*}^{\mathbb{A}^{1}}\mathcal{Y})(X)$ is the homotopy fiber of
$x^{*}$.

Let $\mathcal{Y}\to R_{\mathbb{A}^{1}}\mathcal{Y}$ be a fibrant replacement of
$\mathcal{Y}$ in the $\mathbb{A}^{1}$-model structure on
$\mathrm{Spc}_{*}(k)$. After forgetting the basepoint,
$R_{\mathbb{A}^{1}}\mathcal{Y}$ is fibrant in the $\mathbb{A}^{1}$-model
structure on the category $\mathrm{Spc}(k)$ of unpointed motivic spaces. Since
the singular functor preserves $\mathbb{A}^{1}$-fibrations,
$\mathrm{Sing}^{\mathbb{A}^{1}}R_{\mathbb{A}^{1}}\mathcal{Y}$ is fibrant and
we may assume
$\widetilde{\mathcal{Y}}=\mathrm{Sing}^{\mathbb{A}^{1}}R_{\mathbb{A}^{1}}\mathcal{Y}$.
Moreover, we get the following commutative diagram of simplicial sets which,
by the above argument, is a morphism of homotopy fiber sequences for every
pointed smooth $k$-scheme $(X,x)$.

(144)
$\textstyle{(\mathrm{Sing}_{*}^{\mathbb{A}^{1}}\mathcal{Y})(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\mathrm{Sing}_{*}^{\mathbb{A}^{1}}R_{\mathbb{A}^{1}}\mathcal{Y})(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\mathrm{Sing}^{\mathbb{A}^{1}}\mathcal{Y})(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{x^{*}}$$\textstyle{(\mathrm{Sing}^{\mathbb{A}^{1}}R_{\mathbb{A}^{1}}\mathcal{Y})(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{x^{*}}$$\textstyle{(\mathrm{Sing}^{\mathbb{A}^{1}}\mathcal{Y})(\operatorname{Spec}{k})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\mathrm{Sing}^{\mathbb{A}^{1}}R_{\mathbb{A}^{1}}\mathcal{Y})(\operatorname{Spec}{k})}$

Now we assume that the underlying simplicial presheaf of $\mathcal{Y}$ is
$\mathbb{A}^{1}$-naive and that $X$ is affine. Since $\mathcal{Y}$ is
$\mathbb{A}^{1}$-naive and both $X$ and $\operatorname{Spec}{k}$ are affine,
the horizontal maps in the middle and at the bottom are weak equivalences of
simplicial sets. Thus, since diagram (144) is a morphism of homotopy fiber
sequences, the top horizontal map is a weak equivalence of simplicial sets.
Hence it induces a bijection on $\pi_{0}$. Since
$\pi_{0}((\mathrm{Sing}_{*}^{\mathbb{A}^{1}}R_{\mathbb{A}^{1}}\mathcal{Y})(X))=[X,\mathcal{Y}]^{\mathbb{A}^{1}}$,
this proves the assertion. ∎

######  Lemma 145.

The smooth $k$-schemes $\mathcal{J}$ and $\mathbb{P}^{1}$ are
$\mathbb{A}^{1}$-naive.

###### Proof.

Let $Q_{2}$ be the smooth affine quadric over $\mathbb{Z}$ defined by
$xy=z(z+1)$. By [6, Theorem 4.2.2], $Q_{2}$ is $\mathbb{A}^{1}$-naive. The
scheme endomorphism of $\operatorname{Spec}{\mathbb{Z}[x,y,z]}$ given by the
ring homomorphism defined by sending $x\mapsto z,\,y\mapsto-y,\,z\mapsto-x$
induces an isomorphism $Q_{2}\cong\mathcal{J}$. Hence $\mathcal{J}$ is
$\mathbb{A}^{1}$-naive. By [6, Lemma 4.2.4] an affine torsor bundle over a
base space is $\mathbb{A}^{1}$-naive if and only if the base space is
$\mathbb{A}^{1}$-naive. Since $\mathcal{J}$ is $\mathbb{A}^{1}$-naive and an
affine torsor bundle over $\mathbb{P}^{1}$, it follows that $\mathbb{P}^{1}$
is $\mathbb{A}^{1}$-naive. ∎

######  Proposition 146.

The canonical map
$\nu\colon[X,\mathbb{P}^{1}]^{\mathrm{N}}\xrightarrow{\cong}[X,\mathbb{P}^{1}]^{\mathbb{A}^{1}}$
is a bijection for every affine pointed smooth $k$-scheme $X$.

###### Proof.

The proposition follows from remark 141 and proposition 143, since lemma 145
shows $\mathbb{P}^{1}$ is $\mathbb{A}^{1}$-naive. ∎

For $X=\mathcal{J}$, the previous proposition yields the comparison of the
sets $[\mathcal{J},\mathbb{P}^{1}]^{\mathrm{N}}$ and
$[\mathcal{J},\mathbb{P}^{1}]^{\mathbb{A}^{1}}$ of pointed homotopy classes
that we wanted.

##  Appendix B Facts about the resultant

In sections 2.3 and 5.1 we used the following facts about the resultant for
which we now provide references or proofs.

Throughout this section we let $S$ be an integral domain and let
$A=a_{n}X^{n}+a_{n-1}X^{n-1}+\ldots+a_{0}$ and
$B=b_{n}X^{n}+b_{n-1}X^{n-1}+\ldots+b_{0}$ be polynomials over $S$ in the
indeterminate $X$.

######  Lemma 147 (Remark 4 on page IV.76 in [11]).

Assume $\mathrm{res}(A,B)\in S^{\times}$. Then there exist polynomials $U,V\in
S[X]$ such that $AU+BV=1$.

######  Lemma 148 (Remark 1 on page IV.76 in [11]).

Let $\tilde{A}=a_{n}+a_{n-1}X+\ldots+a_{0}X^{n}$ and
$\tilde{B}=b_{n}+b_{n-1}X+\ldots+b_{0}X^{n}$ be the reversed polynomials of
$A$ and $B$. If $\mathrm{res}(A,B)\in S^{\times}$, then
$\mathrm{res}(\tilde{A},\tilde{B})=(-1)^{n}\mathrm{res}(A,B)\in S^{\times}$.

######  Lemma 149 (Remark 5 on page IV.77 in [11]).

Assume $\mathrm{res}(A,B)\in S^{\times}$. Let $C\in S[X]$ be a polynomial such
that $\deg(A)\geq\deg(BC)$. Then we have
$\mathrm{res}(A+BC,B)=\mathrm{res}(A,B)$.

######  Lemma 150.

Assume $\mathrm{res}(A,B)\in S^{\times}$ and that $A$ is monic. Then we have

$\mathrm{res}\left(AX-\frac{1}{u}B,uA\right)=-u\mathrm{res}(A,B)~{}\text{for
all}~{}u\in S^{\times}.$

###### Proof.

The strategy of the proof is as follows: We determine the Sylvester matrix for
the pair $\left(AX-\frac{1}{u}B,uA\right)$ and will then use elementary row
and column operations to confirm that it has the determinant claimed.

First note that
$\mathrm{res}\left(AX-\frac{1}{u}B,uA\right)=u^{n+1}\mathrm{res}\left(AX-\frac{1}{u}B,A\right)$.
Let $A=\sum_{i=0}^{n}a_{i}X^{i}$, and $B=\sum_{i=0}^{n}b_{i}X^{i}$. Let
$c_{i}=a_{i-1}-\frac{1}{u}b_{i}$ for $0\leq i\leq{n+1}$ and set
$a_{-1}=b_{n+1}=0$. Then $AX-\frac{1}{u}B=\sum_{i=0}^{n+1}c_{i}X^{i}$. The
Sylvester matrix for the pair $\left(AX-\frac{1}{u}B,A\right)$ is

$\begin{pmatrix}c_{n+1}&0&\ldots&0&0&0&\ldots&0\\\
c_{n}&c_{n+1}&&\vdots&a_{n}&0&&\vdots\\\
\vdots&\vdots&\ddots&\vdots&\vdots&\ddots&\\\
c_{1}&c_{2}&\ldots&c_{n+1}&a_{1}&\ldots&a_{n}&0\\\
c_{0}&c_{1}&\ldots&c_{n}&a_{0}&\ldots&a_{n-1}&a_{n}\\\
0&c_{0}&\ldots&c_{n-1}&0&a_{0}&\ldots&a_{n-1}\\\
\vdots&&\ddots&\vdots&\vdots&&\ddots&\vdots\\\
0&0&\ldots&c_{0}&0&0&\ldots&a_{0}\par\end{pmatrix}.$

Since $c_{n+1}=1$, we can remove the first row and first column to obtain a
submatrix with the same determinant, namely

$\begin{pmatrix}c_{n+1}&0&\ldots&0&a_{n}&0&\ldots&0&0\\\
c_{n}&c_{n+1}&&\vdots&a_{n-1}&a_{n}&\ddots&&\vdots\\\
\vdots&\vdots&\ddots&\vdots&\vdots&\ddots&\ddots&\vdots&\\\
c_{2}&c_{3}&\ldots&c_{n+1}&a_{1}&a_{2}&\ldots&a_{n}&0\\\
c_{1}&c_{2}&\ldots&c_{n}&a_{0}&a_{1}&\ldots&a_{n-1}&a_{n}\\\
c_{0}&c_{1}&\ldots&c_{n-1}&0&a_{0}&\ldots&a_{n-2}&a_{n-1}\\\
\vdots&c_{0}&\ddots&\vdots&\vdots&&\ddots&\vdots&\vdots\\\
0&0&\ddots&c_{1}&0&0&\ldots&a_{0}&a_{1}\\\
0&0&\ldots&c_{0}&0&0&\ldots&0&a_{0}\par\end{pmatrix}.$

Subtracting column $1$ from column $n+1$ yields

$\begin{pmatrix}c_{n+1}&0&\ldots&0&0&0&\ldots&0&0\\\
c_{n}&c_{n+1}&&\vdots&\frac{1}{u}b_{n}&a_{n}&\ddots&\ddots&0\\\
\vdots&\vdots&\ddots&\vdots&\vdots&\vdots&\ddots&\\\
c_{2}&c_{3}&\ldots&c_{n+1}&\frac{1}{u}b_{2}&a_{2}&\ldots&a_{n}&0\\\
c_{1}&c_{2}&\ldots&c_{n}&\frac{1}{u}b_{1}&a_{1}&\ldots&a_{n-1}&a_{n}\\\
c_{0}&c_{1}&\ldots&c_{n-1}&\frac{1}{u}b_{0}&a_{0}&\ldots&a_{n-2}&a_{n-1}\\\
\vdots&c_{0}&\ddots&\vdots&\vdots&&\ddots&\vdots&\vdots\\\
0&0&\ddots&c_{1}&0&0&\ldots&a_{0}&a_{1}\\\
0&0&\ldots&c_{0}&0&0&\ldots&0&a_{0}\par\end{pmatrix}.$

Once again, the determinant of this matrix is the same as that of the
submatrix where the first row and first and column removed. We remove them and
obtain

$\begin{pmatrix}c_{n+1}&0&\ldots&0&\frac{1}{u}b_{n}&a_{n}&0&\ldots&0&0\\\
c_{n}&c_{n+1}&&\vdots&\frac{1}{u}b_{n-1}&a_{n-1}&a_{n}&\ddots&&\vdots\\\
\vdots&\vdots&\ddots&\vdots&\vdots&\vdots&\ddots&\ddots&\vdots&\\\
c_{3}&c_{4}&\ldots&c_{n+1}&\frac{1}{u}b_{2}&a_{2}&a_{3}&\ldots&a_{n}&0\\\
c_{2}&c_{3}&\ldots&c_{n}&\frac{1}{u}b_{1}&a_{1}&a_{2}&\ldots&a_{n-1}&a_{n}\\\
c_{1}&c_{2}&\ldots&c_{n-1}&\frac{1}{u}b_{0}&a_{0}&a_{1}&\ldots&a_{n-2}&a_{n-1}\\\
c_{0}&c_{1}&\ldots&c_{n-2}&0&0&a_{0}&\ldots&a_{n-3}&a_{n-2}\\\
\vdots&c_{0}&\ddots&\vdots&\vdots&&&\ddots&\vdots&\vdots\\\
0&0&\ddots&c_{1}&0&0&0&\ldots&a_{0}&a_{1}\\\
0&0&\ldots&c_{0}&0&0&0&\ldots&0&a_{0}\end{pmatrix}.$

We subtract column $n+i$ from column $i$ for each $i<n$, and the result is
that at each entry $c_{i}$, we get instead $c_{i}-a_{i-1}=-\frac{1}{u}b_{i}$.

$\begin{pmatrix}-\frac{1}{u}b_{n+1}&0&\ldots&0&\frac{1}{u}b_{n}&a_{n}&0&\ldots&0&0\\\
-\frac{1}{u}b_{n}&-\frac{1}{u}b_{n+1}&&\vdots&\frac{1}{u}b_{n-1}&a_{n-1}&a_{n}&\ddots&&\vdots\\\
\vdots&\vdots&\ddots&\vdots&\vdots&\vdots&\ddots&\ddots&\vdots&\\\
-\frac{1}{u}b_{3}&-\frac{1}{u}b_{4}&\ldots&-\frac{1}{u}b_{n+1}&\frac{1}{u}b_{2}&a_{2}&a_{3}&\ldots&a_{n}&0\\\
-\frac{1}{u}b_{2}&-\frac{1}{u}b_{3}&\ldots&-\frac{1}{u}b_{n}&\frac{1}{u}b_{1}&a_{1}&a_{2}&\ldots&a_{n-1}&a_{n}\\\
-\frac{1}{u}b_{1}&-\frac{1}{u}b_{2}&\ldots&-\frac{1}{u}b_{n-1}&\frac{1}{u}b_{0}&a_{0}&a_{1}&\ldots&a_{n-2}&a_{n-1}\\\
-\frac{1}{u}b_{0}&-\frac{1}{u}b_{1}&\ldots&-\frac{1}{u}b_{n-2}&0&0&a_{0}&\ldots&a_{n-3}&a_{n-2}\\\
\vdots&-\frac{1}{u}b_{0}&\ddots&\vdots&\vdots&&&\ddots&\vdots&\vdots\\\
0&0&\ddots&-\frac{1}{u}b_{1}&0&0&0&\ldots&a_{0}&a_{1}\\\
0&0&\ldots&-\frac{1}{u}b_{0}&0&0&0&\ldots&0&a_{0}\end{pmatrix}$

Then multiplying the first $n$ columns by $u$ and applying a cyclic
permutation of the $n$ first columns yields the Sylvester matrix of the pair
$(B,A)$. The sign of the permutation is $(-1)^{n-1}$. Interchanging column $i$
with $n+i$ for all $i\leq n$ yields $(A,B)$, and this needed another
permutation of sign $(-1)^{n-1}$, so the signs cancel out. ∎

##  Appendix C Testing compatibility via real realization and signatures

Now we provide the additional evidence for conjecture 90 and a positive answer
to question 91 referred to in section 4.

We assume that $k$ is a subfield of $\R$. Let $\mathcal{H}_{*}(k)$ denote the
homotopy category of pointed smooth $k$-schemes and let $\mathcal{H}_{*}$ be
the homotopy category of pointed topological spaces. By [23] sending a smooth
$k$-scheme $X$ to the topological space $X(\R)$ equipped with its usual
structure of a real manifold extends to a functor
$\Re\colon\mathcal{H}_{*}(k)\rightarrow\mathcal{H}_{*}$, see also [4, page 14]
and [14, Section 5.3].

For a smooth map $f$ between oriented compact smooth manifolds of the same
dimension, let $\deg^{\mathrm{top}}(f)\in\Z$ denote the topological Brouwer
degree of $f$. In [21] Morel describes the analog of the topological degree
map in $\mathbb{A}^{1}$-homotopy theory. For endomorphisms of $\mathbb{P}^{1}$
it defines a homomorphism

$\deg^{\mathbb{A}^{1}}\colon[\mathbb{P}^{1},\mathbb{P}^{1}]^{\mathbb{A}^{1}}\to\mathrm{GW}(k).$

Let $f\colon\mathbb{P}^{1}\to\mathbb{P}^{1}$ be a morphism. Since we assume
that $k$ is a subfield of $\R$, we can form the real realization
$\Re(f)\colon\mathbb{P}^{1}(\R)\to\mathbb{P}^{1}(\R)$. Following Morel, the
signature, denoted $\mathrm{sgn}$, of the quadratic form given by the
$\mathbb{A}^{1}$-Brouwer degree of $f$ equals the topological Brouwer degree
of $\Re(f)$, i.e.,

(151)
$\displaystyle\mathrm{sgn}\left(\deg^{\mathbb{A}^{1}}(f)\right)=\deg^{\mathrm{top}}(\Re(f)).$

We note that, in some form, this was also shown by Eisenbud, Levine, and
Teissier in [15, Theorem 1.2] for the local degree of maps between real affine
spaces. The latter approach has been incorporated into the motivic theory by
Kass and Wickelgren [18].

The motivic Brouwer degree map $\deg^{\mathbb{A}^{1}}$ is a homomorphism for
the conventional group structure $\oplus^{\mathbb{A}^{1}}$ on
$[\mathbb{P}^{1},\mathbb{P}^{1}]^{\mathbb{A}^{1}}$, and the signature is
additive. Hence, for morphisms $f,g\colon\mathbb{P}^{1}\to\mathbb{P}^{1}$ and
their sum $f\oplus^{\mathbb{A}^{1}}g$ in
$[\mathbb{P}^{1},\mathbb{P}^{1}]^{\mathbb{A}^{1}}$, (151) implies

(152)
$\displaystyle\deg^{\mathrm{top}}\left(\Re\left(f\oplus^{\mathbb{A}^{1}}g\right)\right)$
$\displaystyle=\mathrm{sgn}\left(\deg^{\mathbb{A}^{1}}\left(f\oplus^{\mathbb{A}^{1}}g\right)\right)$
$\displaystyle=\deg^{\mathrm{top}}(\Re(f))+\deg^{\mathrm{top}}(\Re(g)).$

We will now use this fact to test the compatibility of the action of
definition 77 and thereby of definition 81 with the conventional group
structure in the following way.

The real points $\mathcal{J}(\R)$ of $\mathcal{J}$ form a surface in $\R^{3}$
given by the equation $x(1-x)-yz=0$. The intersection with the plane defined
by $y=z$ is the circle given by the set of points satisfying $x(1-x)-y^{2}=0$.
Its center is the point $(1/2,0,0)\in\R^{3}$. We parameterize this circle via
the map $\gamma:\mathbb{S}^{1}\to\mathcal{J}(\R)$ given by

$\gamma:\theta\mapsto\Big{(}1/2+\cos(\theta)/2,\sin(\theta)/2,\sin(\theta)/2\Big{)}.$

The real realization of $\mathbb{P}^{1}$ is the topological real projective
line $\mathbb{RP}^{1}$. Hence, for a morphism
$f\colon\mathcal{J}\to\mathbb{P}^{1}$, we may form the composition
$\Re(f)\circ\gamma$ which is a smooth map $\mathbb{S}^{1}\to\mathbb{RP}^{1}$.
We can then apply the topological Brouwer degree to the composition
$\Re(f)\circ\gamma$. Since the real realization of a naive homotopy induces a
homotopy of maps between topological spaces, this induces a well-defined map

$\displaystyle\deg^{\mathrm{top}}(\Re(-)\circ\gamma)\colon[\mathcal{J},\mathbb{P}^{1}]^{\mathrm{N}}$
$\displaystyle\longrightarrow\Z$ $\displaystyle f~{}~{}~{}$
$\displaystyle\longmapsto\deg^{\mathrm{top}}(\Re(f)\circ\gamma).$

######  Lemma 153.

The following diagram commutes.

(160)

###### Proof.

To prove the assertion it suffices to show that both parts of the diagram
commute. The functor $\Re$ commutes with the canonical map
$\nu\colon[\mathcal{J},\mathbb{P}^{1}]^{\mathrm{N}}\to[\mathcal{J},\mathbb{P}^{1}]^{\mathbb{A}^{1}}$.
This implies that the upper part commutes. We verify in example 166 that for
$\pi\colon\mathcal{J}\to\mathbb{P}^{1}$ the composite map
$\Re(\pi)\circ\gamma\colon\mathbb{S}^{1}\to\mathbb{RP}^{1}$ is an orientation
preserving diffeomorphism. Now let $f\colon\mathbb{P}^{1}\to\mathbb{P}^{1}$ be
a morphism. Since the composition with $\Re(\pi)\circ\gamma$ preserves
degrees, we obtain the identity

$\displaystyle\deg^{\mathrm{top}}(\Re(f\circ\pi)\circ\gamma)=\deg^{\mathrm{top}}(\Re(f)\circ\Re(\pi)\circ\gamma)=\deg^{\mathrm{top}}(\Re(f)).$

This implies that the lower part of diagram (160) commutes and finishes the
proof. ∎

This implies the following necessary condition for the compatibility of the
operations $\oplus$ and $\oplus^{\mathbb{A}^{1}}$:

######  Proposition 161.

Assume that $\xi$ is a group homomorphism. Then we have

$\displaystyle\deg^{\mathrm{top}}(\Re(\xi(f\oplus
g)))=\deg^{\mathrm{top}}(\Re(f)\circ\gamma)+\deg^{\mathrm{top}}(\Re(g)\circ\gamma).$

###### Proof.

The assumption that $\xi$ is a group homomorphism implies

$\displaystyle\deg^{\mathrm{top}}(\Re(\xi(f\oplus
g)))=\deg^{\mathrm{top}}(\Re(\xi(f)\oplus^{\mathbb{A}^{1}}\xi(g))).$

Identity (152) implies

$\displaystyle\deg^{\mathrm{top}}(\Re(\xi(f)\oplus^{\mathbb{A}^{1}}\xi(g)))=\deg^{\mathrm{top}}(\Re(\xi(f)))+\deg^{\mathrm{top}}(\Re(\xi(g))).$

Commutativity of diagram (160) implies

$\displaystyle\deg^{\mathrm{top}}(\Re(\xi(f)))+\deg^{\mathrm{top}}(\Re(\xi(g)))=\deg^{\mathrm{top}}(\Re(f)\circ\gamma)+\deg^{\mathrm{top}}(\Re(g)\circ\gamma).$

Putting these identities together yields the assertion. ∎

As a special case, we get the following necessary condition for the
compatibility of $\oplus$ with the conventional group structure:

######  Corollary 162.

Let $F\colon\mathcal{J}\to\mathbb{P}^{1}$ be a pointed morphism. Then, if
$\xi$ is a group homomorphism, we must have

$\displaystyle\deg^{\mathrm{top}}(\Re(F\oplus\pi)\circ\gamma)=\deg^{\mathrm{top}}(\Re(F)\circ\gamma)+1.$

In the following section we will apply corollary 162 in a concrete case in
example 167.

Moreover, we exclude a potential alternative to the operation $\oplus$ of
definition 77 in example 170.

######  Remark 163.

Let $F\colon\mathcal{J}\to\mathbb{P}^{1}$ again be a pointed morphism. Assume
that question 91 has a positive answer, i.e., assume that $\tilde{\pi}$ is
naively homotopic to $-\pi$. Then proposition 161 shows that, if $\xi$ is a
group homomorphism, then we must expect to get

(164)
$\displaystyle\deg^{\mathrm{top}}(\Re(F\oplus\tilde{\pi})\circ\gamma)=\deg^{\mathrm{top}}(\Re(F)\circ\gamma)-1.$

Note that, since we do not know whether $\tilde{\pi}$ is naively homotopic to
$-\pi$, (164) may fail to hold for some $F$ even though $\xi$ is a group
homomorphism.

However, we confirm formula (164) in a concrete case in example 169.

We will now compute the topological degrees and thereby the signatures of
several maps and apply the previous observations.

######  Example 165.

Consider the morphism $g_{1,-1}:\mathcal{J}\rightarrow\mathbb{P}^{1}$ defined
by the unimodular row $(2x-1,2y)$. Its real realization is the map
$\Re(g_{1,-1}):\Re(\mathcal{J})\rightarrow\Re(\mathbb{P}^{1})$ defined by

$\displaystyle\Re(g_{1,-1}):(x,y,z)\mapsto[2x-1:2y].$

Precomposing with $\gamma$ gives

$\displaystyle\Re(g_{1,-1})\circ\gamma:\theta\mapsto\left[\cos(\theta):\sin(\theta)\right],$

which is the usual double cover of $\mathbb{RP}^{1}$ by $\mathbb{S}^{1}$ and
has topological Brouwer degree 2.

As explained in section 2.3, a morphism $f:\mathcal{J}\to\mathbb{P}^{1}$ may
be described by gluing together partially defined maps on open subsets. In the
following examples we will define a morphism
$\Re(f)\circ\gamma:\mathbb{S}^{1}\to\mathbb{RP}^{1}$ by gluing
$\Re(f|_{D(x)})\circ\gamma\colon\gamma^{-1}(\Re(D(x)))\to\mathbb{RP}^{1}$ and
$\Re(f|_{D(1-x)})\circ\gamma\colon\gamma^{-1}(\Re(D(1-x)))\to\mathbb{RP}^{1}$
on their overlaps in the respective domains.

######  Example 166.

The real realization of $\pi:\mathcal{J}\rightarrow\mathbb{P}^{1}$ is defined
on $\Re(D(x))$ by

$\displaystyle\Re(\pi|_{D(x)}):(x,y,z)$ $\displaystyle\mapsto[x:y],$

and on $\Re(D(1-x))$ by

$\displaystyle\Re(\pi|_{D(1-x)}):(x,y,z)$ $\displaystyle\mapsto[z:1-x].$

Precomposing with $\gamma$ gives

$\displaystyle\Re(\pi|_{D(x)})\circ\gamma:\theta$
$\displaystyle\mapsto[1/2+\cos(\theta)/2:\sin(\theta)/2],$
$\displaystyle\Re(\pi|_{D(1-x)})\circ\gamma:\theta$
$\displaystyle\mapsto[\sin(\theta)/2:1/2-\cos(\theta)/2],$

which glue together to give a map of degree 1:

$(\Re(\pi)\circ\gamma)(\theta)=\begin{cases}[1+\cos(\theta):\sin(\theta)]&\theta\neq\pi,\\\
[0:1]&\theta=\pi.\end{cases}$

This shows that $\Re(\pi)\circ\gamma$ is an orientation preserving
diffeomorphism and has topological Brouwer degree $1$.

In the following example we test the necessary condition of corollary 162 in a
concrete case.

######  Example 167.

Recall that the unimodular row $g_{1,-1}=(2x-1,2y)$ can be augmented to the
following matrix with determinant 1:

$m_{1,-1}=\begin{pmatrix}2x-1&-2z\\\ 2y&2x-1\end{pmatrix}.$

The group action of definition 77 yields the map
$F:=g_{1,-1}\oplus\pi=(2x-1,-2z:2y,2x-1)_{1}$.

Taking real realization and precomposing with $\gamma$ yields the map the map
$\Re(F)\circ\gamma\colon\mathbb{S}^{1}\to\mathbb{RP}^{1}$ given by

$(\Re(F)\circ\gamma)(\theta)=\begin{cases}[\cos(\theta)+\cos(2\theta):\sin(\theta)+\sin(2\theta)]&\theta\neq\pi,\\\
[0:1]&\theta=\pi.\end{cases}$

The topological degree of this map is $3$.

Hence our computation confirms that

$\displaystyle\deg^{\mathrm{top}}(\Re(F)\circ\gamma)=3=2+1=\deg^{\mathrm{top}}(\Re(g_{-1,1})\circ\gamma)+\deg^{\mathrm{top}}(\Re(\pi)\circ\gamma),$

as required for the compatibility of $\oplus$ with $\oplus^{\mathbb{A}^{1}}$.

The next example confirms that the signature of the motivic Brouwer degree of
$\tilde{\pi}$ has the value $-1$ as expected if question 91 has a positive
answer.

######  Example 168.

The real realization of the morphism
$\tilde{\pi}=(1,0:0,-1)_{-1}\colon\mathcal{J}\to\mathbb{P}^{1}$ is defined on
$\Re(D(x))$ and $\Re(D(1-x))$ respectively by $\Re(D(x))$ by

$\displaystyle\Re(\tilde{\pi}|_{D(x)}):(x,y,z)$ $\displaystyle\mapsto[x:-z],$
$\displaystyle\Re(\tilde{\pi}|_{D(1-x)}):(x,y,z)$
$\displaystyle\mapsto[y:-1+x].$

Precomposing with $\gamma$ gives

$\displaystyle\Re(\tilde{\pi}|_{D(x)})\circ\gamma:\theta$
$\displaystyle\mapsto[1/2+\cos(\theta)/2:-\sin(\theta)/2],$
$\displaystyle\Re(\tilde{\pi}|_{D(1-x)})\circ\gamma:\theta$
$\displaystyle\mapsto[\sin(\theta)/2:-1/2+\cos(\theta)/2],$

which glue together to give a map of topological Brouwer degree $-1$.

Now we confirm that identity (164) of remark 163 does hold in an example.

######  Example 169.

Consider the unimodular row $F=(2x-1,2z)$ which can be augmented to the
following matrix

$M=\begin{pmatrix}2x-1&-2y\\\ 2z&2x-1\end{pmatrix}$

with determinant $1$. Note that $F$ is homotopic to $g_{1,-1}$ by lemma 66. We
let $F$ act on $\tilde{\pi}$ via the action of definition 77. This yields the
map $L:=F\oplus\tilde{\pi}=(2x-1,2y:2z,-2x+1)_{-1}$. Precomposing its real
realization with $\gamma$ yields the same map as in example 166 where we
showed it has topological Brouwer degree $1$.

Hence our computation confirms

$\displaystyle\deg^{\mathrm{top}}(\Re(L)\circ\gamma)=1=2-1=\deg^{\mathrm{top}}(\Re(F)\circ\gamma)+\deg^{\mathrm{top}}(\Re(\tilde{\pi})\circ\gamma),$

as required in remark 163.

######  Example 170.

Consider now an alternative action $\boxplus$ of
$[\mathcal{J},\mathbb{P}^{1}]^{\mathrm{N}}_{0}$ on
$[\mathcal{J},\mathbb{P}^{1}]^{\mathrm{N}}$ defined as follows. For
$[(A,B)]\in[\mathcal{J},\mathbb{A}^{2}\setminus\\{0\\}]^{\mathrm{N}}$ and
$[s_{0},s_{1}]\in[\mathcal{J},\mathbb{P}^{1}]^{\mathrm{N}}_{n}$, extend the
unimodular row $(A,B)$ to a matrix $M$ in $\mathrm{SL}_{2}(R)$ and define

$[(A,B)]\boxplus[s_{0},s_{1}]:=[M^{T}\cdot(s_{0},s_{1})^{T}].$

Again we look at the unimodular row $F=(A,B)=(2x-1,2z)$ and the matrix $M$ of
example 169. The action $\boxplus$ of $F$ on $\pi$ yields the morphism
$H:=F\boxplus\pi=(2x-1,2z:-2y,2x-1)_{1}=(1,0:0,-1)_{1}$. Taking real
realization and precomposing with $\gamma$ yields the map
$\Re(H)\circ\gamma\colon\mathbb{S}^{1}\to\mathbb{RP}^{1}$ given by

$(\Re(H)\circ\gamma)(\theta)=\begin{cases}[1+\cos(\theta):-\sin(\theta)]&\theta\neq\pi,\\\
[0:1]&\theta=\pi.\end{cases}$

This map has topological Brouwer degree $-1$. Hence our computation shows

$\displaystyle\deg^{\mathrm{top}}(\Re(F\boxplus\pi)\circ\gamma)=-1\neq
3=\deg^{\mathrm{top}}(\Re(F)\circ\gamma)+\deg^{\mathrm{top}}(\Re(\pi)\circ\gamma).$

Thus, by the analogous statement of corollary 162 for $\boxplus$, we see that
$\boxplus$ cannot be used to define an operation compatible with the
conventional group structure.

## References

  * [1] A Ananyevskiy, _On the relation of special linear algebraic cobordism to Witt groups_, Homology Homotopy Appl. 18 (2016) 204–230
  * [2] M Arkowitz, _Introduction to homotopy theory_, Universitext, Springer, New York (2011)
  * [3] A Asok, J Fasel, _Euler class groups and motivic stable cohomotopy_, J. Eur. Math. Soc. (JEMS) 24 (2022) 2775–2822With an appendix by Mrinal Kanti Das
  * [4] A Asok, J Fasel, B Williams, _Motivic spheres and the image of the Suslin–Hurewicz map_, Inventiones mathematicae 219 (2020) 39–73
  * [5] A Asok, M Hoyois, M Wendt, _Affine representability results in $\mathbb{A}^{1}$-homotopy theory, I: vector bundles_, Duke Math. J. 166 (2017) 1923–1953
  * [6] A Asok, M Hoyois, M Wendt, _Affine representability results in $\mathbb{A}^{1}$-homotopy theory, II: Principal bundles and homogeneous spaces_, Geom. Topol. 22 (2018) 1181–1225
  * [7] A Asok, M Hoyois, M Wendt, _Affine representability results in $\mathbb{A}^{1}$-homotopy theory III: finite fields and complements_, Algebr. Geom. 7 (2020) 634–644
  * [8] M F Atiyah, I G Macdonald, _Introduction to commutative algebra_ , Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont. (1969)
  * [9] T Bachmann, M J Hopkins, _$\eta$ -periodic motivic stable homotopy theory over fields_, preprint, arXiv:2005.06778v4 (2020)
  * [10] T Bachmann, K Wickelgren, _Euler classes: six-functors formalism, dualities, integrality and linear subspaces of complete intersections_, J. Inst. Math. Jussieu 22 (2023) 681–746
  * [11] N Bourbaki, _Polynomials and rational fractions_, from: “Algebra II: Chapters 4–7”, Springer Berlin Heidelberg, Berlin, Heidelberg (2003) 1–105
  * [12] C Cazanave, _Théorie homotopique des schémas d’Atiyah et Hitchin_ , PhD thesis, Université Paris-Nord-Paris XIII (2009)
  * [13] C Cazanave, _Algebraic homotopy classes of rational functions_, Annales scientifiques de l’ École Normale Supérieure 45 (2012) 511–534
  * [14] D Dugger, D C Isaksen, _Topological hypercovers and $\mathbb{A}^{1}$-realizations_, Mathematische Zeitschrift 246 (2004) 667–689
  * [15] D Eisenbud, H I Levine, _An algebraic formula for the degree of a $C^{\infty}$ map germ_, Ann. of Math. (2) 106 (1977) 19–44
  * [16] S M Gersten, _Homotopy theory of rings_, J. Algebra 19 (1971) 396–415
  * [17] R Hartshorne, _Algebraic geometry_ , Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg (1977)
  * [18] J L Kass, K Wickelgren, _The class of Eisenbud-Khimshiashvili-Levine is the local $\mathbf{A}^{1}$-Brouwer degree_, Duke Math. J. 168 (2019) 429–469
  * [19] M Levine, A Raksit, _Motivic Gauss-Bonnet formulas_, Algebra Number Theory 14 (2020) 1801–1851
  * [20] J Milnor, D Husemoller, _Symmetric bilinear forms_ , Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 73, Springer-Verlag, New York-Heidelberg (1973)
  * [21] F Morel, _$\mathbb{A}^{1}$ -algebraic topology_, from: “International Congress of Mathematicians. Vol. II”, Eur. Math. Soc., Zürich (2006) 1035–1059
  * [22] F Morel, _$\mathbb{A}^{1}$ -algebraic topology over a field_, volume 2052 of _Lecture Notes in Mathematics_ , Springer, Heidelberg (2012)
  * [23] F Morel, V Voevodsky, _${\bf A}^{1}$ -homotopy theory of schemes_, Inst. Hautes Études Sci. Publ. Math. (1999) 45–143 (2001)
  * [24] R M Switzer, _Algebraic topology—homotopy and homology_, Die Grundlehren der mathematischen Wissenschaften, Band 212, Springer-Verlag, New York-Heidelberg (1975)
  * [25] C A Weibel, _The $K$-book_, volume 145 of _Graduate Studies in Mathematics_ , American Mathematical Society, Providence, RI (2013)