\begin{thm}\label{mainthm}
Suppose that $\Sigma$ is a smooth, projective, geometrically connected curve of genus $g$ defined over $k=\R$, and suppose that $\Sigma(\R)$ is a union of $b+1$ circles, with $b\geq 0$. Then
$$P_t(M_r^d)  =  (-t)^{(g-1)r^2+1} A_{g,r,d} (-t, -t^{\frac{1}{2}}, -t^{\frac{1}{2}},  ..., -t^{\frac{1}{2}}, -1,-1,..,-1) $$
where  $(g-b)$-many entries equal $-t^{\frac{1}{2}}$ and $b$-many equal $-1$.
\end{thm}

\begin{proof}
The homomorphism  $$P_t^{vir}: K_0(Var_\R) \rightarrow \Z[t] \subset \Z((t^{-1}))$$  is constructed by McCrory and Parusinski \cite{MP}.   Since $P_t^{vir}(\mathbb{L} ) = -t$   and $P_t^{vir}(\mathbb{L}^n -1)) = (-t)^n-1$ are both invertible in $\Z((t^{-1}))$, with $((-t)^n-1)^{-1} = (-t)^{-n} (\sum_{k=0}^{\infty} (-t)^{-kn})$, this extends to a homomorphism  $$P_t^{vir}: Mot(\R) \rightarrow   \Z((t^{-1})),$$
which extends naturally to the dimensional completion.
\end{proof}

The zeta function of a variety $X$ is given by the formal power series
$$Z_X(z)  := \sum_{n=0}^\infty  [X^{(n)} ]z^n  \in  Mot(k) [[z]],  $$
where $X^{(n)}  = X^n/S_n $ denotes the $n$-fold symmetric product of $X$.  The zeta function is multiplicative in the sense that for closed $Y \subseteq X$,  $$ Z(X, z) = Z(Y,z) Z(X\setminus Y, z).$$

If $\Sigma/k$ is a smooth projective curve of genus $g$ and $\Sigma(k) \neq \emptyset$, then a theorem of Kapranov (\cite{K}  Theorem 1.1.9) says that

$$ Z_\Sigma(z) =  \frac{P(z)}{(1-z)(1-\mathbb{L}z)} $$
where $P(z)$ is a polynomial of degree $2g$ such that  $Z_\Sigma ( 1/\mathbb{L}z) = \mathbb{L}^{1-g} x^{2-2g} Z_\Sigma(z)$. The homomorphism (\ref{evsigma}) is determined by the property that
$ \prod_{i=1}^g (1-\alpha_i z)(1-\alpha_i^{-1}qz)$ is sent to $P(z)$ in some algebraic extension of $\overline{Mot}(k)$

\section{Betti numbers of $\Sigma^{(n)}(\R)$}\label{Betti numbers of}

Suppose that $\Sigma$ is a smooth, geometrically connected, projective curve of genus $g$ over $\R$. For $n \geq 0$ denote by $\Sigma^{(n)}$ its $n$-th symmetric product. The set of complex points $\Sigma^{(n)}(\C)$ can be identified with the orbit space $$ Sym^n( \Sigma(\C)) := \Sigma(\C)^n/ S_n, $$  while $\Sigma^{(n)}(\R) \subset \Sigma^{(n)}(\C)$ can be identified with $Gal(\C/\R)$ fixed points.

If $\Sigma(\R)$ is empty, then
$$ \Sigma^{(n)}(\R) \cong  \begin{cases}  \emptyset  &  \text{if $n$ is odd} \\  Sym^{n/2}(N_g) & \text{if $n$ is even} \end{cases}$$
where $N_g$ is a non-orientable surface double covered by $\Sigma(\C)$.  We abusively write $ P_t(X)  :=  \sum_{k}  (-t)^k H_{cpt}^k(X;\Z_2)$.  By Macdonald's formula \cite{Mac},
\begin{equation}\label{MacDs}
 \sum_{m=0}^\infty P_t( Sym^{m}(N_g)) z^m = \frac{ (1-tz)^{g+1} }{(1-z)(1-t^2z)},
 \end{equation}
 so

$$ P_t^{vir}(Z_\Sigma (z)) := \sum_{n=0}^{\infty} P_t ( \Sigma^{(n)} ) z^n =  \frac{ (1-tz^2)^{g+1} }{(1-z^2)(1-t^2z^2)}.$$

When $\Sigma(\R) \neq \emptyset$, $P_t(\Sigma^{(n)})$ was calculated in \cite{B}.

\begin{prop}
If $\Sigma(\R)$ is non-empty, with $b+1$ many connected components, then
\begin{equation}\label{zetahom}
 P_t^{vir}(Z_\Sigma (z)) =   \sum_{n=0}^{\infty} P_t ( \Sigma^{(n)}) z^n   =  \frac{(1-tz^2)^{g-b} (1+z)^{b} (1-tz)^{b} }{(1-z)(1+tz)} .
 \end{equation}
\end{prop}

\begin{proof}

Suppose first that $ \Sigma(\C) \setminus \Sigma(\R)$ is connected.  For each pair of integers $k,s \geq 0$ such that $k+2s = n$, $\Sigma^{(n)}(\R)$ has $ { b+1 \choose k}$ connected components with Poincar\'e polynomial isomorphic to $(1-t)^k P_t( Sym^s(N_{g-k}))$.  Therefore, setting $a=b+1$, we have

\begin{eqnarray*}
\sum_{n=0}^{\infty}  P_t(\Sigma^{(n)}) z^m  &=&  \sum_{0 \leq k \leq a} \sum_{s \geq 0}   z^{k+2s} { a \choose k}  (1-t)^k  P_t(  Sym^s(N_{g-k}))\\
&\stackrel{(\ref{MacDs})}{=}&  \sum_{0 \leq k \leq a}   z^{k} { a \choose k}   (1-t)^k \frac{(1-tz^2)^{g-k+1}}{(1-z^2)(1-t^2z^2)}\\
&=& \left(  (z -zt) + (1-tz^2) \right)^a  \frac{(1-tz^2)^{g-a+1}}{(1-z^2)(1-t^2z^2)} \\
&=&  \frac{(1-tz^2)^{g-a+1} (1+z)^{a-1} (1-tz)^{a-1}}{(1-z)(1+tz)} \\
\end{eqnarray*}

In case $\Sigma(\C) \setminus \Sigma(\R)$ is not connected, the calculation is the same except that whenever $k=a$, the corresponding component has Poincar\'e polynomial $(1-t)^{b+1}P_t(Sym^s(\Sigma_{(g-b)/2})) $, where $\Sigma_g$ is an orientable surface of genus $g$. But this equals $(1-t)^{b+1} P_t( Sym^s( N_{g-b-1}))$  (\cite{B} Remark 3.14) so the same formula holds.
\end{proof}

\begin{proof}

Under these conditions $ X^{\C^\times} = \coprod_{i \in I} F_i$  where the $F_i$ are non-singular, projective over $\R$, and we have a Bialycki-Birula stratification $ X = \bigcup_{i \in I} U_i$ where $$U_i(\C) :=  \{x \in X(\C) | lim_{\lambda \mapsto 0} \lambda x  \in F_i \} ,$$ and each $U_i$ is a locally closed subvarity of $X$, isomorphic to a vector bundle of some rank $d_i$ over $F_i$. It follows that

$$ P_t^{vir}(X) = \sum_{i\in I} P_t^{vir}(U_i) =   \sum_{i\in I} P_t^{vir}( F_i) P_t^{vir}(\mathbb{A}^{d_i}) =  \sum_{i\in I} P_t(F_i) (-t)^{d_i}.$$

On the other hand, by a result essentially due to Duistermaat \cite{D}  (see also \cite{BGH}), we know that $ P_t( X) = \sum_i  P_t (U_i) $,  and $U_i(\R)$ deformation retracts onto $F_i(\R)$ and has relative dimension $d_i$, so by Poincar\'e duality,  $P_t(U_i)=P_t(F_i) (-t)^{d_i}$, completing the proof.
\end{proof}

\section{Average Betti numbers}\label{Average Betti numbers}

Our goal in this section is to prove the following somewhat surprising result.

\begin{prop}\label{avgbet}
Suppose that $\Sigma$ is a smooth projective, geometrically connected curve of genus $g$ defined over $\R$,  and $\Sigma(\R)$ has $b+1 >0$ connected components.
Then the Poincar\'e polynomial $P_t( M_r^d)$ is divisible by $2^b (1-t)^g$.
\end{prop}

\begin{proof}
Recall (\ref{Plethlog}):
\begin{equation}
 \sum_{r=1}^\infty H_{g,r} T^r = (z-1)(1-q) \operatorname{Log} \sum_{\mu \in \mathcal{P}} T^{|\mu|} \mathcal{H}_\mu.
\end{equation}

Every non-empty Young diagram $\mu$ includes at least one box with both arm and leg length zero, so $\mathcal{H}_\mu$ contains a factor of $ \prod_{i=1}^g (z - \alpha_i )(1 - \alpha_i^{-1} q)$ in its numerator. Using the combinatorics of the plethystic logarithm  (see e.g.  \cite{HRV08} Theorem 3.5.2.), it follows that every $H_{g,r}$ is divisible by $ \prod_{i=1}^g (z - \alpha_i )(1 - \alpha_i^{-1} q)$ in the localized ring $S^{-1} \Q[ q, z, \alpha_1^{\pm 1},...,\alpha_g^{\pm1}] $ where $S$ is the multiplicative set generated
by  $\{  (z^{a+1} - q^{l} ),  (z^{a}-q^{l+1})|  a,b \geq 0\}$. We know
$$A_{g,r} (q, \alpha_1,..., \alpha_g) = H_{g,r}(q,1,\alpha_1,...,\alpha_g) \in \Z[q, \alpha_1^{\pm 1},..., \alpha_g^{\pm 1} ]$$
so by Gauss' Lemma $\prod_{i=1}^g (1-\alpha_i) (1-\alpha_i^{-1}q)$ divides $  A_{g,r}$ in $ \Z[q, \alpha_1^{\pm 1},..., \alpha_g^{\pm 1} ]$ .  By Theorem \ref{mainthm},   $P_t( M_r^d) = A_{g,r}(-t, -t^{\frac{1}{2}},...,-1,...) $  is divisible by
$$2^b(1-t)^g = (1+t^{\frac{1}{2}})^{g-b}  (1-t^{\frac{1}{2}})^{g-b} (1+1)^b (1-t)^b $$
 in $\Z[t^{\pm \frac{1}{2}}]$, hence also in $ \Z[t]$.
 \end{proof}

\begin{proof}
Substituting  $q = -t$,  $\alpha_1=...= \alpha_{g-b} = -t^{\frac{1}{2}}$ and $\alpha_{g-b+1}= ... =\alpha_g = -1$ into the formal zeta function $$\frac{ \prod_{i=1}^g (1-\alpha_i z)(1-\alpha_i^{-1}qz)}{(1-z)(1-qz)} $$
yields (\ref{zetahom}). Apply (\ref{evfor}).
\end{proof}

\section{On purity of real semi-projective varieties}\label{On purity of real}

A semi-projective variety $Y$ over $\C$ is a smooth, quasiprojective variety equipped with $\C^\times$-action such that the fixed point set $Y^{\C^\times}$ is proper and for ever $y \in Y$ the limit $lim_{\lambda \rightarrow 0} \lambda y$ exists as $\lambda \in \C^\times$ tends to zero.  Mixed Hodge structures of such varieties were studied in Hausel and Rodriguez-Villegas \cite{HRV13} .

A semi-projective variety $X$ over $\R$ is a real variety equipped with an action of $\R^\times$ such that the base change  $X \times_R \C$ is semi-projective over $\C$.  When $\Sigma$ is defined over $\R$,  $M_r^d$ is a semi-projective variety, with respect the $\C^\times$ action of scaling the Higgs field.

\begin{prop}
If $X$ is semi-projective over $\R$ then

$$P_t^{vir} (X) = P_t (X).$$
\end{prop}

\begin{proof}

Under these conditions $ X^{\C^\times} = \coprod_{i \in I} F_i$  where the $F_i$ are non-singular, projective over $\R$, and we have a Bialycki-Birula stratification $ X = \bigcup_{i \in I} U_i$ where $$U_i(\C) :=  \{x \in X(\C) | lim_{\lambda \mapsto 0} \lambda x  \in F_i \} ,$$ and each $U_i$ is a locally closed subvarity of $X$, isomorphic to a vector bundle of some rank $d_i$ over $F_i$. It follows that

$$ P_t^{vir}(X) = \sum_{i\in I} P_t^{vir}(U_i) =   \sum_{i\in I} P_t^{vir}( F_i) P_t^{vir}(\mathbb{A}^{d_i}) =  \sum_{i\in I} P_t(F_i) (-t)^{d_i}.$$

On the other hand, by a result essentially due to Duistermaat \cite{D}  (see also \cite{BGH}), we know that $ P_t( X) = \sum_i  P_t (U_i) $,  and $U_i(\R)$ deformation retracts onto $F_i(\R)$ and has relative dimension $d_i$, so by Poincar\'e duality,  $P_t(U_i)=P_t(F_i) (-t)^{d_i}$, completing the proof.
\end{proof}

Let $\Sigma/k$ be a smooth, projective, geometrically connected curve of genus $g$ over a field $k$ admitting a $k$-rational point, and denote by $M_r^d$ the moduli space of stable Higgs bundles of rank $r$ and degree $d$ over $\Sigma$. When $gcd(d,r)=1$, which we assume henceforth, $M_r^d$ is a smooth, quasi-projective variety.

For $g,r,d \in \mathbb{Z}$ and $g, r \geq 0$, Schiffmann \cite{S} defines a Laurent polynomial
$$ A_{g,r,d} (q, \alpha_1,..., \alpha_g) \in \mathbb{Z}[q, \alpha_1^{\pm 1},..., \alpha_g^{\pm 1} ]$$
which is symmetric under permutations of the $\alpha_i$ and under $\alpha_i \mapsto q \alpha_i^{-1}$. If $k = \mathbb{F}_q$ and $\{ \alpha_i, \alpha_i^{-1}q|i=1,..,g\}$ are set to the eigenvalues of the Frobenius action on $H^1_{et}(\Sigma, \Q_l)$ for $(q,l)=1$, then $A_{g,r,d}$ counts indecomposable bundles of rank $r$ and degree $d$ over $\Sigma$, and $q^{1+(g-1)r^2}A_{g,r,d}$ counts points in $M_r^d(\mathbb{F}_q)$. Using the Artin-Grothendieck Comparision Theorem, it follows that that when $k = \mathbb{C}$, the Poincar\'e polynomial for the compactly supported rational Betti numbers of $M_r^d(\mathbb{C})$ equals

$$ \sum_k (-1)^kt^k \dim_\mathbb{Q} H^k_{cpt}(M_r^d(\mathbb{C});\mathbb{Q}) = t^{2(g-1)r^2+2}A_{g,r,d} (t^2, t,t,t,...,t,t) $$

Our main result is that a similar formula calculates $\Z_2$ Betti numbers of $M_r^d(\mathbb{R})$ when $k =\mathbb{R}$. For a variety $X/\mathbb{R}$, denote its compactly supported Poincar\'e polynomial by
 $$P_t( X) := \sum_k (-1)^k t^k \dim_{\Z_2} H^k_{cpt}(M_r^d(\mathbb{R}); \Z_2).$$

Let $\Sigma/k$ be a smooth, projective, geometrically connected curve of genus $g$ over a field $k$ admitting a $k$-rational point, and denote by $M_r^d$ the moduli space of stable Higgs bundles of rank $r$ and degree $d$ over $\Sigma$. When $gcd(d,r)=1$, which we assume henceforth, $M_r^d$ is a smooth, quasi-projective variety.

For $g,r,d \in \Z$ and $g, r \geq 0$, Schiffmann \cite{S} defines a Laurent polynomial
$$ A_{g,r,d} (q, \alpha_1,..., \alpha_g) \in \Z[q, \alpha_1^{\pm 1},..., \alpha_g^{\pm 1} ]$$
which is symmetric under permutations of the $\alpha_i$ and under $\alpha_i \mapsto q \alpha_i^{-1}$. If $k = \mathbb{F}_q$ and $\{ \alpha_i, \alpha_i^{-1}q|i=1,..,g\}$ are set to the eigenvalues of the Frobenius action on $H^1_{et}(\Sigma, \Q_l)$ for $(q,l)=1$, then $A_{g,r,d}$ counts indecomposable bundles of rank $r$ and degree $d$ over $\Sigma$, and $q^{1+(g-1)r^2}A_{g,r,d}$ counts points in $M_r^d(\mathbb{F}_q)$. Using the Artin-Grothendieck Comparision Theorem, it follows that that when $k = \C$, the Poincar\'e polynomial for the compactly supported rational Betti numbers of $M_r^d(\C)$ equals

Our main result is that a similar formula calculates $\Z_2$ Betti numbers of $M_r^d(\R)$ when $k =\R$. For a variety $X/\R$, denote its compactly supported Poincar\'e polynomial by
 $$P_t( X) := \sum_k (-1)^k t^k \dim_{\Z_2} H^k_{cpt}(M_r^d(\R); \Z_2).$$

For example, for $r=2$, we obtain the explicit formula $$P_{t}(M_2^1) = 2^b t^{4(g-1)+1}(1-t)^g f(t^{1/2},1)$$ where

\begin{equation}\label{evsigma}
 ev_\Sigma : R_g \rightarrow \overline{Mot}(k),
 \end{equation} where $R_g$ is the ring of Laurent polynomials in $q, \alpha_1^{\pm 1},...,\alpha_g^{\pm 1}$ which are invariant under permutations of $\alpha_i$ and under $\alpha_i \mapsto q \alpha_i^{-1}$. This $ev_\Sigma$ sends $q$ to the Tate class $\mathbb{L}$ and the extension to $R_g [[z]] \rightarrow \overline{Mot}(k)[[z]]$ sends the formal zeta function of the curve to the motivic zeta function of $\Sigma$ :

Suppose that $\Sigma$ is a smooth, geometrically connected, projective curve of genus $g$ over $\R$. For $n \geq 0$ denote by $\Sigma^{(n)}$ its $n$-th symmetric product. The set of complex points $\Sigma^{(n)}(\C)$ can be identified with the orbit space $$ Sym^n( \Sigma(\C)) := \Sigma(\C)^n/ S_n, $$ while $\Sigma^{(n)}(\R) \subset \Sigma^{(n)}(\C)$ can be identified with $Gal(\C/\R)$ fixed points.

\begin{prop}
If $\Sigma(\R)$ is non-empty, with $b+1$ many connected components, then
\begin{equation}\label{zetahom}
 P_t^{vir}(Z_\Sigma (z)) = \sum_{n=0}^{\infty} P_t ( \Sigma^{(n)}) z^n = \frac{(1-tz^2)^{g-b} (1+z)^{b} (1-tz)^{b} }{(1-z)(1+tz)} .
 \end{equation}
\end{prop}

\begin{prop}\label{avgbet}
Suppose that $\Sigma$ is a smooth projective, geometrically connected curve of genus $g$ defined over $\R$, and $\Sigma(\R)$ has $b+1 >0$ connected components.
Then the Poincar\'e polynomial $P_t( M_r^d)$ is divisible by $2^b (1-t)^g$.
\end{prop}

For example, for $r=2$, we obtain the explicit formula $$P_{t}(M_2^1) =  2^b t^{4(g-1)+1}(1-t)^g f(t^{1/2},1)$$ where

\begin{equation*}
\begin{split}
f(t^{\frac{1}{2}},z) &=  \frac{(z^2+t^{\frac{1}{2}})^{g-b} (z-t^{\frac{1}{2}})^{g-b} (z^2+1)^b(z-t)^b}{(z^2-1)(z+t)} + \frac{(z-t^{\frac{3}{2}})^{g-b} (1+t^{\frac{3}{2}})^{g-b} (z-t)^b(1+t^2)^b}{(z+t)(1-t^2)}\\
  &- \frac{(z+t^{\frac{1}{2}})^{g-b} (1-t^{\frac{1}{2}})^{g-b} (z+1)^b(1-t)^b}{2(z-1)(1+t)} - \frac{(z-t^{\frac{1}{2}})^{g-b} (1+t^{\frac{1}{2}})^{g-b} (z-1)^b(1+t)^b}{2(z+1)(1-t)}.
\end{split}
\end{equation*}
A different formula for $P_t(M_2^1)$ was obtained in \cite{B}.  Formulas for $P_t(M_r^d)$ when $r\geq 3$ are new.

We use a construction of $A_{g,r,d}$ due to Mellit \cite{M}.  Let $\mathcal{P}$ denote the set of Young diagrams.  Given $\mu \in \mathcal{P}$, define the rational function $\mathcal{H}_\mu$ in variables $ q, z, \alpha_1,..., \alpha_g$,

$$\mathcal{H}_\mu : =  \prod_{\square \in \mu}  \frac{\prod_{i=1}^g (z^{a(\square)+1} - \alpha_i q^{l(\square)})(z^{a(\square)} - \alpha_i^{-1} q^{l(\square)+1})}{(z^{a(\square)+1} - q^{l(\square)})(z^{a(\square)} - q^{l(\square)+1})}$$
where $a(\square)$ and $l(\square)$ denote the arm and leg lengths of boxes in the diagram.  For $g \geq 1$, define rational functions $H_{g,r}$ by
\begin{equation} \label{Plethlog}
 \sum_{r=1}^\infty H_{g,r} T^r = (z-1)(1-q) \operatorname{Log} \sum_{\mu \in \mathcal{P}} T^{|\mu|} \mathcal{H}_\mu,
\end{equation}
where $\operatorname{Log}$ denotes the plethystic logarithm.  Mellit proved that for all $r \geq 1$, $H_{g,r}$ is a Laurent polynomial in $q$, $z$, and $\alpha_1, \ldots, \alpha_g$, and for all $d$,
\[
A_{g,r,d}(q, \alpha_1, \ldots, \alpha_g) = H_{g,r}(q, 1, \alpha_1, \ldots, \alpha_g).
\]

Note that $A_{g,r,d}$ is independent of $d$, so we denote it $A_{g,r}$ henceforth.   When applied to $k=\mathbb{C}$, Mellit's result proved a conjectural formula of Hausel and Rodriguez-Villegas for the $\mathbb{Q}$-Betti numbers of $M_r^d$.   Theorem \ref{mainthm} can be thought of as a Hausel Rodriguez-Villegas style formula for the $\Z_2$-Betti number of the moduli space of real Higgs bundles over a real curve.

Our proof of Theorem \ref{mainthm} can be outlined as follows.  Fedorov-Soibelman-Soibelman  \cite{FSS} proved a motivic version of Schiffmann's formula which, when $k$ is a field of characteristic zero, identifies the class $[M_r^d]$ in $\overline{Mot}(k)$, the dimensional completion of the Grothendieck ring of Artin stacks over $k$.  Mellit (\cite{M} \S 6), identifies their class $[M_r^d]$ as the image of $q^{(g-1)r^2+1} A_{r,d}$  under a homomorphism

\begin{equation}\label{evsigma}
 ev_\Sigma  :  R_g  \rightarrow \overline{Mot}(k),
 \end{equation} where $R_g$ is the ring of Laurent polynomials in $q, \alpha_1^{\pm 1},...,\alpha_g^{\pm 1}$ which are invariant under permutations of $\alpha_i$ and under $\alpha_i \mapsto q \alpha_i^{-1}$.  This $ev_\Sigma$ sends $q$ to the Tate class $\mathbb{L}$ and the extension to $R_g [[z]] \rightarrow \overline{Mot}(k)[[z]]$ sends the formal zeta function of the curve to the motivic zeta function of $\Sigma$ :

\begin{equation}\label{evfor}
 ev_\Sigma \left( \frac{ \prod_{i=1}^g (1-\alpha_i z)(1-\alpha_i^{-1}qz)}{(1-z)(1-qz)}\right) = \sum_{n=0}^{\infty} z^n  \Sigma^{(n)} = Z_\Sigma (z).
 \end{equation}

When $k= \mathbb{R}$, one can define a virtual Poincar\'e polynomial homomorphism $$P_t^{vir}:  \overline{Mot}(\mathbb{R}) \rightarrow \mathbb{Z}((t^{-1})) ,$$ which sends the class $[X]$ of a smooth, projective variety $X$ to the Poincar\'e polynomial  $  P_t^{vir}([X]) = P_t(X(\mathbb{R}))$.

To calculate $P_t^{vir}( [M_r^d])$, it only remains  to determine the composition $P_t^{vir} \circ ev_\Sigma$, which boils down to calculating $P_t^{vir} ( Z_\Sigma(z))$. We carry this out in \S \ref{Betti numbers of}.  Then in \S \ref{On purity of real} we prove that the virtual Poincar\'e polynomial $M_r^d$ agrees with its topological Poincar\'e polynomial, completing the proof of Theorem \ref{mainthm}.  In section \ref{Average Betti numbers} we show that the Poincar\'e polynomial $P_t(M_r^d)$ is divisible by $P_t(Pic^0(\Sigma))$, which has the surprising implication that average value of the Poincar\'e polynomial of the connected components of $M_r^d$ is a polynomial with integer coefficients.

Our proof of Theorem~\ref{mainthm} can be outlined as follows.

- Fedorov--Soibelman--Soibelman \cite{FSS} prove a motivic version of Schiffmann's formula which (in characteristic zero) "identifies the class $[M_r^d]$ in $\overline{Mot}(k)$," the dimensional completion of the Grothendieck ring of Artin stacks.

- Mellit (\cite{M} \S 6) "identifies their class $[M_r^d]$ as the image of $q^{(g-1)r^2+1} A_{r,d}$ under" the homomorphism
  \[
    ev_\Sigma: R_g \to \overline{Mot}(k),
  \]
  where $R_g$ is the ring of Laurent polynomials in $q,\alpha_1^{\pm1},\ldots,\alpha_g^{\pm1}$ invariant under permutations of the $\alpha_i$ and under $\alpha_i\mapsto q\alpha_i^{-1}$. This map sends $q$ to the Tate class $\mathbb{L}$, and its extension to $R_g[[z]]$ sends the "formal zeta function of the curve" to the motivic zeta function $Z_\Sigma(z)$ via
  \[
  ev_\Sigma\left(\frac{\prod_{i=1}^g(1-\alpha_i z)(1-\alpha_i^{-1}qz)}{(1-z)(1-qz)}\right)=\sum_{n\ge0} z^n\Sigma^{(n)}=Z_\Sigma(z).
  \]

- For $k=\mathbb{R}$, define the virtual Poincar\'e polynomial homomorphism
  \[
  P_t^{vir}:\overline{Mot}(\mathbb{R})\to\mathbb{Z}((t^{-1})),
  \]
  which sends a smooth projective variety $X$ to $P_t^{vir}([X])=P_t(X(\mathbb{R}))$.

- "To calculate $P_t^{vir}([M_r^d])$, it only remains to determine the composition $P_t^{vir}\circ ev_\Sigma$, which boils down to calculating $P_t^{vir}(Z_\Sigma(z))$." This is carried out in \S~\ref{Betti numbers of}.

- Then, in \S~\ref{On purity of real}, they "prove that the virtual Poincar\'e polynomial $M_r^d$ agrees with its topological Poincar\'e polynomial, completing the proof of Theorem~\ref{mainthm}."

Our proof of Theorem~\ref{mainthm} can be outlined as follows.

- Fedorov--Soibelman--Soibelman \cite{FSS} prove a motivic version of Schiffmann's formula which (in characteristic zero) "identifies the class $[M_r^d]$ in $\overline{Mot}(k)$," the dimensional completion of the Grothendieck ring of Artin stacks.

- Mellit (\cite{M} \S 6) "identifies their class $[M_r^d]$ as the image of $q^{(g-1)r^2+1} A_{r,d}$ under" the homomorphism
  \[
    ev_\Sigma: R_g \to \overline{Mot}(k),
  \]
  where $R_g$ is the ring of Laurent polynomials in $q,\alpha_1^{\pm1},\ldots,\alpha_g^{\pm1}$ invariant under permutations of the $\alpha_i$ and under $\alpha_i\mapsto q\alpha_i^{-1}$. This map sends $q$ to the Tate class $\mathbb{L}$, and its extension to $R_g[[z]]$ sends the "formal zeta function of the curve" to the motivic zeta function $Z_\Sigma(z)$ via
  \[
  ev_\Sigma\left(\frac{\prod_{i=1}^g(1-\alpha_i z)(1-\alpha_i^{-1}qz)}{(1-z)(1-qz)}\right)=\sum_{n\ge0} z^n\Sigma^{(n)}=Z_\Sigma(z).
  \]

- For $k=\mathbb{R}$, define the virtual Poincar\'e polynomial homomorphism
  \[
  P_t^{vir}: \overline{Mot}(\mathbb{R})\to\mathbb{Z}((t^{-1})),
  \]
  which sends a smooth projective variety $X$ to $P_t^{vir}([X])=P_t(X(\mathbb{R}))$. More precisely, one constructs
  \[
  P_t^{vir}: K_0(Var_\R) \rightarrow \Z[t] \subset \Z((t^{-1}))
  \]
  (McCrory and Parusinski \cite{MP}). Since $P_t^{vir}(\mathbb{L} ) = -t$ and $P_t^{vir}(\mathbb{L}^n -1)) = (-t)^n-1$ are both invertible in $\Z((t^{-1}))$, with $((-t)^n-1)^{-1} = (-t)^{-n} (\sum_{k=0}^{\infty} (-t)^{-kn})$, this extends to a homomorphism
  \[
  P_t^{vir}: Mot(\R) \rightarrow   \Z((t^{-1})),
  \]
  which extends naturally to the dimensional completion.

- To calculate $P_t^{vir}([M_r^d])$, it only remains to determine the composition $P_t^{vir}\circ ev_\Sigma$, which boils down to calculating $P_t^{vir}(Z_\Sigma(z))$. Next we carry out this calculation as follows.

  The zeta function of a variety $X$ is given by the formal power series
  \[
  Z_X(z)  := \sum_{n=0}^\infty  [X^{(n)} ]z^n  \in  Mot(k) [[z]],
  \]
  where $X^{(n)}  = X^n/S_n $ denotes the $n$-fold symmetric product of $X$. The zeta function is multiplicative in the sense that for closed $Y \subseteq X$,
  \[
  Z(X, z) = Z(Y,z) Z(X\setminus Y, z).
  \]

  If $\Sigma/k$ is a smooth projective curve of genus $g$ and $\Sigma(k) \neq \emptyset$, then a theorem of Kapranov (\cite{K}  Theorem 1.1.9) says that
  \[
  Z_\Sigma(z) =  \frac{P(z)}{(1-z)(1-\mathbb{L}z)}
  \]
  where $P(z)$ is a polynomial of degree $2g$ such that $Z_\Sigma ( 1/\mathbb{L}z) = \mathbb{L}^{1-g} x^{2-2g} Z_\Sigma(z)$.

  We then compute $P_t^{vir}(Z_\Sigma(z))$ by analyzing the real loci $\Sigma^{(n)}(\R)$. In the case $\Sigma(\R)=\emptyset$ one has
  \[
  \Sigma^{(n)}(\R) \cong  \begin{cases}  \emptyset  &  \text{if $n$ is odd} \\  Sym^{n/2}(N_g) & \text{if $n$ is even} \end{cases}
  \]
  (with $N_g$ a non-orientable surface double covered by $\Sigma(\C)$), and writing abusively $ P_t(X)  :=  \sum_{k}  (-t)^k H_{cpt}^k(X;\Z_2)$, Macdonald's formula (\cite{Mac}) gives
  \begin{equation}\label{MacDs}
   \sum_{m=0}^\infty P_t( Sym^{m}(N_g)) z^m = \frac{ (1-tz)^{g+1} }{(1-z)(1-t^2z)},
   \end{equation}
  hence
  \[
   P_t^{vir}(Z_\Sigma (z)) := \sum_{n=0}^{\infty} P_t ( \Sigma^{(n)} ) z^n =  \frac{ (1-tz^2)^{g+1} }{(1-z^2)(1-t^2z^2)}.
  \]

  When $\Sigma(\R) \neq \emptyset$, with $b+1$ connected components, one has the explicit identity
  \begin{equation}\label{zetahom}
   P_t^{vir}(Z_\Sigma (z)) =   \sum_{n=0}^{\infty} P_t ( \Sigma^{(n)}) z^n   =  \frac{(1-tz^2)^{g-b} (1+z)^{b} (1-tz)^{b} }{(1-z)(1+tz)} .
   \end{equation}

- Finally, we use purity for real semi-projective varieties. We first use the following proposition: if $X$ is semi-projective over $\R$ then

$$P_t^{vir} (X) = P_t (X).$$

Since, when $\Sigma$ is defined over $\R$, $M_r^d$ is semi-projective (with respect the $\C^\times$ action of scaling the Higgs field), this shows that the virtual Poincar\'e polynomial of $M_r^d$ agrees with its topological Poincar\'e polynomial. This completes the proof of the main theorem.

\label{mainthm}
Suppose that $\Sigma$ is a smooth, projective, geometrically connected curve of genus $g$ defined over $k=\mathbb{R}$, and suppose that $\Sigma(\mathbb{R})$ is a union of $b+1$ circles, with $b\geq 0$. Then
$$P_t(M_r^d)  =  (-t)^{(g-1)r^2+1} A_{g,r,d} (-t, -t^{\frac{1}{2}}, -t^{\frac{1}{2}},  ..., -t^{\frac{1}{2}}, -1,-1,..,-1) $$
where  $(g-b)$-many entries equal $-t^{\frac{1}{2}}$ and $b$-many equal $-1$.,

Question:
Let \Sigma/\mathbb{R}\u001f be a smooth, projective, geometrically connected curve of genus \(g\), and let \(M_r^d\) denote the moduli space of stable Higgs bundles of rank \(r\) and degree \(d\) over \(\Sigma\). For a real variety \(X/\mathbb{R}\), define its compactly supported Poincare9 polynomial by
\[
P_t(X)=\sum_k (-1)^k t^k \dim_{\mathbb{Z}_2} H^k_{cpt}(X(\mathbb{R});\mathbb{Z}_2).
\]
Also let \(A_{g,r,d}(q,\alpha_1,\dots,\alpha_g)\in \mathbb{Z}[q,\alpha_1^{\pm1},\dots,\alpha_g^{\pm1}]\) be Schiffmanns Laurent polynomial, which is symmetric in the \(\alpha_i\). Suppose that \(\Sigma(\mathbb{R})\) is a union of \(b+1\) circles, with \(b\ge 0\). Which statement holds for every such curve \(\Sigma\)?

\[
P_t(M_r^d)=(-t)^{(g-1)r^2+1}\,A_{g,r,d}\bigl(-t,\underbrace{-t^{1/2},\dots,-t^{1/2}}_{g-b\text{ entries}},\underbrace{-1,\dots,-1}_{b\text{ entries}}\bigr).
\]

Choices:
B.
\[
P_t(M_r^d)=(-t)^{(g-1)r^2+1}\,A_{g,r,d}\bigl(-t,\underbrace{-t^{1/2},\dots,-t^{1/2}}_{g\text{ entries}}\bigr).
\]
C.
\[
P_t(M_r^d)=(-t)^{(g-1)r^2+1}\,A_{g,r,d}\bigl(-t,\alpha_1,\dots,\alpha_g\bigr)
\]
for some ordering of \(\alpha_1,\dots,\alpha_g\in\{-t^{1/2},-1\}\) with exactly \(g-b\) of them equal to \(-t^{1/2}\) and exactly \(b\) of them equal to \(-1\).
D.
\[
P_t(M_r^d)=(-t)^{(g-1)r^2+1}\,A_{g,r,d}\bigl(t,\underbrace{-t^{1/2},\dots,-t^{1/2}}_{g-b\text{ entries}},\underbrace{-1,\dots,-1}_{b\text{ entries}}\bigr).
\]
E.
\[
P_t(M_r^d)=(-t)^{(g-1)r^2+1}\,A_{g,r,d}\bigl(-t,\underbrace{-1,\dots,-1}_{g-b\text{ entries}},\underbrace{-t^{1/2},\dots,-t^{1/2}}_{b\text{ entries}}\bigr).
\]

dependence on the real-topology parameter b

dropped the explicit block ordering of the specialized \alpha_i

the evaluation of q under \(P_t^{vir}(\mathbb{L})=-t\)

assignment of the counts \(g-b\) and \(b\) to \(-t^{1/2}\) versus \(-1\)

The stem does not state the formula or uniquely signal the correct choice. It introduces the geometric setup and asks for the explicit specialization, but the exact signs and multiplicities are not leaked.

This is essentially a direct recall of a specific theorem/formula: the question asks which explicit formula holds for the Poincare polynomial. It does not substantially reframe the result or ask for a derived consequence.

There is some pressure to discriminate among close alternatives (swapped multiplicities, sign errors, wrong number of specialization slots, weaker-but-true formulation), but the task mainly tests theorem recall rather than genuine derivation or generative reasoning.

The distractors are strong and mathematically plausible: B swaps the counts of the two specialization values, D introduces a natural sign/realization confusion, E violates the required number of parameters, and C is a weaker true statement that fails the demand for an explicit formula.

A solid multiple-choice item for testing precise theorem recall, with high-quality distractors and little answer leakage, but it is largely tautological and only moderately probes reasoning.


\begin{theorem} \label{thm homeo}
The map
\begin{equation*}
 \Psi_{\ge0}\colon Y_{\ge 0}\rightarrow X(\Sigma)_{\ge0}
 \quad ; \quad
 gB \mapsto [\Delta_{\varpi_1}(g),\ldots,\Delta_{\varpi_{\rkg}}(g);\qp{1}(g),\ldots,\qp{n}(g)]
\end{equation*}
is a homeomorphism satisfying
\begin{equation*}
\Psi_{\ge 0}(\RY{K}{J;>0}) = \Xp{K,J}
\end{equation*}
for $K\subseteq J \subseteq I$.
\end{theorem}

\begin{proposition}\label{prop description of YKJ}
For $K\subseteq J\subseteq I$, we have
\begin{align*}
 \RY{K}{J}
 = \left\{ x\dot{w}_J B\in G/B \ \left| \ x \in (U_J)^{e_J} \text{ and }
 \begin{cases}
 \Delta_{\varpi_i}(x\dot{w}_J)=0 \quad \text{if $i\in K$}, \\
 \Delta_{\varpi_i}(x\dot{w}_J)\ne0 \quad \text{if $i\in I-K$}
 \end{cases}
 \right.
 \right\},
\end{align*}
where $(U_J)^{e_J}=U_J\cap Z_{G_J}(e_J)$.
\end{proposition}

\begin{proposition}\label{prop restrict}
The morphism $\Psi \colon Y\to X(\fan)$ restricts to a continuous map
\begin{equation*}
\Psi_{\ge 0} \colon Y_{\ge 0}\to X(\fan)_{\ge 0}
\end{equation*}
which sends $\RY{K}{J;>0}$ to $\Xp{K,J}$ for $K\subseteq J \subseteq  I$.
\end{proposition}

\begin{proposition}\label{prop Delta is nonnegative}
For $x\in U_{\ge0}$ and $w\in W$, we have
\begin{equation*}
 \Delta_{\varpi_i}(x\dot{w}) \ge0
 \qquad (i\in I),
\end{equation*}
where $\Delta_{\varpi_i}$ is the function defined in \eqref{eq def of Delta}.
\end{proposition}

\begin{proposition}\label{prop description for R KI>0 2}
Let $K\subseteq J\subseteq I$. Then, we have
\begin{equation*}
\RY{K}{J;>0}=\left\{x\dot{w}_JB \ \left| \
\begin{matrix}
\text{
\rm $x\in (U_J)^{e_J}_{\ge0}$ and}
~\begin{cases}\Delta_{\varpi_i}(x\dot{w}_J)=0\quad\text{if}\quad i\in K\\
\Delta_{\varpi_i}(x\dot{w}_J)> 0\quad\text{if}\quad i\in J-K \\
\Delta_{\varpi_i}(x\dot{w}_J)=1 \quad\text{if}\quad i\in I-J
\end{cases}
\end{matrix}
\right.
\right\}.
\end{equation*}
\end{proposition}

\begin{proposition}\label{prop LR and our}
Let $i\in I$. For all $x\in U^e_{\ge0}$, we have
\begin{align*}
 \Delta_i([\dot{w}_0^{-1}x\dot{w}_0]) = (\Delta_{\varpi_i}(x\dot{w}_0))^{m_i}.
\end{align*}
\end{proposition}

\begin{theorem} \label{thm main}
    The totally nonnegative part of the Peterson variety is homeomorphic to the strongly dominant weight polytope as a cell-decomposed space. In particular, it is a regular CW-complex, which is homeomorphic to a cube.
\end{theorem}

\begin{proposition}\label{prop nonnengative stratum parametrization}
For $K\subseteq J\subseteq I$, the map
\begin{align*}
 (\R^{I})_{K,J;>0} \rightarrow \Xp{K,J}
 \quad ; \quad
 (x_1,\ldots,x_{\rkg}) \mapsto [x_1,\ldots,x_{\rkg}\hspace{2pt};\delta_1,\ldots,\delta_{\rkg}]
\end{align*}
is a bijection, where $(\delta_1,\ldots,\delta_{\rkg})\in\R^{I}$ is the element given by
\begin{align*}
\begin{cases}
 \delta_i=0 \quad (i\in I-J) \\
 \delta_i=1 \quad (i\in J) .
\end{cases}
\end{align*}
\end{proposition}

\label{sec Introduction}
Let $G$ be a semisimple algebraic group over $\mathbb C$ with a Borel subgroup $B\subset G$. The Peterson variety $Y$ is a certain remarkable subvarieties of the flag variety $G/B$, introduced by Dale Peterson \cite{peterson1997quantum} to realize quantum cohomology rings of all Langlands dual partial flag varieties geometrically. By using the geometry of $Y$, he discovered a connection of the quantum cohomology of those flag varieties with the homology of the affine Grassmannian $\mathcal{G} r_{G^{\vee}}$ of the Langlands dual group $G^{\vee}$, which was verified by Lam--Shimozono in \cite{lam2010quantum}. It is also closely related to the wonderful compactification of a certain unipotent subgroup of $G$ \cite{Bualibanu2017Peterson}. The geometry and topology of the Peterson variety have been studied extensively, see, for example, \cite{abe2024geometry,abe2023peterson,Bualibanu2017Peterson,Goldin2024positivity,gui2025structure,harada2015equivariant,kostant1996flag}.

The theory of total positivity for reductive algebraic groups was pioneered by Lusztig \cite{lusztig1994total} as a broad extension of the classical theory of Schoenberg and Gantmacher--Krein on total positivity for matrices. It has close connections to cluster algebras \cite{fomin2010total}, KP equations \cite{kodama2014kp,kodama2011kp}, Poisson geometry \cite{lu2020bott}, and the physics of scattering amplitudes \cite{williams2021positive}. Lusztig \cite{lusztig1994total,lusztig1998total} also defined the totally positive and the totally nonnegative parts of the flag variety, which naturally induces the corresponding definitions for subvarieties of the flag variety. Of particular interests are so-called {\it regularity theorems} on CW-complex structures on the totally non-negative parts of these varieties, see \cite{bao2024product,galashin2022regularity,hersh2014regular} for the most recent developments. Note that convex polytopes are prototypical examples of regular CW
complexes and the topology of a regular CW complex is completely determined by the combinatorial
structure of its associated cell closure poset \cite{bjorner1984posets}.

The interaction bewteen the Peterson variety and the total positivity was first studied by Rietsch. In \cite{rietsch2003totally}, she used Peterson's theory in type A to obtain a parametrization of totally nonnegative Toeplitz matrices. In \cite{Rietsch2006}, she gave a mirror construction of the totally nonnegative part of the Peterson variety $Y_{\geq 0}$ in type A and obtained its cell decomposition. She also showed in \cite{Rietsch2006} that the totally nonnegative part of the Peterson variety in type A is contractible. Based on the structure of the cells, she conjectured that as a cell decomposed space $Y_{\geq 0}$ is homeomorphic to a cube. In \cite{abe2025totally}, the first and the third authors of this paper gave a proof of Rietsch's conjecture in type A by using toric geometry closely related to the Peterson variety and concrete computations.

In this paper, we study the totally nonnegative part of the Peterson variety in general Lie types. Now we summarize our main results. By intersecting the Bruhat and opposite Bruhat decompositions of the flag variety $G/B$ and the Peterson variety $Y$, one can obtain the Richardson stratification of $Y$. In Proposition \ref{prop description of YKJ}, we give a description of the Richardson strata of $Y$ in terms of certain functions $\Delta_{\varpi_i}$. Using the positivity of these functions on the totally nonnegative part (Proposition \ref{prop Delta is nonnegative}), we give a description of the Richardson strata of the totally nonnegative part $Y_{\geq 0}$ (Proposition \ref{prop description for R KI>0 2}). Through a particular morphism $\Psi: Y \rightarrow X(\Sigma)$ constructed by the first and the third authors of this paper in \cite{abe2023peterson}, the Peterson variety $Y$ is connected to a particular projective toric orbifold $X(\Sigma)$. We show that this morphism can be restricted to the nonnegative parts $\Psi_{\geq 0}: Y_{\geq 0} \rightarrow X(\Sigma)_{\geq 0}$, which sends the Richardson strata of $Y_{\geq 0}$ to the toric orbit strata of $X(\Sigma)_{\geq 0}$ (Proposition \ref{prop restrict}). We prove that $\Psi_{\geq 0}: Y_{\geq 0} \rightarrow X(\Sigma)_{\geq 0}$ is actually a homeomorphism (Theorem \ref{thm homeo}). Since the torus orbit decomposition of the a projective toric variety gives rise to a cell decomposition of its nonnegative part (see Proposition \ref{prop nonnengative stratum parametrization} in our case), it follows that the Richardson stratification of $Y_{\geq 0}$ is actually a cell decomposition. Note that the moment map restricts to a cell-preserving homeomorphism between $X(\Sigma)_{\geq 0}$ and its moment polytope. Here, the moment polytope of $X(\Sigma)$ is the strongly dominant weight polytope, which is defined to be the intersection of the dominant Weyl chamber and a weight polytope associated with a regular weight. Since the strongly dominant weight polytope is proved to be combinatorially equivalent to a cube\footnote{Topologically, this is equivalent to the existence of a (piecewise linear) homeomorphism between the strongly dominant weight polytope and the standard cube, which restricts to homeomorphisms between their facets (and hence all the faces).} \cite{burrull2023strongly}, we deduce the following main theorem of this paper, which conforms a conjecture of Rietsch\footnote{We thank Konstanze Rietsch for private communication of her conjecture.} for arbitrary Lie type.

\begin{document}
\text{}\vspace{-10pt}
\title[Peterson variety and strongly dominant weight polytope]{Totally nonnegative Peterson variety \\and strongly dominant weight polytope}

In this paper, we study the totally nonnegative part of the Peterson variety in general Lie types. Now we summarize our main results. By intersecting the Bruhat and opposite Bruhat decompositions of the flag variety $G/B$ and the Peterson variety $Y$, one can obtain the Richardson stratification of $Y$. In Proposition \ref{prop description of YKJ}, we give a description of the Richardson strata of $Y$ in terms of certain functions $\Delta_{\varpi_i}$. Using the positivity of these functions on the totally nonnegative part (Proposition \ref{prop Delta is nonnegative}), we give a description of the Richardson strata of the totally nonnegative part $Y_{\geq 0}$ (Proposition \ref{prop description for R KI>0 2}). Through a particular morphism $\Psi: Y \rightarrow X(\Sigma)$ constructed by the first and the third authors of this paper in \cite{abe2023peterson}, the Peterson variety $Y$ is connected to a particular projective toric orbifold $X(\Sigma)$. We show that this morphism can be restricted to the nonnegative parts $\Psi_{\geq 0}: Y_{\geq 0} \rightarrow X(\Sigma)_{\geq 0}$, which sends the Richardson strata of $Y_{\geq 0}$ to the toric orbit strata of $X(\Sigma)_{\geq 0}$ (Proposition \ref{prop restrict}). We prove that $\Psi_{\geq 0}: Y_{\geq 0} \rightarrow X(\Sigma)_{\geq 0}$ is actually a homeomorphism (Theorem \ref{thm homeo}). Since the torus orbit decomposition of the a projective toric variety gives rise to a cell decomposition of its nonnegative part (see Proposition \ref{prop nonnengative stratum parametrization} in our case), it follows that the Richardson stratification of $Y_{\geq 0}$ is actually a cell decomposition. Note that the moment map restricts to a cell-preserving homeomorphism between $X(\Sigma)_{\geq 0}$ and its moment polytope. Here, the moment polytope of $X(\Sigma)$ is the strongly dominant weight polytope, which is defined to be the intersection of the dominant Weyl chamber and a weight polytope associated with a regular weight. Since the strongly dominant weight polytope is proved to be combinatorially equivalent to a cube\footnote{Topologically, this is equivalent to the existence of a (piecewise linear) homeomorphism between the strongly dominant weight polytope and the standard cube, which restricts to homeomorphisms between their facets (and hence all the faces).} \cite{burrull2023strongly}, we deduce the following main theorem of this paper, which conforms a conjecture of Rietsch\footnote{We thank Konstanze Rietsch for private communication of her conjecture.} for arbitrary Lie type.

Now we give some final remarks. Firstly we remark that the functions $\Delta_{\varpi_{i}}$ and $\qp{i}$ appearing in our map $\Psi_{\geq 0}: Y_{\geq 0} \rightarrow X(\Sigma)_{\geq 0}$ are actually affine Schubert classes and quantum parameters in Peterson's theory, see \cite[Remark 6.5]{lam2016total} and \cite[Remark~3.3.7]{rietsch2008mirror}. In \cite[Theorem 7.3]{lam2016total}, Lam--Rietsch essentially used the functions $\Delta_{\varpi_{i}}$ (up to a certain power because they work on adjoint type while we work on the simply-connected case, see Proposition \ref{prop LR and our}) to give a parametrization of the totally nonnegative part of an affine piece of the Peterson variety. We remark that this parametrization is important in the proof of the bijectivity of $\Psi_{\geq 0}: Y_{\geq 0} \rightarrow X(\Sigma)_{\geq 0}$. By analyzing a crucial connection with the affine Grassmannian and geometric Satake equivalence, Lam--Rietsch also showed that different notions of positivity---quantum Schubert and quantum parameter positivity, Lusztig's total positivity, and affine Schubert positivity---on an affine part of the Peterson variety all coincide. Finally, note that $Y_{\geq 0}$ and $X(\Sigma)_{\geq 0}$ are defined in quite different ways; the totally nonnegative part $Y_{\geq 0}$ is defined in terms of root datum of $G$ whereas $X(\Sigma)_{\geq 0}$ is defined in terms of semigroup algebras. It is quite remarkable that so many apparently different notions of positivity coincide in this setting.

This paper is organized as follows. In Section \ref{sec Notation}, we fix some notations which we use throughout this paper. In Section \ref{sec Richardson strata}, we recall the definition of the Peterson variety $Y$ and the construction of its Richardson strata. In Section \ref{sec Totally nonnegative Richardson strata}, we study the totally nonnegative part $Y_{\geq 0}$ and give a description of its Richardson strata. In Section \ref{sec toric}, we study the toric orbifold $X(\Sigma)$ whose moment polytope is the strongly dominant weight polytope. In Section \ref{sec map}, we recall the definition of the morphism $\Psi: Y \rightarrow X(\Sigma)$, and reduce the fact that it induces a homeomorphism $\Psi_{\geq 0}: Y_{\geq 0} \rightarrow X(\Sigma)_{\geq 0}$ to the key claim (Theorem \ref{thm LR in our setting}). Sections \ref{sec splittings} and \ref{sec LR in our setting} are devoted to give a proof of this key claim, which completes the proof of Theorem \ref{thm main}. In the Appendix, we provide proofs of two well-known lemmas for the reader because of the lack of suitable references.

Let $\lambda\in\WL=\Hom(T,\C^{\times})$ be a regular dominant weight. Namely, we have $\lambda=\sum_{i\in I} a_i \varpi_i$ for some positive coefficients $a_i\in\R_{>0}$ $(i\in I)$.
Regarding $\lambda$ as an element of $\mathfrak{t}^*_{\R}\coloneqq \WL\otimes_{\Z}\R$, the weight polytope associated to $\lambda$ is the convex hull of the $W$-orbit of $\lambda$ :
\begin{align*}
 \text{Conv}(W\cdot \lambda) \subseteq \mathfrak{t}^*_{\R}.
\end{align*}
For its relations to the irreducible representation $V_{\lambda}$ with highest weight $\lambda$, see e.g.\ \cite[Sect.~10.1]{hall2015gtm222}.
According to \cite{burrull2023strongly}, we define the \textit{strongly dominant weight polytope} $P^{\lambda}$ by the intersection
\begin{align*}
 P^{\lambda} \coloneqq \text{Conv}(W\cdot \lambda) \cap \sigma_+ \subseteq \mathfrak{t}^*_{\R},
\end{align*}
where
$
 \sigma_+ \coloneqq \{ a_1 \varpi_1 + \cdots + a_{\rkg} \varpi_{\rkg} \mid a_i\ge0 \ (i\in I) \}
$
is the (closed) dominant Weyl chamber in $\mathfrak{t}^*_{\R}$.
In loc.\ cit., $P^{\lambda}$ is proved to be a rational\footnote{This means that the vertices of $P^{\lambda}$ lie in $\WL\otimes_{\Z}\Q$.} combinatorial cube of dimension $n(=|I|)$. Topologically, this is equivalent to the existence of a (piecewise linear) homeomorphism between $P^\lambda$ and the standard $n(=|I|)$-cube, which restricts to homeomorphisms between their facets (and hence all the faces), e.g.\ \cite[Sect.\ 2.2]{ziegler1995lectures}. In particular, its combinatorial structure is independent of the choice of the regular dominant weight $\lambda$.
We review some facts about $P^{\lambda}$ from loc.\ cit.
We set
\begin{align*}
 &H_i \coloneqq \{ \mu \in \mathfrak{t}^*_{\R} \mid \alpha^{\vee}_i(\mu)= 0 \}, \\
 &H^{\lambda}_i \coloneqq \{ \lambda - \mu \in \mathfrak{t}^*_{\R} \mid \varpi^{\vee}_i(\mu)= 0 \}.
\end{align*}
Then the set of facets of $P^{\lambda}$ are given by $P^{\lambda}\cap H_i$ and $P^{\lambda}\cap H^{\lambda}_i$ for $i\in I$ with outward normal vectors $-\alpha^{\vee}_i$ $(i\in I)$ and $\varpi^{\vee}_i$ $(i\in I)$, respectively.
Moreover, for $K, J\subseteq I$, the intersection
\begin{align}\label{eq faces of SDWP}
 F_{K,J} \coloneqq P^{\lambda}\cap \bigcap_{j\in K} H_i \cap \bigcap_{j\notin J} H^{\lambda}_i
\end{align}
is non-empty if and only if $K\subseteq J$ (equivalently, $K\cap (I-J)=\emptyset$).
See \cite[Sect.~2]{burrull2023strongly} for details.
This means that the normal fan of $P^{\lambda}$ is precisely our fan $\Sigma$ given in Definition~\ref{def fan}.

Now we give some final remarks. Firstly we remark that the functions $\Delta_{\varpi_{i}}$ and $\qp{i}$ appearing in our map $\Psi_{\geq 0}: Y_{\geq 0} \rightarrow X(\Sigma)_{\geq 0}$ are actually affine Schubert classes and quantum parameters in Peterson's theory, see \cite[Remark 6.5]{lam2016total} and \cite[Remark~3.3.7]{rietsch2008mirror}. In \cite[Theorem 7.3]{lam2016total}, Lam--Rietsch essentially used the functions $\Delta_{\varpi_{i}}$ (up to a certain power because they work on adjoint type while we work on the simply-connected case, see Proposition \ref{prop LR and our}) to give a parametrization of the totally nonnegative part of an affine piece of the Peterson variety. We remark that this parametrization is important in the proof of the bijectivity of $\Psi_{\geq 0}: Y_{\geq 0} \rightarrow X(\Sigma)_{\geq 0}$. By analyzing a crucial connection with the affine Grassmannian and geometric Satake equivalence, Lam--Rietsch also showed that different notions of positivity---quantum Schubert and quantum parameter positivity, Lusztig's total positivity, and affine Schubert positivity---on an affine part of the Peterson variety all coincide. Finally, note that $Y_{\geq 0}$ and $X(\Sigma)_{\geq 0}$ are defined in quite different ways; the totally nonnegative part $Y_{\geq 0}$ is defined in terms of root datum of $G$ whereas $X(\Sigma)_{\geq 0}$ is defined in terms of semigroup algebras. It is quite remarkable that so many apparently different notions of positivity coincide in this setting.

This paper is organized as follows. In Section \ref{sec Notation}, we fix some notations which we use throughout this paper. In Section \ref{sec Richardson strata}, we recall the definition of the Peterson variety $Y$ and the construction of its Richardson strata. In Section \ref{sec Totally nonnegative Richardson strata}, we study the totally nonnegative part $Y_{\geq 0}$ and give a description of its Richardson strata. In Section \ref{sec toric}, we study the toric orbifold $X(\Sigma)$ whose moment polytope is the strongly dominant weight polytope. In Section \ref{sec map}, we recall the definition of the morphism $\Psi: Y \rightarrow X(\Sigma)$, and reduce the fact that it induces a homeomorphism $\Psi_{\geq 0}: Y_{\geq 0} \rightarrow X(\Sigma)_{\geq 0}$ to the key claim (Theorem \ref{thm LR in our setting}). Sections \ref{sec splittings} and \ref{sec LR in our setting} are devoted to give a proof of this key claim, which completes the proof of Theorem \ref{thm main}. In the Appendix, we provide proofs of two well-known lemmas for the reader because of the lack of suitable references.

\vspace{10pt}

\subsection{Acknowledgments}
We are grateful to Thomas Lam, Changzheng Li, and Zhi L$\ddot{\text{u}}$ for valuable discussions. We would particularly like to thank Konstanze Rietsch for telling us her conjecture. This research was partially done while the authors were visiting the  school of Mathematical Sciences in Fudan University.
The first author is supported in part by JSPS Grant-in-Aid for Scientific Research(C): 23K03102.
The second author is supported in part by NSFC:  12471309.
The third author is supported in part by NSFC:  11901218.

\vspace{30pt}

The post-theorem introduction indicates that the proof of Theorem~\ref{thm main} proceeds by constructing the morphism $\Psi: Y \rightarrow X(\Sigma)$ and then showing that it induces a homeomorphism $\Psi_{\geq 0}: Y_{\geq 0} \rightarrow X(\Sigma)_{\geq 0}$. The text says they "reduce the fact that it induces a homeomorphism $\Psi_{\geq 0}: Y_{\geq 0} \rightarrow X(\Sigma)_{\geq 0}$ to the key claim (Theorem \ref{thm LR in our setting})", and that Sections \ref{sec splittings} and \ref{sec LR in our setting} "give a proof of this key claim, which completes the proof of Theorem \ref{thm main}." It is also remarked that a Lam--Rietsch parametrization using the functions $\Delta_{\varpi_i}$ is "important in the proof of the bijectivity of $\Psi_{\geq 0}: Y_{\geq 0} \rightarrow X(\Sigma)_{\geq 0}$."

The post-theorem introduction indicates that, in establishing the main theorem, the proof proceeds by constructing the morphism $\Psi: Y \rightarrow X(\Sigma)$ and then showing that it induces a homeomorphism $\Psi_{\geq 0}: Y_{\geq 0} \rightarrow X(\Sigma)_{\geq 0}$. The text says they reduce the fact that it induces a homeomorphism $\Psi_{\geq 0}: Y_{\geq 0} \rightarrow X(\Sigma)_{\geq 0}$ to the key claim:

\begin{theorem}\label{thm LR in our setting}
The map
\begin{align*}
 U^e_{\ge0} \rightarrow \R^{I}_{\ge0}
 \quad ; \quad
 x \mapsto
 (\Delta_{\varpi_1}(x\dot{w}_0), \ldots, \Delta_{\varpi_{\rkg}}(x\dot{w}_0))
\end{align*}
is a homeomorphism.
 \end{theorem}

It is then explained that the subsequent parts of the paper give a proof of this key claim, which completes the proof of the main theorem. It is also remarked that a Lam--Rietsch parametrization using the functions $\Delta_{\varpi_i}$ is important in the proof of the bijectivity of $\Psi_{\geq 0}: Y_{\geq 0} \rightarrow X(\Sigma)_{\geq 0}$.

Question:
Let $G$ be a semisimple algebraic group over $\mathbb C$, let $B\subset G$ be a Borel subgroup, let $T\subset B$ be a maximal torus, and let $Y\subset G/B$ be the Peterson variety. Let $Y_{\ge 0}$ denote the totally nonnegative part of $Y$. Fix a regular dominant weight $\lambda=\sum_{i\in I} a_i\varpi_i\in \operatorname{Hom}(T,\mathbb C^{\times})$ with all $a_i>0$, let $W$ be the Weyl group, and define the strongly dominant weight polytope by
\[
P^{\lambda}=\operatorname{Conv}(W\cdot \lambda)\cap \sigma_+,
\qquad
\sigma_+=\Bigl\{\sum_{i\in I} b_i\varpi_i\mid b_i\ge 0\Bigr\}.
\]
Viewing $P^{\lambda}$ with its face decomposition, and interpreting “homeomorphic as a cell-decomposed space” to mean a homeomorphism that respects the cell decompositions, which existence statement holds?

There exists a homeomorphism of cell-decomposed spaces from $Y_{\ge 0}$ onto $P^{\lambda}$. Consequently, $Y_{\ge 0}$ is a regular CW-complex and is homeomorphic to a cube.

Choices:
B. There exists a homeomorphism of cell-decomposed spaces from $Y_{\ge 0}$ onto $\operatorname{Conv}(W\cdot \lambda)$. Consequently, $Y_{\ge 0}$ is a regular CW-complex and is homeomorphic to the full weight polytope $\operatorname{Conv}(W\cdot \lambda)$.
C. There exists a homeomorphism from $Y_{\ge 0}$ onto $P^{\lambda}$. Consequently, $Y_{\ge 0}$ is homeomorphic to a cube.
D. For every regular dominant weight $\lambda$, there exists a homeomorphism of cell-decomposed spaces from $Y_{\ge 0}$ onto $P^{\lambda}$ that is induced by the coordinate map $x\mapsto (\Delta_{\varpi_1}(x\dot w_0),\ldots,\Delta_{\varpi_{|I|}}(x\dot w_0))$. Consequently, $Y_{\ge 0}$ is a regular CW-complex and is homeomorphic to a cube.
E. There exists a homeomorphism of cell-decomposed spaces from $Y_{\ge 0}$ onto $P^{\lambda}$ if and only if the chosen regular dominant weight $\lambda$ has rational coefficients $a_i\in \mathbb{Q}_{>0}$. Consequently, only for such $\lambda$ is $Y_{\ge 0}$ a regular CW-complex homeomorphic to a cube.

target polytope is the strongly dominant truncation, not the full weight polytope

drops the requirement that the homeomorphism respect the cell decompositions

confuses the Lam--Rietsch coordinate homeomorphism on $U^e_{\ge0}\cong \mathbb R^I_{\ge0}$ with the global cell-decomposed-space homeomorphism $Y_{\ge0}\to P^\lambda$

replaces independence of the combinatorial type from regular dominant $\lambda$ by a spurious rationality restriction

The stem gives definitions and asks for the correct conclusion, but it does not explicitly reveal the key claims about regular CW-structure, cell-preserving homeomorphism, or cube homeomorphism.

The item is very close to asking for the exact theorem statement about the topology of the totally nonnegative Peterson variety, with the correct option essentially reproducing that result. The competing choices add nuance, but the question still largely tests theorem recall.

Some reasoning is needed to distinguish subtle variants: full weight polytope vs. dominant part, plain homeomorphism vs. cell-decomposed homeomorphism, and regular vs. non-regular CW conclusions. However, success depends mostly on recognizing the precise theorem rather than generating a conclusion from mathematical reasoning.

Several distractors are plausible and target natural confusions, especially about the target polytope and the strength of the topological conclusion. But choice C is a weaker true statement, which makes the single-best-answer format somewhat ambiguous and lowers distractor quality.

A technically sophisticated MCQ with little answer leakage and mostly plausible distractors, but it is close to theorem restatement and is weakened by an arguably true weaker option, so it tests precise recall more than generative reasoning.


Question:
Let $G$ be a semisimple algebraic group over $\mathbb C$, let $B\subset G$ be a Borel subgroup, let $T\subset B$ be a maximal torus, and let $Y\subset G/B$ be the Peterson variety. Let $Y_{\ge 0}$ denote the totally nonnegative part of $Y$. Fix a regular dominant weight $\lambda=\sum_{i\in I} a_i\varpi_i\in \operatorname{Hom}(T,\mathbb C^{\times})$ with all $a_i>0$, let $W$ be the Weyl group, and define the strongly dominant weight polytope by
\[
P^{\lambda}=\operatorname{Conv}(W\cdot \lambda)\cap \sigma_+,
\qquad
\sigma_+=\Bigl\{\sum_{i\in I} b_i\varpi_i\mid b_i\ge 0\Bigr\}.
\]
Viewing $P^{\lambda}$ with its face decomposition, and interpreting “homeomorphic as a cell-decomposed space” to mean a homeomorphism that respects the cell decompositions, which existence statement holds?

Choices:
B. There exists a homeomorphism of cell-decomposed spaces from $Y_{\ge 0}$ onto $\operatorname{Conv}(W\cdot \lambda)$. Consequently, $Y_{\ge 0}$ is a regular CW-complex and is homeomorphic to the full weight polytope $\operatorname{Conv}(W\cdot \lambda)$.
C. There exists a homeomorphism from $Y_{\ge 0}$ onto $P^{\lambda}$. Consequently, $Y_{\ge 0}$ is homeomorphic to a cube.
D. For every regular dominant weight $\lambda$, there exists a homeomorphism of cell-decomposed spaces from $Y_{\ge 0}$ onto $P^{\lambda}$ that is induced by the coordinate map $x\mapsto (\Delta_{\varpi_1}(x\dot w_0),\ldots,\Delta_{\varpi_{|I|}}(x\dot w_0))$. Consequently, $Y_{\ge 0}$ is a regular CW-complex and is homeomorphic to a cube.
E. There exists a homeomorphism of cell-decomposed spaces from $Y_{\ge 0}$ onto $P^{\lambda}$ if and only if the chosen regular dominant weight $\lambda$ has rational coefficients $a_i\in \mathbb{Q}_{>0}$. Consequently, only for such $\lambda$ is $Y_{\ge 0}$ a regular CW-complex homeomorphic to a cube.

The stem does not explicitly reveal the correct option, and it does not strongly hint at the precise conclusion among the nuanced variants. The reader must distinguish between several closely related statements.

The item is very close to asking for the precise statement of a known theorem, so it is partly a theorem-recall task. However, it does involve choosing among strengthened, weakened, and distorted formulations rather than simply restating a definition.

There is some reasoning pressure because the options differ in subtle but important ways: the target polytope, whether the homeomorphism respects cell structure, whether it is induced by a specific coordinate map, and whether spurious rationality hypotheses are introduced. Still, this is more precision/recall than genuinely generative mathematical reasoning.

Most distractors are mathematically plausible and reflect realistic failure modes, especially confusing the full weight polytope with the truncated one or overclaiming the source of the homeomorphism. However, choice C is a weaker statement that appears to be true if A is true, so the options are not cleanly mutually exclusive, which weakens distractor quality.

A technically sophisticated but somewhat flawed MCQ: it avoids answer leakage and uses plausible distractors, but it mostly tests theorem-statement recognition rather than deep reasoning, and it is weakened by the presence of a weaker true option among the distractors.


\label{sec1}

Let $G$ be a compact Lie group, and $P$ a principal $G$-bundle over the $4$-dimensional sphere $S^4$. The $G$-gauge group over the base space $S^4$ is the topological group of $G$-bundle automorphisms of $P$. Suppose that $G$ is simply-connected and simple. Its classifying space $BG$ is $3$-connected and $\pi_4(BG)\cong \mathbb{Z}$. Let us denote the homotopy class of a map $f$ by the same symbol $f$. Then, the principal $G$-bundle $P$ is classified by a map $k\colon S^4 \to BG$ in $\pi_4(BG)\cong \mathbb{Z}$ and there are infinitely many isomorphism classes of principal $G$-bundles over $S^4$. In \cite{kono-1991}, Kono classified the homotopy types of $SU(2)$-gauge groups over $S^4$ and showed that there are six homotopy types of these groups.

Since then, many classification results on the homotopy types of gauge groups of low-rank simple Lie groups have been obtained. Among compact Lie groups, $SU(2)$ is the rank $1$ simply-connected simple Lie group. There are three simply-connected simple compact Lie groups of rank $2$: $SU(3)$, $Sp(2)$, and $G_2$. For $G=SU(3)$, Hamanaka and Kono \cite{hamanaka-kono-2006} proved that $\mathcal{G}_k\simeq \mathcal{G}_{k'}$ if and only if $(k, 24)=(k', 24)$, where $(m,n)$ is the greatest common divisor of $m$ and $n$. For $G=Sp(2)$, Theriault \cite{theriault-2010} showed $\mathcal{G}_k\simeq_{(p)} \mathcal{G}_{k'}$ at any prime $p$ if and only if $(k, 40)=(k', 40)$, where $\simeq_{(p)}$ means $p$-local homotopy equivalence. Then, for $G=G_2$, Kishimoto, Theriault and Tsutaya \cite{ktt-2017} showed that $\mathcal{G}_k\simeq_{(p)} \mathcal{G}_{k'}$ at any prime $p$ if $(k, 168)=(k', 168)$ and if $\mathcal{G}_k \simeq_{(p)} \mathcal{G}_{k'}$ at any prime $p$, then $(k,84)=(k', 84)$. The discrepancy between $(k, 168)=(k',168)$ and $(k, 84)=(k', 84)$ has remained as an open problem.

In this paper, we complete the classification for rank $2$ simple Lie groups up to $p$-local homotopy equivalence at any prime $p$ by proving the following.

In this paper, we complete the classification for rank $2$ simple Lie groups up to $p$-local homotopy equivalence at any prime $p$ by proving the following.

Theorem~\ref{theorem:1.1} (2) is equivalent to the following local form. \begin{proposition}\label{proposition:1.2}
Let $\mathcal{G}_k$ be the gauge group of a principal $G_2$-bundle over $S^4$ whose classifying map is $k\in \mathbb{Z}\cong \pi_{4}(BG_2)$.
Then, we have
\begin{itemize}
\item[{\rm (1)}] $\mathcal{G}_k\simeq_{(2)} \mathcal{G}_{k'}$ if $(k, 4)=(k', 4)$,
\item[{\rm (2)}]
$\mathcal{G}_k\simeq_{(3)} \mathcal{G}_{k'}$ if $(k, 3)=(k', 3)$,
\item[{\rm (3)}] $\mathcal{G}_k\simeq_{(7)} \mathcal{G}_{k'}$ if $(k, 7)=(k', 7)$
and
\item[{\rm (4)}] $\mathcal{G}_k\simeq_{(p)} \mathcal{G}_{k'} \simeq _{(p)} G_2 \times \Omega_0^4 G_2$ for $p\not=2,3,7$.
\end{itemize}
\end{proposition}

First, we recall the basic method for determining the homotopy types of $G$-gauge groups over $S^4$, which relates their homotopy types to Samelson products. We begin with the following homotopy fiber sequence of mapping spaces.
\[
\Omega {\mathrm{Map}} (S^4, BG)_k \to G \to {\mathrm{Map}}_* (S^4, BG)_k \to {\mathrm{Map}} (S^4, BG)_k \stackrel{\mathrm{ev}}{\longrightarrow} BG,
\]
where $\mathrm{Map} (S^4, BG)_k$ is the connected component of the space of continuous maps containing the map $k\colon S^4\to BG$, $\mathrm{Map}_* (S^4, BG)_k$ is its subspace consisting of base point preserving maps, and $\mathrm{ev}$ is the evaluation map.
There is a homotopy equivalence $\Omega_0^3 G\simeq \mathrm{Map}_{*}(S^4, BG)_0 \simeq \mathrm{Map}_{*}(S^4, BG)_k$ and Gottlieb \cite{gottlieb-1972} showed that the classifying space of the gauge group $B\mathcal{G}_k$ is homotopy equivalent to the mapping space $ \mathop{\mathrm{Map}} (S^4, BG)_k$. Therefore, we have the following fiber sequence.
\[
\mathcal{G}_k \to G \stackrel{\partial_k}{\longrightarrow} \Omega_0^3 G \to B\mathcal{G}_k \to BG.
\]
Thus, the $G$-gauge group $\mathcal{G}_k$ is homotopy equivalent to the homotopy fiber of the map $\partial_k$. Let $i_3\colon S^3 \to G$ be the inclusion map of the bottom cell. Then, Lang \cite{lang-1973} proved that the map $\partial_k$ is the triple adjoint of the Samelson product $\langle k\cdot i_3, 1\rangle$ where $1$ is the identity map of $G$. By the linearity of the Samelson product, we have $\langle k\cdot i_3, 1\rangle\simeq k \cdot \langle i_3, 1\rangle$. Furthermore, in \cite{theriault-2010}, Theriault showed that $\mathcal{G}_k$ is $p$-locally homotopy equivalent to $\mathcal{G}_{k'}$ at all prime $p$ if $(k, m)=(k',m)$ where $m$ is the order of the Samelson product $\langle i_3, 1\rangle$.

\begin{proposition}\label{proposition:2.1}
The following holds.
\begin{itemize}
\item[{\rm (1)}] $\pi_{8}(S^{6})=\mathbb{Z}/2\{ \eta_{6}^{2}\}$, $\pi_{11}(S^{6})=\mathbb{Z}\{ \Delta(\iota_{13})\}$, $\pi_{14}(S^{6})_{{(2)}}=\mathbb{Z}/8\{\bar{\nu}_6 \} \oplus \mathbb{Z}/2\{ \varepsilon_6 \}$,
\item[{\rm (2)}] $\pi_{14}(S^{8})=\mathbb{Z}/2\{ \nu_{8}^{2} \}$, $\pi_{14}(S^{9})=\{0\}$, $\pi_{14}(S^{10})=\{0\}$.
\item[{\rm (3)}] $\eta_{n}\nu_{n+1}\simeq 0$ for $n\geq 5$.
\end{itemize}
\end{proposition}

\begin{proposition}[Mimura] \label{proposition:2.2}
Let $i\colon SU(3)\to G_2$ be the inclusion map and
$p\colon G_2\to S^6=G_2/SU(3)$ the projection map.
By $\langle \alpha \rangle$, we denote an element in $\pi_i(G_2)$ such that $p_*(\langle \alpha\rangle)\simeq \alpha$.
The following holds.
\begin{itemize}
\item[{\rm (1)}] $\pi_{11}(G_{2})=\mathbb{Z}\{ \langle 2 \Delta(\iota_{13})\rangle\} \oplus \mathbb{Z}/2\{ i_{*}([\nu_{5}^{2}])\}$.
\item[{\rm (2)}] $\pi_{14}(G_{2})_{(2)}=\mathbb{Z}/8\{ \langle \bar{\nu}_{6}+\varepsilon_{6}\rangle\} \oplus \mathbb{Z}/2\{ i_{*}([\nu_{5}^{2}])\circ \nu_{11}\}$
\end{itemize}
\end{proposition}

\begin{theorem}[{\oshima} \cite{oshima-2005}*{Theorem 2.1}] \label{theorem:3.1}
There is a map $\gamma \colon S^{11}\to G_2$ such that
\begin{itemize}
\item[{\rm (1)}]
$p\circ \gamma\simeq 2\Delta(\iota_{13})$,
\item[{\rm (2)}] $\langle i_3, \gamma\rangle\in \pi_{14}(G_2)$ has order $21=3\cdot 7$.
\end{itemize}
\end{theorem}

\begin{proposition}\label{proposition:3.2}
Let $i_{8}\colon S^8\to P^9(2)$ be the inclusion map of the bottom cell. Then, the following hold.
\begin{itemize}
\item[{\rm (1)}] ${i_8}_{*}\colon \pi_{14}(S^8)\to \pi_{14}(P^9(2))$ is an isomorphism.
\item[{\rm (2)}] For maps $f\colon S^{14}\to S^8$, and $g\colon S^{8}\to S^{6}$,
$g \circ f \simeq 0$ in $\pi_{14}(S^6)$.
\item[{\rm (3)}] If a map $f\colon S^{14}\to S^{6}$ factors through $P^9(2)$, then $f\simeq 0$ in $\pi_{14}(S^{6})$.
\end{itemize}
\end{proposition}

\begin{proposition}\label{proposition:3.3}
The following holds.
\begin{itemize}
\item[{\rm (1)}] $\pi_{11}(C)\simeq \mathbb{Z}\{ \bar{i}_{6}\circ \Delta(\iota_{13})\} \oplus \mathbb{Z}\{ \bar{s}\}$,
\item[{\rm (2)}] $\mathop{\mathrm{Ker}} \bar{p}_{6\; *} \colon \pi_{11}(C)\to \pi_{11}(S^6)$ is generated by $w\simeq a \bar{i}_{6}\circ \Delta(\iota_{13})+b \bar{s}$ where $a\in \mathbb{Z}$ and $b=1$ or $a$ is odd and $b=2$.
\item[{\rm (3)}] $j\circ q\circ \gamma\simeq cw$ where $c\in \{ \pm 1, \pm 2\}$.
\item[{\rm (4)}] $p_{11}\circ q\circ \gamma \simeq bc (\iota_{11})$ where $bc \in \{ \pm 1, \pm 2, \pm 4\}$.
\end{itemize}
\end{proposition}

\begin{theorem}\label{theorem:1.1}
Let $\mathcal{G}_k$ be the gauge group of a principal $G_2$-bundle over $S^4$ whose classifying map is $k\in \mathbb{Z}\cong \pi_{4}(BG_2)$. The following holds.
\begin{itemize}
\item[{\rm (1)}]
If $\mathcal{G}_k\simeq \mathcal{G}_{k'}$, then $(k, 84)=(k', 84)$ and
\item[{\rm (2)}]  $\mathcal{G}_k\simeq_{(p)} \mathcal{G}_{k'}$ at any prime $p$ if $(k, 84)=(k', 84)$.
\end{itemize}
\end{theorem}

Theorem~\ref{theorem:1.1} (2) is equivalent to the following local form. \begin{proposition}\label{proposition:1.2}
Let $\mathcal{G}_k$ be the gauge group of a principal $G_2$-bundle over $S^4$ whose classifying map is $k\in \mathbb{Z}\cong \pi_{4}(BG_2)$.
Then, we have
\begin{itemize}
\item[{\rm (1)}] $\mathcal{G}_k\simeq_{(2)} \mathcal{G}_{k'}$  if $(k, 4)=(k', 4)$,
\item[{\rm (2)}]
$\mathcal{G}_k\simeq_{(3)} \mathcal{G}_{k'}$  if $(k, 3)=(k', 3)$,
\item[{\rm (3)}] $\mathcal{G}_k\simeq_{(7)} \mathcal{G}_{k'}$  if $(k, 7)=(k', 7)$
and
\item[{\rm (4)}] $\mathcal{G}_k\simeq_{(p)} \mathcal{G}_{k'} \simeq _{(p)} G_2 \times \Omega_0^4 G_2$ for $p\not=2,3,7$.
\end{itemize}
\end{proposition}

Theorem~\ref{theorem:1.1} (1) was established in \cite{ktt-2017}*{Theorem 1.2 (a)} and Proposition~\ref{proposition:1.2} (2), (3), (4) were established in \cite{ktt-2017}*{Propositions 4.6 and 3.2}. In this paper, we prove Proposition~\ref{proposition:1.2} (1) and complete the proof of Theorem~\ref{theorem:1.1} assuming the results of \cite{ktt-2017}.

First, we recall the basic method for determining the homotopy types of $G$-gauge groups over $S^4$, which relates their homotopy types to Samelson products. We begin with the following homotopy fiber sequence of mapping spaces.
\[
\Omega {\mathrm{Map}} (S^4, BG)_k  \to G \to  {\mathrm{Map}}_* (S^4, BG)_k \to {\mathrm{Map}} (S^4, BG)_k \stackrel{\mathrm{ev}}{\longrightarrow} BG,
\]
where $\mathrm{Map} (S^4, BG)_k$ is the connected component of the space of continuous maps containing the map $k\colon S^4\to BG$, $\mathrm{Map}_* (S^4, BG)_k$ is its subspace consisting of base point preserving maps,  and $\mathrm{ev}$ is the evaluation map.
There is a homotopy equivalence  $\Omega_0^3 G\simeq \mathrm{Map}_{*}(S^4, BG)_0 \simeq \mathrm{Map}_{*}(S^4, BG)_k$ and Gottlieb \cite{gottlieb-1972} showed that the classifying space of the gauge group $B\mathcal{G}_k$ is homotopy equivalent to the mapping space $ \mathop{\mathrm{Map}} (S^4, BG)_k$. Therefore, we have the following fiber sequence.
\[
\mathcal{G}_k  \to G \stackrel{\partial_k}{\longrightarrow} \Omega_0^3 G \to B\mathcal{G}_k \to BG.
\]
Thus, the $G$-gauge group $\mathcal{G}_k$ is homotopy equivalent to the homotopy fiber of the map $\partial_k$. Let $i_3\colon S^3 \to G$ be the inclusion map of the bottom cell. Then, Lang \cite{lang-1973} proved that the map $\partial_k$ is the triple adjoint of the Samelson product $\langle k\cdot i_3, 1\rangle$ where $1$ is the identity map of $G$. By the linearity of the Samelson product, we have $\langle k\cdot i_3, 1\rangle\simeq k \cdot \langle i_3, 1\rangle$. Furthermore, in \cite{theriault-2010}, Theriault showed that $\mathcal{G}_k$ is $p$-locally homotopy equivalent to $\mathcal{G}_{k'}$ at all prime $p$ if $(k, m)=(k',m)$ where $m$ is the order of the Samelson product $\langle i_3, 1\rangle$.

For an abelian group $A$, we denote by $A_{(p)}$ its localization at the prime $p$. If $A$ is finite, then $A_{(p)}$ is the $p$-primary subgroup of $A$. By \cite{ktt-2017}*{Corollary 6.6}, the order of $\langle i_3, 1\rangle$ in $[\Sigma^3 G_2, G_2]_{(2)}$ is at least $4$. Therefore, to show Proposition~\ref{proposition:1.2} (1), it suffices to prove the following proposition.

\begin{proposition}\label{proposition:1.3}  The order of the Samelson product $\langle i_3, 1\rangle$ in $[\Sigma^3 G_2, G_2]_{(2)}$ is at most $4$.
\end{proposition}

As for the order of the Samelson product $\langle i_3, 1\rangle$ in $[\Sigma^3 G_2, G_2]$, together with \cite{ktt-2017}*{Lemma 3.1, Proposition 4.5 and Corollary 6.6}, by Proposition~\ref{proposition:1.3}, we have the following theorem.

\begin{theorem}\label{theorem:1.4}
The order of the Samelson product $\langle i_3, 1\rangle$ in $[\Sigma^3 G_2, G_2]$ is $84$.
\end{theorem}

Next, we recall some properties of a space $V:=G_2/SU(2)$, which played an important role in \cite{ktt-2017} and is crucial in this paper. Let $q\colon G_2\to V$ be the obvious projection map. By \cite{ktt-2017}*{Lemmas 7.1 and 7.2}, the Samelson product $3 \cdot \langle i_3, 1\rangle\colon \Sigma^3 G_2\to G_2$ factors through $\Sigma^3 q \colon \Sigma^3 G_2 \to \Sigma^3 V$.
Let us write this factorization as follows.
 \[
3\cdot \langle i_3, 1\rangle\simeq \psi\circ \Sigma^3 q.
\]
Let $P^{n+1}(2)$ be the mapping cone of the degree $2$ map $\times 2 \colon S^n\to S^n$.
The $6$-skeleton of $V=S^5\cup e^6\cup e^{11}$ is $P^6(2)$ and, by \cite{ktt-2017}{Lemma 7.4}, there is a homotopy equivalence $\Sigma^3 V\simeq P^9(2) \vee S^{14}$.  Let \[
p_{11}\colon V \to S^{11}
\]
 be the pinch map to the top cell, collapsing the $6$-skeleton to the base point. We denote by
 \[
 i_6\colon P^6(2) \to V
 \]
  the inclusion map. Then, the above homotopy equivalence provides maps $s'\colon \Sigma^3 V \to P^9(2)$ and $s''\colon S^{14}\to \Sigma^3V$ such that the identity map of $\Sigma^3 V$ is homotopic to
  \[
  \Sigma^3 i_6 \circ s' + s'' \circ \Sigma^3 p_{11}.
  \] The following proposition is what we prove in the rest of this paper.

\begin{proposition} \label{proposition:1.5}
We have  $4(\psi\circ s'')\simeq 0$  in $\pi_{14}(G_2)_{(2)}$. \end{proposition}

\begin{proof}[Proof of Proposition~\ref{proposition:1.3}] We consider the following decomposition in $[\Sigma^3 G_2, G_2]$. \[ 3 \cdot \langle i_3, 1\rangle\simeq \psi \circ \Sigma^3 i_6\circ s' \circ \Sigma^3 q +\psi \circ s'' \circ \Sigma^3 (p_{11}\circ q). \]
By \cite{ktt-2017}*{Lemma 5.2 (a) and (b)}, the order of the identity map of $P^9(2)$ is $4$. Hence, we have
$4 (\Sigma^3 i_6\circ s' )\simeq 0$. Therefore, by Proposition~\ref{proposition:1.5}, the order of $ \langle i_3, 1\rangle$ in $[\Sigma^3 G_2, G_2]_{(2)}$ is at most $4$.  \end{proof}

In what follows, we use the symbol $p$ to express the map $p\colon G_2\to S^6\simeq G_2/SU(3)$ only. When we need to indicate the generator of a cyclic group, we write $\mathbb{Z}/m\{ a\}$ for the cyclic group of order $m$ generated by $a$.

This paper is organized as follows. In Section \ref{sec2}, we collect some facts on homotopy groups of $G_2$ and $S^6$ and reduce the problem on the $2$-primary homotopy group $\pi_{14}(G_2)_{(2)}$ to that of $\pi_{14}(S^6)_{(2)}$. In Section \ref{sec3}, we prove Proposition~\ref{proposition:1.5}.

To complete Theorem~\ref{theorem:1.1} (2) it suffices to prove the local statement Proposition~\ref{proposition:1.2}. The paper reduces this to proving Proposition~\ref{proposition:1.2} (1) (the $2$-local case), since (2),(3),(4) are cited from \cite{ktt-2017}.

The method recalled is: using Gottlieb’s identification $B\mathcal{G}_k\simeq \mathrm{Map}(S^4,BG)_k$ and the resulting fiber sequence
\[
\mathcal{G}_k\to G\xrightarrow{\partial_k} \Omega_0^3 G\to B\mathcal{G}_k\to BG,
\]
so $\mathcal{G}_k$ is the homotopy fiber of $\partial_k$. By Lang, $\partial_k$ is the triple adjoint of the Samelson product $\langle k\cdot i_3,1\rangle$, hence by linearity $\langle k\cdot i_3,1\rangle\simeq k\cdot\langle i_3,1\rangle$. Theriault’s result is then invoked: $\mathcal{G}_k\simeq_{(p)}\mathcal{G}_{k'}$ for all primes $p$ if $(k,m)=(k',m)$ where $m$ is the order of $\langle i_3,1\rangle$.

For $G=G_2$ at $p=2$, \cite{ktt-2017} gives that the order of $\langle i_3,1\rangle$ in $[\Sigma^3G_2,G_2]_{(2)}$ is at least $4$, so “to show Proposition~\ref{proposition:1.2} (1), it suffices” to prove Proposition~\ref{proposition:1.3}: the order is at most $4$.

The proof sketch for Proposition~\ref{proposition:1.3} is given: using that $3\cdot\langle i_3,1\rangle$ factors through $\Sigma^3q\colon \Sigma^3G_2\to \Sigma^3V$ (with $V=G_2/SU(2)$), write
\[
3\cdot\langle i_3,1\rangle\simeq \psi\circ \Sigma^3q.
\]
With the homotopy decomposition $\Sigma^3V\simeq P^9(2)\vee S^{14}$, choose maps $s'\colon \Sigma^3V\to P^9(2)$ and $s''\colon S^{14}\to \Sigma^3V$ so that
\[
\mathrm{id}_{\Sigma^3V}\simeq \Sigma^3i_6\circ s'\; +\; s''\circ \Sigma^3p_{11}.
\]
Then
\[
3\cdot\langle i_3,1\rangle\simeq \psi\circ \Sigma^3i_6\circ s'\circ \Sigma^3q\; +\; \psi\circ s''\circ \Sigma^3(p_{11}\circ q).
\]
Using \cite{ktt-2017} (Lemma 5.2(a),(b)) that “the order of the identity map of $P^9(2)$ is $4$,” one gets $4(\Sigma^3i_6\circ s')\simeq 0$. The remaining summand is controlled by Proposition~\ref{proposition:1.5}, namely $4(\psi\circ s'')\simeq 0$ in $\pi_{14}(G_2)_{(2)}$. Combining these shows “the order of $\langle i_3,1\rangle$ in $[\Sigma^3G_2,G_2]_{(2)}$ is at most $4$,” hence Proposition~\ref{proposition:1.2}(1), and therefore Theorem~\ref{theorem:1.1}(2) via the Samelson-product/Thériault criterion.

(They additionally note that with Proposition~\ref{proposition:1.3} and results of \cite{ktt-2017}, the global order is computed as $84$ (Theorem~\ref{theorem:1.4}).)

To establish the main theorem (2) it suffices to prove the local statement

\begin{proposition}\label{proposition:1.2}
Let $\mathcal{G}_k$ be the gauge group of a principal $G_2$-bundle over $S^4$ whose classifying map is $k\in \mathbb{Z}\cong \pi_{4}(BG_2)$.
Then, we have
\begin{itemize}
\item[{\rm (1)}] $\mathcal{G}_k\simeq_{(2)} \mathcal{G}_{k'}$  if $(k, 4)=(k', 4)$,
\item[{\rm (2)}]
$\mathcal{G}_k\simeq_{(3)} \mathcal{G}_{k'}$  if $(k, 3)=(k', 3)$,
\item[{\rm (3)}] $\mathcal{G}_k\simeq_{(7)} \mathcal{G}_{k'}$  if $(k, 7)=(k', 7)$
and
\item[{\rm (4)}] $\mathcal{G}_k\simeq_{(p)} \mathcal{G}_{k'} \simeq _{(p)} G_2 \times \Omega_0^4 G_2$ for $p\not=2,3,7$.
\end{itemize}
\end{proposition}

The paper reduces this to proving the preceding proposition (1) (the $2$-local case), since (2),(3),(4) are cited from \cite{ktt-2017}.

The method recalled is: using Gottlieb’s identification $B\mathcal{G}_k\simeq \mathrm{Map}(S^4,BG)_k$ and the resulting fiber sequence
\[
\mathcal{G}_k\to G\xrightarrow{\partial_k} \Omega_0^3 G\to B\mathcal{G}_k\to BG,
\]
so $\mathcal{G}_k$ is the homotopy fiber of $\partial_k$. By Lang, $\partial_k$ is the triple adjoint of the Samelson product $\langle k\cdot i_3,1\rangle$, hence by linearity $\langle k\cdot i_3,1\rangle\simeq k\cdot\langle i_3,1\rangle$. Theriault’s result is then invoked: $\mathcal{G}_k\simeq_{(p)}\mathcal{G}_{k'}$ for all primes $p$ if $(k,m)=(k',m)$ where $m$ is the order of $\langle i_3,1\rangle$.

For $G=G_2$ at $p=2$, \cite{ktt-2017} gives that the order of $\langle i_3,1\rangle$ in $[\Sigma^3G_2,G_2]_{(2)}$ is at least $4$, so to show the preceding proposition (1), it suffices to prove the following.

\begin{proposition}\label{proposition:1.3}  The order of the Samelson product $\langle i_3, 1\rangle$ in $[\Sigma^3 G_2, G_2]_{(2)}$ is at most $4$.
\end{proposition}

The proof sketch for the preceding proposition is given: using that $3\cdot\langle i_3,1\rangle$ factors through $\Sigma^3q\colon \Sigma^3G_2\to \Sigma^3V$ (with $V=G_2/SU(2)$), write
\[
3\cdot\langle i_3,1\rangle\simeq \psi\circ \Sigma^3q.
\]
With the homotopy decomposition $\Sigma^3V\simeq P^9(2)\vee S^{14}$, choose maps $s'\colon \Sigma^3V\to P^9(2)$ and $s''\colon S^{14}\to \Sigma^3V$ so that
\[
\mathrm{id}_{\Sigma^3V}\simeq \Sigma^3i_6\circ s'\; +\; s''\circ \Sigma^3p_{11}.
\]
Then
\[
3\cdot\langle i_3,1\rangle\simeq \psi\circ \Sigma^3i_6\circ s'\circ \Sigma^3q\; +\; \psi\circ s''\circ \Sigma^3(p_{11}\circ q).
\]
Using \cite{ktt-2017} (Lemma 5.2(a),(b)) that “the order of the identity map of $P^9(2)$ is $4$,” one gets $4(\Sigma^3i_6\circ s')\simeq 0$. The remaining summand is controlled by

\begin{proposition} \label{proposition:1.5}
We have  $4(\psi\circ s'')\simeq 0$  in $\pi_{14}(G_2)_{(2)}$. \end{proposition}

Combining these shows that the order of $\langle i_3,1\rangle$ in $[\Sigma^3G_2,G_2]_{(2)}$ is at most $4$, hence the preceding proposition (1), and therefore the main theorem (2) via the Samelson-product/Thériault criterion.

(They additionally note that with the proposition above and results of \cite{ktt-2017}, the global order is computed as $84$ (Theorem~\ref{theorem:1.4}).)

Question: Let \(\mathcal{G}_k\) denote the gauge group of a principal \(G_2\)-bundle over \(S^4\) whose classifying map represents \(k\in \pi_4(BG_2)\cong \mathbb Z\). Write \((m,n)\) for the greatest common divisor of integers \(m\) and \(n\), write \(\mathcal{G}_k\simeq \mathcal{G}_{k'}\) for homotopy equivalence, and write \(\mathcal{G}_k\simeq_{(p)} \mathcal{G}_{k'}\) for \(p\)-local homotopy equivalence. Which statement holds for integers \(k\) and \(k'\)?

If \(\mathcal{G}_k\simeq \mathcal{G}_{k'}\), then \((k,84)=(k',84)\). Moreover, if \((k,84)=(k',84)\), then \(\mathcal{G}_k\simeq_{(p)} \mathcal{G}_{k'}\) for every prime \(p\).

Choices:
B. If \(\mathcal{G}_k\simeq \mathcal{G}_{k'}\), then \((k,168)=(k',168)\). Moreover, if \((k,168)=(k',168)\), then \(\mathcal{G}_k\simeq_{(p)} \mathcal{G}_{k'}\) for every prime \(p\).
C. If \((k,84)=(k',84)\), then \(\mathcal{G}_k\simeq_{(2)} \mathcal{G}_{k'}\), \(\mathcal{G}_k\simeq_{(3)} \mathcal{G}_{k'}\), and \(\mathcal{G}_k\simeq_{(7)} \mathcal{G}_{k'}\).
D. If \(\mathcal{G}_k\simeq \mathcal{G}_{k'}\), then \((k,84)=(k',84)\). Moreover, if \((k,84)=(k',84)\), then \(\mathcal{G}_k\simeq \mathcal{G}_{k'}\).
E. If \(\mathcal{G}_k\simeq \mathcal{G}_{k'}\), then \((k,84)=(k',84)\). Moreover, if \((k,4)=(k',4)\), then \(\mathcal{G}_k\simeq_{(p)} \mathcal{G}_{k'}\) for every prime \(p\).

The stem only introduces notation and the setup; it does not reveal the key invariant 84 or otherwise point directly to choice A.

The item is essentially a theorem-identification question: the correct option states a specific classification result rather than asking the student to derive it from intermediate facts. The competing options prevent complete tautology, but it still largely tests recall of the exact statement.

There is some reasoning pressure in comparing necessity vs sufficiency, integral vs p-local equivalence, and 84 vs 168. However, the main demand is recognizing the known theorem rather than generating a conclusion from mathematical reasoning in the stem.

The distractors are strong: B perturbs the key arithmetic invariant, C gives a weaker true-looking statement, D improperly strengthens p-local equivalence to global equivalence, and E introduces a misleading reduced gcd condition. These are plausible and mathematically distinct.

A solid MCQ with no answer leakage and high-quality distractors, but it primarily tests recall of a specific theorem statement rather than genuinely generative mathematical reasoning.