\begin{introtheorem}\label{intthm:character}
    If $H$ is a type B or type C Hessenberg space, there exist non-negative integers $a,b \in \mathbb{N}$, integers $c,d \in \{0,1\}$, and a subset $I \subseteq [n]$, each determined by $S(H)$, such that
    \[
     \lrep{H}_1 = a\,\mathbb{1} + \sum_{i\in I} \fh_i + b\,\fh_1 + c\,\fs + d\,\bm{\delta}
    \]
    and
    \[
     \rrep{H}_1 = \bm{\chi} + \sum_{i\in I} \left(\fh_i - \mathbb{1}\right) + b\left(\fh_1-\mathbb{1}\right) + c \left(
    \fs-\mathbb{1}\right) + d\,\bm{\delta},
    \]
    where
    \begin{itemize}
        \item $\bm{\chi}$ is the character of the defining representation of $\Wn$ on $\C^n$
        \item $\fh_i$ is the character of the action of $\Wn$ on the cosets of $\mathfrak{S}_i\times \mathfrak{W}_{n-i}$,
        \item $\fs$ is the character of the action of $\Wn$ on the cosets of $\mathfrak{W}_1\times \mathfrak{W}_{n-1}$,
        \item $\mathbb{1}$ is the character of the trivial representation, and
        \item $\bm{\delta}$ is the character $w\mapsto (-1)^{\abs{\Neg(w)}}$ where $\Neg(w) = \{w(i) \mid i \in [n],\; w(i) < 0\}$.
    \end{itemize}
    Moreover, $c$ and $d$ are never simultaneously $1$, and $d$ is always $0$ in type B.
\end{introtheorem}

\begin{proposition}\label{prop:permutohedral}
    Let $\bm{\chi}$ be the character of the defining representation of $\Wn$ (i.e. the action on $\C^n$). Let $\fh_i$ be the character of the action of $\Wn$ on the cosets of $\mathfrak{S}_i \times \mathfrak{W}_{n-i}$. Let $\mathbb{1}$ be the character of the trivial representation. Then
   \[
     \lrep{\Delta}_1 = \sum_{i=1}^n \fh_i - \bm{\chi}
    \hspace{1cm}\text{and}\hspace{1cm}
    \rrep{\Delta}_1 = \sum_{i=1}^n \fh_i - n\,\mathbb{1}.
    \]
\end{proposition}

\begin{definition}\label{def:itypes}
Consider $S(H)$ and $i \in [n]$. We call $i$\begin{itemize}
    \item \emph{uncovered} if $\begin{cases} \{t_{i-1},t_i\} \cap S(H) = \emptyset &\text{and } i \neq n-1\text{, or} \\ \{t_{n-1},t_{n-1},t_{n}\} \cap S(H) = \emptyset &\text{and } i = n-1\end{cases}$
    \item \emph{surrounded} if $i \in [n-2]$ and $\{t_{i-1},t_i\} \cap S(H) = \{t_{i-1}\}$ and $t_m \in S(H)\text{ for some } m>i$,
    \item \emph{shaded} if $t_i \in S(H)$ or if $t_n \in S(H)$ and $i=n-1$.
\end{itemize}
\end{definition}

\begin{theorem}\label{thm:character}
    Say $\Delta \subsetneq H$. Let $\bm{\chi}$ be the character of the defining representation of $\Wn$ (i.e. the action on $\C^n$). Let $\fh_i$ be the character of the action of $\Wn$ on the cosets of $\mathfrak{S}_i\times \mathfrak{W}_{n-i}$. Let $\fs$ be the character of the action of $\Wn$ on cosets of $\mathfrak{W}_1 \times \mathfrak{W}_{n-1}$. Let $\mathbb{1}$ be the character of the trivial representation and $\bm{\delta}$ be the character $w\mapsto (-1)^{\abs{\Neg(w)}}$. Then
    \begin{align*}
     \lrep{H}_1 = \abs{\{i \in [n] \mid i\text{ shaded}\,\}}\mathbb{1} &+ \sum_{i\text{ uncovered}} \fh_i \\[6pt]&+ \abs{\{i \in [n-2] \mid i\text{ surrounded}\,\}}\fh_1 \\[6pt]&+ \begin{cases}
    \fs &\text{if $t_n \notin S(H)$}\\
    \bm{\delta} &\text{if }\{t_{n-1},t_n\} \cap S(H) = \{t_{n}\}\\
    0 &\text{otherwise.}
\end{cases}
    \end{align*}
    and
        \begin{align*}
     \rrep{H}_1 = \bm{\chi} &+ \sum_{i\text{ uncovered}} \left(\fh_i - \mathbb{1}\right) \\[6pt]&+ \abs{\{i \in [n-2] \mid i\text{ surrounded}\,\}}\left(\fh_1-\mathbb{1}\right) \\[6pt]&+ \begin{cases}
    \fs-\mathbb{1} &\text{if $t_n \notin S(H)$}\\
    \bm{\delta} &\text{if }\{t_{n-1},t_n\} \cap S(H) = \{t_{n}\}\\
    0 &\text{otherwise.}
\end{cases}
    \end{align*}
\end{theorem}

\label{sec:intro}
Let $\mathfrak{W}_n$ be the group of signed permutations, which are permutations $w$ on the set $\{1,\ldots,n,-n,\ldots,-1\}$ such that $w(-i) = -w(i)$. The group $\mathfrak{W}_n$ is also the Weyl group of types B and C, and so is associated with two distinct root systems. A Hessenberg space $H$ is a particular subset of roots, which determines a set $S(H)$ of transpositions in $\mathfrak{W}_n$. Given a Hessenberg space, one can define the \emph{graded ring of splines} $\mathcal{M}_{H}$ on $\mathfrak{W}_n$. This ring has two $\C[x_\bullet]\coloneqq \C[x_1,\ldots,x_{n}]$-module structures and a $\mathfrak{W}_n$-module structure. This paper determines the algebraic and combinatorial properties of the degree-one graded piece of $\mathcal{M}_{H}$ from the combinatorics of $S(H)$.\par
Let $(i,j)$ be the unique transposition in $\mathfrak{W}_n$ that switches $i$ with $j$ and $-i$ with $-j$ (note $i=-j$ is possible). If $H$ is a Hessenberg space, then the ring of splines is
 \[
 \mathcal{M}_{H} = \bigoplus_{i \geq 0} \mathcal{M}_{H}^i \coloneqq \left\{ \left.\bm{\rho} \in \prod_{w \in \mathfrak{W}_n} \C[x_\bullet]\; \right|\; \bm{\rho}(w) - \bm{\rho}(w(i,j)) \in \left\langle x_{w(i)}-x_{w(j)} \right\rangle\text{ if } (i,j) \in S(H) \right\}\!,
 \]
with (graded) $\mathfrak{W}_n$-module structure $w\cdot \bm{\rho}(v) = w\bm{\rho}(w^{-1}v)$ and (graded) $\C[x_\bullet]$-module structure given by multiplication. \par
The ring $\mathcal{M}_{H}$ is isomorphic to the $T$-equivariant cohomology of a smooth subvariety of the full flag variety $G/B$ called a \emph{regular semisimple Hessenberg variety}. The connection between this combinatorial ring and equivariant cohomology is called \emph{GKM Theory} \cite{GKM_theory}, and it allows for the study of this cohomology in purely combinatorial and algebraic terms. \par
The $\mathfrak{W}_n$-module structure on $\mathcal{M}_{H}$ was first defined as the \emph{dot action} on equivariant cohomology by Tymoczko in \cite{tymoczko2008permutation}. There are two natural $\mathfrak{W}_n$-equivariant quotients, called the left $\mathrm{L}_{H}$ and right $\mathrm{R}_{H}$ quotients, of $\mathcal{M}_{H}$ that are in fact graded $\mathbb{C}$-vector spaces (the left quotient corresponds to ordinary cohomology). The graded $\mathfrak{W}_n$-module structure of $\mathcal{M}_{H}$ induces graded $\mathfrak{W}_n$-representations on the quotients. The primary goal of this paper is to compute the characters of the degree-one pieces of
\[
\mathrm{L}_{H} = \bigoplus_{i\geq 0} \left(\mathrm{L}_{H}\right)_i \;\; \text{ and }\;\;\mathrm{R}_{H} = \bigoplus_{i\geq 0} \left(\mathrm{R}_{H}\right)_i
\] from the combinatorial data of $S(H)$. In doing so, we compute the dot action on second (equivariant and ordinary) cohomology for all regular semisimple Hessenberg varieties in types B and C. This generalizes what is known in type A \cite{chow_linearp,chohonglee_second_cohom,ayzenberg2022second}.\par
These graded representations are of interest to algebraic combinatorists in part because they are the $\mathfrak{W}_n$-equivalents of very well-studied $\mathfrak{S}_n$-representations, which have connections to chromatic symmetric functions \cite{guaypaquet2016shar_wachs_conj,SW2016chromaticquasisymmetric,brosnan_chow_dotactn_is_chromsym} and LLT polynomials \cite{guaypaquet2016shar_wachs_conj,Ayzenberg2018isospectral, ALEXANDERSSON2018LLTchromsym}. We detail some of these connections in the Appendix. From these works, in type A there are combinatorial formulas to compute the characters of both $\mathrm{L}_{H}$ and $\mathrm{R}_{H}$ in at least one basis for the character space of $\mathfrak{S}_n$. This is \textbf{not} the case in types B or C.\par
For one specific family of $H$ (those corresponding to simple roots), the character of $\mathrm{L}_{H}$ was computed by Stembridge \cite{Stembridge_Permuto} in general type. Stembridge's results are via a connection to \emph{permutohedra}, as the associated Hessenberg varieties are in fact toric varieties. While it is possible to compute the left and right characters for general $H$ in types B and C, even partial results are rare and rely on sophisticated geometric tools \cite{BalibanuCrooks_sheaves,PrecupSommers_sheaves}. Formulas are non-combinatorial (and usually non-positive), require the computation of type B/C Green polynomials and Poincar\'e polynomials of other varieties, and proofs require the geometry of perverse sheaves. \par
In contrast, for the degree-one graded piece, the main result of this paper applies to all Hessenberg spaces $H$ and is entirely computable from the combinatorics of roots and transpositions in $\mathfrak{W}_n$. Moreover, we compute the degree-one pieces of the characters $\mathfrak{ch}\left(\mathrm{L}_{H}\right)$ and $\mathfrak{ch}\left(\mathrm{R}_{H}\right)$ in a positive manner.

\setcounter{tocdepth}{1}

Our theorem is the first type B/C result to provide an explicit non-negative expansion for a graded piece of $\lrep{H}$ or $\rrep{H}$ in terms of characters of irreducible- and/or permutation-representations for all Hessenberg spaces. It is also the first to compute the characters for any graded piece of $\lrep{H}$ and $\rrep{H}$ by directly providing a basis for the representation, and in doing so gives a family of characters with which one can attempt to expand the catalog of results and connections that exist in type A to types B and C.\par

The right character follows from the general procedure of ``removing" from any permutation representation with permutation basis $\{v_i\}$ the trivial sub-representation $\C\left\{ \sum v_i \right\}$. In particular, if $\mathfrak{p}$ is the character of a representation with permutation basis $\{v_i\}$, then the character on the complementary subspace $\C\{v_i-v_{i+1}\}$ is $\mathfrak{p}-\mathbb{1}$.
\end{proof}
In type B, $t_n \in S(H)$ implies that $t_{n-1} \in S(H)$, and so it is impossible to get the $\bm{\delta}$ term from Theorem \ref{thm:character}. In type C the opposite is true: $t_{n-1} \in S(H)$ implies that $t_n \in S(H)$, and so $\delta$ may appear. This is the ``moreover" part of Theorem \ref{intthm:character}.
\begin{example}
 The table below gives every $\lrep{H}_1$ and $\lrep{H}_1$ for $n=4$.
\begin{table}[H]
\centering
\renewcommand{\arraystretch}{1.5}
\begin{tabular}{|c|p{2in}|p{2in}|p{0.3in}|}\hline
 { $t_i \in S(H)$} & {\large $\lrep{H}_1$} & \large $\rrep{H}_1$ &\large dim\\\hline
 $\emptyset$ & $\fh_1 +\fh_2+\fh_3+\fh_4-\bchi$ & $\fh_1 +\fh_2+\fh_3+\fh_4-4\triv$ & $76$ \\\hline
 $\{t_1\}$ & $\triv + \fh_3 + \fh_4 + \fs$ &$\bchi + \fh_3 + \fh_4 + \fs -3\triv$ & $53$\\\hline
 $\{t_2\}$ & $\triv+\fh_1+\fh_4+\fs $ &$\bchi+\fh_1+\fh_4+\fs -3\triv$ &$29$\\\hline
 $\{t_3\}$ & $\triv+\fh_1+\fh_2+\fs $ &$\bchi+\fh_1+\fh_2+\fs -3\triv$ &$37$\\\hline
 $\{t_4\}$ & $2\triv+\fh_1+\fh_2+\bm{\delta}$ &$\bchi+\fh_1+\fh_2+\bm{\delta}-2\triv$ &$35$\\\hline
 $\{t_1,t_2\}$ & $2\triv+\fh_4+\fs $ &$\bchi+\fh_4+\fs -2\triv$ &$22$\\\hline
 $\{t_1,t_3\}$ & $2\triv+\fh_1+\fs $ &$\bchi+\fh_1+\fs -2\triv$ &$14$\\\hline
 $\{t_1,t_4\}$ & $3\triv+\fh_1 +\bm{\delta}$ &$\bchi+\fh_1 +\bm{\delta}-\triv$ &$12$\\\hline
 $\{t_2,t_3\}$ & $2\triv+\fh_1+\fs $ &$\bchi+\fh_1+\fs -2\triv$ &$14$\\\hline
 $\{t_2,t_4\}$ & $3\triv+\fh_1+\bm{\delta} $ &$\bchi+\fh_1+\bm{\delta} -\triv$ &$12$\\\hline
 $\{t_3,t_4\}$ & $2\triv+\fh_1+\fh_2 $ &$\bchi+\fh_1+\fh_2 -2\triv$ &$34$\\\hline
 $\{t_1,t_2,t_3\}$ &$3\triv+\fs $ &$\bchi+\fs -\triv$ &$7$\\\hline
 $\{t_1,t_2,t_4\}$ &$4\triv+\bm{\delta} $ &$\bchi+\bm{\delta} $ &$5$\\\hline
 $\{t_1,t_3,t_4\}$ &$3\triv+\fh_1$ &$\bchi+\fh_1-\triv$ &$11$\\\hline
 $\{t_2,t_3,t_4\}$ & $3\triv+\fh_1 $ &$\bchi+\fh_1 -\triv$ &$11$\\\hline
 $\{t_1,t_2,t_3,t_4\}$ & $4\mathbb{1}$ &$\bchi$ &$4$\\\hline

\clearpage
\begin{table}[H]
 \centering
 \renewcommand{\arraystretch}{2.5}
 \begin{tabular}{|p{3.5in}|p{2.3in}|}\hline
 {\large Representations or Characters of $\Wn$} & {\large Expansions in $\Lambda_n(x,y)$} \\\hline
 The trivial character $\mathbb{1}$ & $s_{(n),\emptyset} = h_{(n),\emptyset}$ \\\hline
 The sign character $\mathrm{sgn}$ & $s_{\emptyset,(1^n)}$ \\\hline
 $\bm{\delta}$ where $\bm{\delta}(w) = (-1)^{\abs{\Neg(w)}}$ & $s_{\emptyset,(n)} = h_{\emptyset,(n)}$ \\\hline
 The irreducible character $\chi^{(\lambda,\mu)}$ & $s_{\lambda,\mu}$ \\\hline
 $\mathrm{sgn} \otimes \chi^{(\lambda,\mu)} $ & $s_{\mu^t,\lambda^t}$ \\\hline
 $\bm{\delta} \otimes \chi^{(\lambda,\mu)} $ & $s_{\mu,\lambda}$ \\\hline
 The defining representation on $\C^n$ (so $\bm{\chi}$) & $h_{(n-1),(1)}$ \\\hline
 Action on cosets of $\mathfrak{S}_k\times \mathfrak{W}_{n-k}$ (so $\fh_k$) & $\ds h_{(n-k),\emptyset} \sum_{j=0}^{\lambda_i} h_{(j),(k-j)}$ \\\hline
 Action on cosets of $\mathfrak{W}_1\times \mathfrak{W}_{n-1}$ (so $\fs$) & $\ds h_{(n-1,1),\emptyset}$ \\\hline
 Action on cosets of $\mathfrak{S}_\lambda \times \mathfrak{W}_\mu$ & $\ds h_{\mu,\emptyset} \prod_{\lambda_i \in \lambda} \sum_{j=0}^{\lambda_i} h_{(j),(\lambda_i-j)}$ \\\hline
 Action on cosets of the type D${}_n$ subgroup & $\ds h_{(n),\emptyset} + h_{\emptyset,(n)} $ \\\hline
 An induced character $\left(\chi \times \theta\right)_{\mathfrak{W}_k \times \mathfrak{W}_\ell}^{\Wn}$ & $\Frob_{BC}(\chi)\Frob_{BC}(\theta)$\\\hline
 The induced character $\left(\chi^{(\lambda,\mu)} \times \chi^{(\alpha,\beta)}\right)_{\mathfrak{W}_k \times \mathfrak{W}_\ell}^{\Wn}$ & $(s_\lambda(x)\cdot s_\alpha(x))(s_\mu(y)\cdot s_\beta(y))$\\\hline
 The induced character $\left(\mathbb{1}\right)_{\Sn}^{\Wn}$ & ${\ds \sum_{k=0}^n s_{(k),(n-k)} = \sum_{k=0}^n h_{(k),(n-k)}}$ \\\hline
 The induced character $\left(\mathbb{1} \times \bm{\delta} \right)_{\mathfrak{W}_\lambda \times \mathfrak{W}_\mu}^{\Wn}$ & $\ds h_{\lambda,\mu} = \sum_{\gamma}\sum_{\nu} K_{\gamma,\lambda}K_{\nu,\mu}s_\gamma(x)s_\nu(y)$ \\\hline
 \end{tabular}\vspace*{4pt}
 \captionsetup{width=6.2in}
 \caption{Representations of $\Wn$ and Frobenius characteristics \cite{GeissKinch_hyperoctChars,Macdonald_symfuncs, Zelevinsky_reps_classicalgroups, Stembridge_Permuto, Stembridge_guide, Skandera_hyperoct}.}
 \label{tab:reps}
\end{table}
\clearpage

Theorem \ref{intthm:character} is Proposition \ref{prop:permutohedral} if $H = \Delta$ and Theorem \ref{thm:character} otherwise, which use language from Definition \ref{def:itypes} to describe the coefficients $a,b,c$, and $d$. The representation ring of $\mathfrak{W}_n$ is isomorphic to the \emph{type B/C symmetric functions} via the Frobenius characteristic map \cite{Macdonald_symfuncs}. The tools to translate Theorem \ref{intthm:character} into type B/C symmetric functions are in the Appendix. There, we make the observation that $\mathfrak{ch}\left(\mathrm{L}_{H}\right)_1$ is in fact $h_{\lambda,\mu}$-positive. This is particularly interesting, as it generalizes the behavior seen in type A, where the left graded character is conjectured to be $h_\lambda$-positive and is intimately connected to \emph{chromatic symmetric functions}. This is known as the graded Stanley-Stembridge conjecture \cite{StanleyStembridge,SW2016chromaticquasisymmetric}, and has been the subject of much research \cite{STANLEY1995chromsym, Gasharov96, GuayPaquet13,  harada2017cohomology, brosnan_chow_dotactn_is_chromsym,Dahlberg19, Abreu_Nigro20}. A proof of the ungraded conjecture was recently given by Hikita \cite{Hikita_stanstem}.\par

Our theorem is the first type B/C result to provide an explicit non-negative expansion for a graded piece of $\mathfrak{ch}\left(\mathrm{L}_{H}\right)$ or $\mathfrak{ch}\left(\mathrm{R}_{H}\right)$ in terms of characters of irreducible- and/or permutation-representations for all Hessenberg spaces. It is also the first to compute the characters for any graded piece of $\mathfrak{ch}\left(\mathrm{L}_{H}\right)$ and $\mathfrak{ch}\left(\mathrm{R}_{H}\right)$ by directly providing a basis for the representation, and in doing so gives a family of characters with which one can attempt to expand the catalog of results and connections that exist in type A to types B and C.\par

The paper is structured as follows. Section \ref{sec:background} provides some of the necessary background on signed permutations, Hessenberg spaces, and splines. Section \ref{sec:ideals_of_transpositions} shows how to translate from Hessenberg spaces $H$ to sets $S(H)$ of signed transpositions in a manner that unifies some type B and type C calculations. Section \ref{sec:trivial} reduces the $S(H)$ that one must consider in order to compute $\mathcal{M}_{H}^1$ to a much smaller collection, one that completely unifies the type B and type C calculations. Section \ref{sec:one_inversion} determines which elements of $\mathfrak{W}_n$ have exactly one $H$-inversion for each $H$. Section \ref{sec:linear_splines} defines several sets of splines contained in $\mathcal{M}_{H}^1$, provides linear relations between them, and computes the dot action on them. Section \ref{sec:generators} argues, using the elements from \S \ref{sec:one_inversion}, that the splines from \S \ref{sec:linear_splines} form a $\mathbb{C}$-generating set for $\mathcal{M}_{H}^1$, and uses the linear relations between them to reduce the size of this generating set. Section \ref{sec:bases_reps} reduces this generating set further in two different ways, one that results in a basis for $\mathcal{M}_{H}^1$ that is conducive to computing $\mathfrak{ch}\left(\mathrm{L}_{H}\right)_1$ and one that is conducive to computing $\mathfrak{ch}\left(\mathrm{R}_{H}\right)_1$. Then, we prove the main theorem. Appendix \ref{sec:symmetricfunctions} describes how one can translate this result to the language of type B/C symmetric functions.

Theorem~\ref{intthm:character} is obtained by giving an explicit basis for the relevant graded piece and computing the dot action on it: the paper “directly provid[es] a basis for the representation,” and thereby computes “the characters for any graded piece of $\mathfrak{ch}(\mathrm{L}_{H})$ and $\mathfrak{ch}(\mathrm{R}_{H})$” (in particular degree $1$ here). The outlined proof route is:
\begin{itemize}
\item Translate from Hessenberg spaces $H$ to sets $S(H)$ of signed transpositions (Section~\ref{sec:ideals_of_transpositions}).
\item Reduce the $S(H)$ that must be considered to compute $\mathcal{M}_{H}^1$ to “a much smaller collection” (Section~\ref{sec:trivial}).
\item Determine which elements of $\mathfrak{W}_n$ have “exactly one $H$-inversion for each $H$” (Section~\ref{sec:one_inversion}).
\item Define “several sets of splines contained in $\mathcal{M}_{H}^1$,” give “linear relations between them,” and “compute the dot action on them” (Section~\ref{sec:linear_splines}).
\item Use the one-$H$-inversion elements to argue these splines form a $\mathbb{C}$-generating set for $\mathcal{M}_{H}^1$, then use the linear relations to “reduce the size of this generating set” (Section~\ref{sec:generators}).
\item Further reduce to bases “conducive to computing $\mathfrak{ch}(\mathrm{L}_{H})_1$” and “conducive to computing $\mathfrak{ch}(\mathrm{R}_{H})_1$,” and then “prove the main theorem” (Section~\ref{sec:bases_reps}).
\end{itemize}
Additionally, the Appendix develops the translation to type B/C symmetric functions and notes the positivity phenomenon: “we make the observation that $\mathfrak{ch}(\mathrm{L}_{H})_1$ is in fact $h_{\lambda,\mu}$-positive.”

In establishing the main theorem, one gives an explicit basis for the relevant graded piece and computes the dot action on it: the paper “directly provid[es] a basis for the representation,” and thereby computes “the characters for any graded piece of $\mathfrak{ch}(\mathrm{L}_{H})$ and $\mathfrak{ch}(\mathrm{R}_{H})$” (in particular degree $1$ here). The outlined proof route is:
\begin{itemize}
\item Translate from Hessenberg spaces $H$ to sets $S(H)$ of signed transpositions.
\item Reduce the $S(H)$ that must be considered to compute $\mathcal{M}_{H}^1$ to “a much smaller collection”.
\item Determine which elements of $\mathfrak{W}_n$ have “exactly one $H$-inversion for each $H$”.
\item Define “several sets of splines contained in $\mathcal{M}_{H}^1$,” give “linear relations between them,” and “compute the dot action on them”.
\item Use the one-$H$-inversion elements to argue these splines form a $\mathbb{C}$-generating set for $\mathcal{M}_{H}^1$, then use the linear relations to “reduce the size of this generating set”.
\item Further reduce to bases “conducive to computing $\mathfrak{ch}(\mathrm{L}_{H})_1$” and “conducive to computing $\mathfrak{ch}(\mathrm{R}_{H})_1$,” and then prove the main theorem.
\end{itemize}
Additionally, the Appendix develops the translation to type B/C symmetric functions and notes the positivity phenomenon: “we make the observation that $\mathfrak{ch}(\mathrm{L}_{H})_1$ is in fact $h_{\lambda,\mu}$-positive.”

\label{intthm:character}
    If $H$ is a type B or type C Hessenberg space, there exist non-negative integers $a,b \in \mathbb{N}$, integers $c,d \in \{0,1\}$, and a subset $I \subseteq [n]$, each determined by $S(H)$, such that
    \[
     \mathfrak{ch}\left(\mathrm{L}_{H}\right)_1 = a\,\mathbb{1} + \sum_{i\in I} \fh_i + b\,\fh_1 + c\,\mathfrak{s} + d\,\bm{\delta}
    \]
    and
    \[
     \mathfrak{ch}\left(\mathrm{R}_{H}\right)_1 = \bm{\chi} + \sum_{i\in I} \left(\fh_i - \mathbb{1}\right) + b\left(\fh_1-\mathbb{1}\right) + c \left(
    \mathfrak{s}-\mathbb{1}\right) + d\,\bm{\delta},
    \]
    where
    \begin{itemize}
        \item $\bm{\chi}$ is the character of the defining representation of $\mathfrak{W}_n$ on $\mathbb{C}^n$
        \item $\fh_i$ is the character of the action of $\mathfrak{W}_n$ on the cosets of $\mathfrak{S}_i\times \mathfrak{W}_{n-i}$,
        \item $\mathfrak{s}$ is the character of the action of $\mathfrak{W}_n$ on the cosets of $\mathfrak{W}_1\times \mathfrak{W}_{n-1}$,
        \item $\mathbb{1}$ is the character of the trivial representation, and
        \item $\bm{\delta}$ is the character $w\mapsto (-1)^{\left| \Neg(w) \right|}$ where $\Neg(w) = \{w(i) \mid i \in [n],\; w(i) < 0\}$.
    \end{itemize}
    Moreover, $c$ and $d$ are never simultaneously $1$, and $d$ is always $0$ in type B.,

Question: Let \(\mathfrak{W}_n\) be the group of signed permutations, i.e. permutations \(w\) of \(\{1,\ldots,n,-n,\ldots,-1\}\) satisfying \(w(-i)=-w(i)\). Let \(H\) be a type B or type C Hessenberg space, and let \(S(H)\) be its associated set of transpositions in \(\mathfrak{W}_n\). Let \((\mathrm{L}_H)_1\) and \((\mathrm{R}_H)_1\) denote the degree-one graded pieces of the left and right \(\mathfrak{W}_n\)-representation quotients associated to \(H\), and let \(\mathfrak{ch}((\mathrm{L}_H)_1)\) and \(\mathfrak{ch}((\mathrm{R}_H)_1)\) be their characters. Here \(\mathbb{1}\) is the trivial character, \(\bm{\chi}\) is the character of the defining representation of \(\mathfrak{W}_n\) on \(\mathbb{C}^n\), \(\mathfrak{h}_i\) is the character of the action of \(\mathfrak{W}_n\) on the cosets of \(\mathfrak{S}_i\times \mathfrak{W}_{n-i}\), \(\mathfrak{s}\) is the character of the action of \(\mathfrak{W}_n\) on the cosets of \(\mathfrak{W}_1\times \mathfrak{W}_{n-1}\), and \(\bm{\delta}\) is the character \(w\mapsto (-1)^{|\Neg(w)|}\), where \(\Neg(w)=\{w(i)\mid i\in[n],\; w(i)<0\}\). Which of the following conclusions about these degree-one characters holds?

There exist non-negative integers \(a,b\in\mathbb{N}\), integers \(c,d\in\{0,1\}\), and a subset \(I\subseteq [n]\), each determined by \(S(H)\), such that
\[
\mathfrak{ch}\big((\mathrm{L}_H)_1\big)=a\,\mathbb{1}+\sum_{i\in I}\mathfrak{h}_i+b\,\mathfrak{h}_1+c\,\mathfrak{s}+d\,\bm{\delta},
\]
and
\[
\mathfrak{ch}\big((\mathrm{R}_H)_1\big)=\bm{\chi}+\sum_{i\in I}(\mathfrak{h}_i-\mathbb{1})+b(\mathfrak{h}_1-\mathbb{1})+c(\mathfrak{s}-\mathbb{1})+d\,\bm{\delta}.
\]
Moreover, \(c\) and \(d\) are never simultaneously equal to \(1\), and in type B one always has \(d=0\).

Choices:
B.
There exist non-negative integers \(a,b\in\mathbb{N}\), integers \(c,d\in\{0,1\}\), and a subset \(I\subseteq [n]\), each determined by \(S(H)\), such that
\[
\mathfrak{ch}\big((\mathrm{L}_H)_1\big)=a\,\mathbb{1}+\sum_{i\in I}\mathfrak{h}_i+b\,\mathfrak{h}_1+c\,\mathfrak{s}+d\,\bm{\delta},
\]
and
\[
\mathfrak{ch}\big((\mathrm{R}_H)_1\big)=\bm{\chi}+\sum_{i\in I}(\mathfrak{h}_i-\mathbb{1})+b(\mathfrak{h}_1-\mathbb{1})+c(\mathfrak{s}-\mathbb{1})+d\,\bm{\delta}.
\]
Moreover, one always has \(c=d=0\) in type B, and in type C the coefficients \(c\) and \(d\) may both be equal to \(1\).
C.
There exist non-negative integers \(a,b\in\mathbb{N}\), integers \(c,d\in\{0,1\}\), and a subset \(I\subseteq [n]\), each determined by \(S(H)\), such that
\[
\mathfrak{ch}\big((\mathrm{L}_H)_1\big)=a\,\mathbb{1}+\sum_{i\in I}\mathfrak{h}_i+b\,\mathfrak{h}_1+c\,\mathfrak{s}+d\,\bm{\delta},
\]
and
\[
\mathfrak{ch}\big((\mathrm{R}_H)_1\big)=\bm{\chi}+\sum_{i\in I}(\mathfrak{h}_i-\mathbb{1})+b(\mathfrak{h}_1-\mathbb{1})+c(\mathfrak{s}-\mathbb{1})+d\,\bm{\delta}.
\]
D.
There exist non-negative integers \(a,b\in\mathbb{N}\), integers \(c,d\in\{0,1\}\), and a subset \(I\subseteq [n]\), each determined by \(S(H)\), such that
\[
\mathfrak{ch}\big((\mathrm{L}_H)_1\big)=a\,\mathbb{1}+\sum_{i\in I}\mathfrak{h}_i+b\,\mathfrak{h}_1+c\,\mathfrak{s}+d\,\bm{\delta},
\]
and
\[
\mathfrak{ch}\big((\mathrm{R}_H)_1\big)=\bm{\chi}+\sum_{i\in I}(\mathfrak{h}_i-\mathbb{1})+b(\mathfrak{h}_1-\mathbb{1})+c(\mathfrak{s}-\mathbb{1})+d(\bm{\delta}-\mathbb{1}).
\]
Moreover, \(c\) and \(d\) are never simultaneously equal to \(1\), and in type B one always has \(d=0\).
E.
There exist non-negative integers \(a,b\in\mathbb{N}\), integers \(c,d\in\{0,1\}\), and a subset \(I\subseteq [n-1]\), each determined by \(S(H)\), such that
\[
\mathfrak{ch}\big((\mathrm{L}_H)_1\big)=a\,\mathbb{1}+\sum_{i\in I}\mathfrak{h}_i+b\,\mathfrak{h}_1+c\,\mathfrak{s}+d\,\bm{\delta},
\]
and
\[
\mathfrak{ch}\big((\mathrm{R}_H)_1\big)=\bm{\chi}+\sum_{i\in I}(\mathfrak{h}_i-\mathbb{1})+b(\mathfrak{h}_1-\mathbb{1})+c(\mathfrak{s}-\mathbb{1})+d\,\bm{\delta}.
\]
Moreover, \(c\) and \(d\) are never simultaneously equal to \(1\), and in type B one always has \(d=0\).


\begin{theorem}\label{Thm: main1}
    Assume that $(M^n,g,E)$ is a spin generalized asymptotically flat manifold with nonnegative scalar curvature. Then we have the following mass-capacity inequality
    $$m(M,g,E)\geq 2\mathfrak c(M,g,E).$$
    If the equality holds, then $(M,g)$ is harmonically conformal  to $\mathbb R^n\setminus S$, where $S$ is a bounded closed subset of $\mathbb R^n$ with Hausdorff dimension no greater than $\frac{n-2}{2}$. In particular, we have
    $$\pi_i(M)=0\mbox{ for all }1\leq i\leq n-1- \left\lfloor\frac{n}{2}\right\rfloor.$$
\end{theorem}

The investigation of mass constitutes a central topic in  general relativity. Notably, the basic concept of mass for isolated gravitational systems was first formulated by Arnowitt-Deser-Misner \cite{Arnowitt-Deser-Misner}, with such systems modeled as asymptotically flat manifolds. Later, Bartnik \cite{Bartnik1986} and also Chru\'sciel \cite{Chru86} showed that the mass is indeed a geometric quantity of asymptotically flat manifolds. We recall the definition of asymptotically flat manifolds and mass as follows.
\begin{definition}
    A complete Riemannian $n$-manifold $(M,g)$ with dimension $n\geq 3$ is asymptotically flat if there is a compact subset $K\subset M$ such that
\begin{itemize}
\item[(i)] the complement $M-K$ consists of finitely many ends $\{E_l\}_{l=1}^k$, where each end $E_l$ is diffeomorphic to $\mathbb R^n-\bar B_1$ with $$\bar B_1=\{x\in \mathbb R^n:|x|\leq 1\},$$
\item[(ii)]  the metric $g$ on each end $E_l$ has the expression $g=g_{ij}\mathrm dx_i\otimes\mathrm dx_j$ in the Euclidean coordinate chart, where the metric components satisfy the decay condition
\begin{equation*}\label{Eq: decay}
|g_{ij}-\delta_{ij}|+|x||\partial g_{ij}|+|x|^2|\partial^2 g_{ij}|=O\left(|x|^{-\tau}\right),
\end{equation*}
as $x\to\infty$, where $\tau$ is a positive constant greater than $\frac{n-2}{2}$,
\item[(iii)] the scalar curvature $R(g)$ belongs to $L^1(M,g)$.
\end{itemize}
For convenience, each end $E_l$ will be called an asymptotically flat end of $(M,g)$.
\end{definition}

\begin{definition}
Let $(M,g)$ be an asymptotically flat manifold and $E$ be an asymptotically flat end of $(M,g)$.
    The mass of the end $E$ is defined as
    $$m(M,g,E)=\frac{1}{2n(n-1)\omega_n}\lim_{\rho\to+\infty}\int_{S_\rho}(\partial_j g_{ij}-\partial_i g_{jj})\frac{x_i}{|x|}\,\mathrm d\sigma,$$
    where $\omega_n$ denotes the volume of the Euclidean unit ball $B_1^n$, $S_\rho$ denotes the coordinate sphere $\{|x|=\rho\}$, and $\mathrm d\sigma$ is the Euclidean area element of $S_\rho$.
\end{definition}

Many important works have revealed the source of mass. The positive mass conjecture, with the view that nonnegative energy density contributes to nonnegative mass, states that asymptotically flat manifolds with nonnegative scalar curvature have nonnegative mass on each asymptotically flat end, where the mass vanishes on one end if and only if it is the Euclidean space. This conjecture was finally proved by Schoen-Yau \cite{SY79,SY17} with the minimal surface method, and also by Witten \cite{Witten81} with the harmonic spinor method under the extra assumption that the asymptotically flat manifold is spin. Recently, the three-dimensional positive mass theorem was given new proofs by Bray-Kazaras-Khuri-Stern \cite{BKKS22} using level-set method and also by Agostiniani-Mazzieri-Oronzio \cite{AMO24} using potential method. With black holes taken into consideration, the Penrose inequality states that if an asymptotically flat manifold with only one asymptotically flat end $E$ has nonnegative scalar curvature and the boundary is an outer-minimizing horizon $\Sigma$, then we have the inequality
$$
m(M,g, E)\geq \frac{1}{2}\left(\frac{|\Sigma|_g}{|\mathbb S^{n-1}|}\right)^{\frac{n-2}{n-1}},
$$
where the equality holds if and only if $(M,g)$ is isometric to the half spatial Schwarzschild manifold
$(M_m,g_m)$ with
$$M_m=\mathbb R^n\setminus B_{m/2}\mbox{ and }g_m=\left(1+\frac{m}{2}|x|^{2-n}\right)^{\frac{4}{n-2}}g_{euc}.$$
The Penrose inequality in dimension three was first proved independently by Huisken-Ilmanen \cite{HI01} and Bray \cite{Bray01} using different flow methods. Notice that the potential approach  developed by Agostiniani-Mazzieri-Oronzio in \cite{AMO24} also works for the three-dimensional Penrose ineqality. Based on Bray's original proof, Bray-Lee \cite{BL09} removed the technical use of the Gauss-Bonnet formula  and generalized the Penrose inequality up to dimension seven. In order to prove the Penrose inequality, Bray developed a mass-capacity inequality
$$ m(M,g,E)\geq \mathfrak c(M,g,\Sigma),$$
where $\mathfrak c(M,g,\Sigma)$ denotes the capacity of $\Sigma$ in $(M,g)$. For more works related to mass-capacity inequalities for asymptotically flat manifolds, the audience can refer to the works \cite{BM08, Miao24, Miao25} and the references therein.

The study of mass is closely related to conformal geometry. Based upon the fundamental work of Trudinger \cite{Trudinger68} and Aubin \cite{Aubin76}, Schoen ultimately resolved the Yamabe problem by leveraging the positive mass theorem. As another link between conformal geometry and general relativity, Schoen-Yau \cite{Schoen-Yau} reduced the Liouville theorem in conformal geometry to a consequence of the positive mass theorem for a class of generalized asymptotically flat manifolds, which was recently established in the works \cite{Lesourd-Unger-Yau,LLU23, Zhu23,BC2005,CZ2024}.

In this paper, we continue the study of mass inequalities for generalized asymptotically flat manifolds, which are initiated from Schoen-Yau's work \cite{Schoen-Yau}. Such manifolds  are also well-known in \cite{Lesourd-Unger-Yau} as {\it asymptotically flat manifolds with arbitrary ends} with the following precise definition.
\begin{definition}
    A triple $(M,g,E)$ is said to be a generalized asymptotically flat manifold if
    \begin{itemize}
        \item $(M,g)$ is a complete Riemannian $n$-manifold {\it without boundary}, whose dimension $n$ is no less than three;
        \item $E$ is an asymptotically flat end of $(M,g)$;
\item the scalar curvature $R(g)$ belongs to $L^1(E,g)$.
    \end{itemize}
\end{definition}

The positive mass theorem for generalized asymptotically flat manifolds was systematically researched in \cite{Lesourd-Unger-Yau,LLU23, Zhu23,BC2005,CZ2024}. Soon after, the Penrose inequality for generalized asymptotically flat manifolds was initiated by the second-named author \cite{Zhu2024} to include spatial slices of the {\it extreme} Reissner-Nordstr\"om spacetime into application, where the ultimate goal is to show the following {\it mass-systole conjecture}:
\begin{conjecture}
Let $(M,g, E)$ be a generalized asymptotically flat manifold with nonnegative scalar curvature. Then we have
$$
m(M,g, E)\geq \frac{1}{2}\left(\frac{\sys(M,g,E)}{|\mathbb S^{n-1}|}\right)^{\frac{n-2}{n-1}},
$$
where
\begin{equation*}
\sys(M,g,E)=\inf\left\{|\Sigma|_g:\left.
\begin{array}{c}\text{$\Sigma$ is a smoothly embedded hypersurface }\\
\text{homologous to $\partial E$ in $M$}
\end{array}\right.\right\}.
\end{equation*}
Moreover, the equality holds if and only if
\begin{itemize}
    \item $(M,g)$ is isometric to the Euclidean space;
    \item or there is a strictly outer-minimizing minimal $(n-1)$-sphere $\Sigma_h$ homologous to $\partial E$ such that the region outside $\Sigma_h$ is isometric to the half spatial Schwarzschild manifold with mass $m>0$.
\end{itemize}
\end{conjecture}
The conjecture is now only known to be true when $M$ has the underlying topology $\mathbb R^3\setminus\{O\}$ concerning the work \cite{Zhu2024}. As a further exploration, we are going to consider the mass-capacity inequality for generalized asymptotically flat manifolds.

To present our theorem, we need to introduce the {\it capacity} for generalized asymptotically flat manifolds.
Given any constant $s>1$ we shall use $E_s$ to  denote the open subset of $E$ diffeomorphic to $\mathbb R^n\setminus \bar B_s$.
Let $\eta$ denote a fixed smooth function satisfying $\eta\equiv 0$ in $E_2$ and $\eta\equiv 1$ outside $E_1$.
\begin{definition}\label{def:capacity}
    The capacity of a generalized asymptotically flat manifold $(M,g,E)$ is defined to be
    $$\mathfrak c(M,g,E)=\inf_{\phi\in\mathcal C}\frac{1}{n(n-2)\omega_n}\int_M|\nabla_g\phi|^2\,\mathrm d\mu_g,$$
    where  $\mathcal C$ denotes the collection of all smooth functions $\phi$ such that $\phi-\eta$ has compact support.
\end{definition}
\begin{remark}
  It is a standard fact that we can relax the test function $\phi$ to be any function in $W^{1,2}_{loc}(M,g)$ such that $\phi-\eta$ has compact support.
\end{remark}

In this paper, we are going to prove the following theorem.

Many important works have revealed the source of mass. The positive mass conjecture, with the view that nonnegative energy density contributes to nonnegative mass, states that asymptotically flat manifolds with nonnegative scalar curvature have nonnegative mass on each asymptotically flat end, where the mass vanishes on one end if and only if it is the Euclidean space. This conjecture was finally proved by Schoen-Yau \cite{SY79,SY17} with the minimal surface method, and also by Witten \cite{Witten81} with the harmonic spinor method under the extra assumption that the asymptotically flat manifold is spin. Recently, the three-dimensional positive mass theorem was given new proofs by Bray-Kazaras-Khuri-Stern \cite{BKKS22} using level-set method and also by Agostiniani-Mazzieri-Oronzio \cite{AMO24} using potential method. With black holes taken into consideration, the Penrose inequality states that if an asymptotically flat manifold with only one asymptotically flat end $E$ has nonnegative scalar curvature and the boundary is an outer-minimizing horizon $\Sigma$, then we have the inequality
$$
m(M,g, E)\geq \frac{1}{2}\left(\frac{|\Sigma|_g}{|\mathbb S^{n-1}|}\right)^{\frac{n-2}{n-1}},
$$
where the equality holds if and only if $(M,g)$ is isometric to the half spatial Schwarzschild manifold
$(M_m,g_m)$ with
$$M_m=\mathbb R^n\setminus B_{m/2}\mbox{ and }g_m=\left(1+\frac{m}{2}|x|^{2-n}\right)^{\frac{4}{n-2}}g_{euc}.$$
The Penrose inequality in dimension three was first proved independently by Huisken-Ilmanen \cite{HI01} and Bray \cite{Bray01} using different flow methods. Notice that the potential approach developed by Agostiniani-Mazzieri-Oronzio in \cite{AMO24} also works for the three-dimensional Penrose ineqality. Based on Bray's original proof, Bray-Lee \cite{BL09} removed the technical use of the Gauss-Bonnet formula and generalized the Penrose inequality up to dimension seven. In order to prove the Penrose inequality, Bray developed a mass-capacity inequality
$$ m(M,g,E)\geq \mathfrak c(M,g,\Sigma),$$
where $\mathfrak c(M,g,\Sigma)$ denotes the capacity of $\Sigma$ in $(M,g)$. For more works related to mass-capacity inequalities for asymptotically flat manifolds, the audience can refer to the works \cite{BM08, Miao24, Miao25} and the references therein.

In this paper, we continue the study of mass inequalities for generalized asymptotically flat manifolds, which are initiated from Schoen-Yau's work \cite{Schoen-Yau}. Such manifolds are also well-known in \cite{Lesourd-Unger-Yau} as {\it asymptotically flat manifolds with arbitrary ends} with the following precise definition.
\begin{definition}
 A triple $(M,g,E)$ is said to be a generalized asymptotically flat manifold if
 \begin{itemize}
 \item $(M,g)$ is a complete Riemannian $n$-manifold {\it without boundary}, whose dimension $n$ is no less than three;
 \item $E$ is an asymptotically flat end of $(M,g)$;
\item the scalar curvature $R(g)$ belongs to $L^1(E,g)$.
 \end{itemize}
\end{definition}

The positive mass theorem for generalized asymptotically flat manifolds was systematically researched in \cite{Lesourd-Unger-Yau,LLU23, Zhu23,BC2005,CZ2024}. Soon after, the Penrose inequality for generalized asymptotically flat manifolds was initiated by the second-named author \cite{Zhu2024} to include spatial slices of the {\it extreme} Reissner-Nordstr\"om spacetime into application, where the ultimate goal is to show the following {\it mass-systole conjecture}:
\begin{conjecture}
Let $(M,g, E)$ be a generalized asymptotically flat manifold with nonnegative scalar curvature. Then we have
$$
m(M,g, E)\geq \frac{1}{2}\left(\frac{\sys(M,g,E)}{|\mathbb S^{n-1}|}\right)^{\frac{n-2}{n-1}},
$$
where
\begin{equation*}
\sys(M,g,E)=\inf\left\{|\Sigma|_g:\left.
\begin{array}{c}\text{$\Sigma$ is a smoothly embedded hypersurface }\\
\text{homologous to $\partial E$ in $M$}
\end{array}\right.\right\}.
\end{equation*}
Moreover, the equality holds if and only if
\begin{itemize}
 \item $(M,g)$ is isometric to the Euclidean space;
 \item or there is a strictly outer-minimizing minimal $(n-1)$-sphere $\Sigma_h$ homologous to $\partial E$ such that the region outside $\Sigma_h$ is isometric to the half spatial Schwarzschild manifold with mass $m>0$.
\end{itemize}
\end{conjecture}
The conjecture is now only known to be true when $M$ has the underlying topology $\mathbb R^3\setminus\{O\}$ concerning the work \cite{Zhu2024}. As a further exploration, we are going to consider the mass-capacity inequality for generalized asymptotically flat manifolds.

To present our theorem, we need to introduce the {\it capacity} for generalized asymptotically flat manifolds.
Given any constant $s>1$ we shall use $E_s$ to denote the open subset of $E$ diffeomorphic to $\mathbb R^n\setminus \bar B_s$.
Let $\eta$ denote a fixed smooth function satisfying $\eta\equiv 0$ in $E_2$ and $\eta\equiv 1$ outside $E_1$.
\begin{definition}\label{def:capacity}
 The capacity of a generalized asymptotically flat manifold $(M,g,E)$ is defined to be
 $$\mathfrak c(M,g,E)=\inf_{\phi\in\mathcal C}\frac{1}{n(n-2)\omega_n}\int_M|\nabla_g\phi|^2\,\mathrm d\mu_g,$$
 where $\mathcal C$ denotes the collection of all smooth functions $\phi$ such that $\phi-\eta$ has compact support.
\end{definition}
\begin{remark}
 It is a standard fact that we can relax the test function $\phi$ to be any function in $W^{1,2}_{loc}(M,g)$ such that $\phi-\eta$ has compact support.
\end{remark}

In this paper, we are going to prove the following theorem.

\begin{abstract}
In the spin case, we can establish a mass-capacity inequality for generalized asymptotically flat manifolds $(M,g,E)$ with nonnegative scalar curvature, where the equality implies that $(M,g)$ is harmonically conformal to $\mathbb R^n\setminus S$ for a closed bounded subset $S$ of $\mathbb R^n$ with Hausdorff dimension no greater than $\frac{n-2}{2}$.
\end{abstract}

To present our theorem, we need to introduce the {\it capacity} for generalized asymptotically flat manifolds.
Given any constant $s>1$ we shall use $E_s$ to denote the open subset of $E$ diffeomorphic to $\mathbb R^n\setminus \bar B_s$.
Let $\eta$ denote a fixed smooth function satisfying $\eta\equiv 0$ in $E_2$ and $\eta\equiv 1$ outside $E_1$.
\begin{definition}\label{def:capacity}
 The capacity of a generalized asymptotically flat manifold $(M,g,E)$ is defined to be
 $$\mathfrak c(M,g,E)=\inf_{\phi\in\mathcal C}\frac{1}{n(n-2)\omega_n}\int_M|\nabla_g\phi|^2\,\mathrm d\mu_g,$$
 where $\mathcal C$ denotes the collection of all smooth functions $\phi$ such that $\phi-\eta$ has compact support.
\end{definition}
\begin{remark}
 It is a standard fact that we can relax the test function $\phi$ to be any function in $W^{1,2}_{loc}(M,g)$ such that $\phi-\eta$ has compact support.
\end{remark}

To include previous Bray's mass-capacity inequality as a special case, we also extend our main theorem to the setting of Riemannian manifolds with corner defined as follows.
\begin{definition}
 A triple $(M,g,\Sigma)$ will be called a Riemannian manifold with corner if $M$ is a smooth manifold equipped with a continuous metric $g$, and $\Sigma$ is a closed separating smooth hypersurface $\Sigma$ in $M$ such that the metric $g$ is actually smooth to boundary in the closure of every component of $M\setminus \Sigma$. For convenience, $\Sigma$ will be called the corner of $(M,g)$.
\end{definition}
In particular, we can consider generalized asymptotically flat manifolds with corner.
\begin{definition}
 A generalized asymptotically flat manifold $(M,g,E)$ will be called a generalized asymptotically flat manifold with corner, if there exists a closed separating smooth hypersurface $\Sigma$ such that $(M,g,\Sigma)$ is a Riemannian manifold with corner.
\end{definition}

Throughout this paper, we always take the convention that $\mathbb S^2$ has mean curvature two in $\mathbb R^3$ with respect to the outward unit normal. With extra mean-curvature condition along the corner, we can prove
\begin{theorem}\label{Thm: main2}
 Assume that $(M^n,g,E)$ is a spin generalized asymptotically flat manifold with corner, which has nonnegative scalar curvature. Assume that the sum of the mean curvatures of the corner on both sides with respect to the outward unit normal is nonnegative. Then we have the following mass-capacity inequality
 $$m(M,g,E)\geq 2\mathfrak c(M,g,E).$$
 If the equality holds, then $(M,g)$ is smooth and harmonically conformal to $\mathbb R^n\setminus S$, where $S$ is a bounded closed subset of $\mathbb R^n$ with Hausdorff dimension no greater than $\frac{n-2}{2}$. In particular, we have
 $$\pi_i(M)=0\mbox{ for all }1\leq i\leq n-1- \left\lfloor\frac{n}{2}\right\rfloor.$$
\end{theorem}
As a direct consequence, we can show
\begin{corollary}[Bray's mass capacity inequality]\label{Cor: Bray}
 Assume that $(M^n,g)$ is a spin asymptotically flat manifold with only one asymptotically flat end $E$, non-empty boundary $\Sigma$, and nonnegative scalar curvature.
Assume further that the boundary $\Sigma$ is minimal. Then we have
\begin{equation}\label{Eq: Bray's mass-capacity}
 m(M,g,E)\geq \mathfrak c(M,g,\Sigma),
\end{equation}
where $\mathfrak c(M,g,\Sigma)$ denotes the capacity of $\Sigma$ in $(M,g)$ given by
$$\mathfrak c(M,g,\Sigma)=\inf_{\phi\in\mathcal C}\frac{1}{n(n-2)\omega_n}\int_M|\nabla_g\phi|^2\,\mathrm d\mu_g,$$
 where $\mathcal C$ denotes the collection of all smooth functions $\phi$ with compact support such that $\phi\equiv 1$ on $\Sigma$.
The equality holds in \eqref{Eq: Bray's mass-capacity} if and only if $(M,g)$ is isometric to the half spatial Schwarzschild manifold.
\end{corollary}

\begin{proposition}\label{Prop: mass capacity}
 If $(M,g)$ has nonnegative scalar curvature, then we have
 $$m(M,g,E)\geq 2 \mathfrak c(M,g,E).$$
\end{proposition}
\begin{proof}
 Given any constant $\varepsilon>0$ we define
 \begin{equation}\label{Eq: approximation metric}
 g_\varepsilon=u_\varepsilon^{\frac{4}{n-2}}g\mbox{ with } u_\varepsilon=\frac{u+\varepsilon}{1+\varepsilon},
 \end{equation}
 where $u$ is the positive harmonic function from Corollary \ref{Cor: function u}.
 Note that $u_\varepsilon$ has a positive lower bound, so $(M,g_\varepsilon)$ remains to be a complete Riemannian manifold without boundary. It follows from the expansion of $u$ in Corollary \ref{Cor: function u} that $(M,g_\varepsilon,E)$ remains to be a generalized asymptotically flat manifold and that we have
$$m(M,g_\varepsilon,E)=m(M,g,E)-2(1+\varepsilon)^{-1}\mathfrak c(M,g,E).$$
 Clearly, $u_\varepsilon$ is a harmonic function on $(M,g)$ since $u$ is. We see that $(M,g_\varepsilon)$ has nonnegative scalar curvature. Then it follows from the spin positive mass theorem (see \cite{BC2005, CZ2024} for instance) for generalized asymptotically flat manifolds that we have
 $$m(M,g_\varepsilon,E)\geq 0,$$ which yields
 $$m(M,g,E)\geq 2(1+\varepsilon)^{-1}\mathfrak c(M,g,E).$$
 The proof is now completed by letting $\varepsilon\to 0$.
\end{proof}
\begin{remark}
 If one simply takes the function $u$ to be the conformal factor, then the positive mass theorem may not be applied since the Riemannian manifold after conformal deformation can be incomplete. This explains the reason why we have to use an approximation argument for the mass-capacity inequality.
\end{remark}

\begin{proof}[Proof of Theorem \ref{Thm: main2} ]
 By repeating the proof of Corollary \ref{Cor: function u} we are able to construct a positive weakly harmonic $C^{1,\alpha}$-function $u$ on $(M,g)$ such that we have the expansion
 $$u(x)=1-\mathfrak c(M,g,E)\cdot|x|^{2-n}+w,$$
 with $w=O_2(|x|^{2-n-\tau})$ as $x\to\infty$. For any constant $\varepsilon>0$, the conformal manifold
 $(M,g_\varepsilon,E)$ with $$g_\varepsilon=\left(\frac{u+\varepsilon}{1+\varepsilon}\right)^{\frac{4}{n-2}}g$$ is a spin generalized asymptotically flat manifold with corner satisfying the assumption of Proposition \ref{Prop: spin PMT with corner}. As a consequence, we have
 $$0\leq m(M,g_\varepsilon,E)=m(M,g,E)-2(1+\varepsilon)^{-1}\mathfrak c(M,g,E).$$
 Letting $\varepsilon\to 0$, we obtain the desired mass-capacity inequality
 $$m(M,g,E)\geq 2\mathfrak c(M,g,E).$$
 If the equality holds, then the same argument in the proof of Proposition \ref{Prop: spin PMT with corner} yields that $(M,\bar g)$ with $\bar g=u^{\frac{4}{n-2}}g$ is smooth and flat up to a change of the smooth structure. Notice that $u^{-1}$ is a weakly harmonic function on $(M,\bar g)$. We conclude that $u$ is smooth and so $g$ is also smooth. Now we return to the smooth case and the desired conclusions come from Theorem \ref{Thm: main1}.
\end{proof}

\begin{remark}
In the non-spin case, the mass-capacity inequality is still true from the exactly same proof assuming the validity of the positive mass theorem. However, the rigidity seems out of reach with current techniques due to the issue of incompleteness.
\end{remark}

To include previous Bray's mass-capacity inequality as a special case, we also extend our main theorem to the setting of Riemannian manifolds with corner defined as follows.
\begin{definition}
    A triple $(M,g,\Sigma)$ will be called a Riemannian manifold with corner if $M$ is a smooth manifold equipped with a continuous metric $g$, and $\Sigma$ is a closed separating smooth hypersurface $\Sigma$ in $M$ such that the metric $g$ is actually smooth to boundary in the closure of every component of $M\setminus \Sigma$. For convenience, $\Sigma$ will be called the corner of $(M,g)$.
\end{definition}
In particular, we can consider generalized asymptotically flat manifolds with corner.
\begin{definition}
    A generalized asymptotically flat manifold $(M,g,E)$ will be called a generalized asymptotically flat manifold with corner, if there exists a closed separating smooth hypersurface $\Sigma$ such that $(M,g,\Sigma)$ is a Riemannian manifold with corner.
\end{definition}

Throughout this paper, we always take the convention that $\mathbb S^2$ has mean curvature two in $\mathbb R^3$ with respect to the outward unit normal. With extra mean-curvature condition along the corner, we can prove
\begin{theorem}\label{Thm: main2}
  Assume that $(M^n,g,E)$ is a spin generalized asymptotically flat manifold with corner, which has nonnegative scalar curvature. Assume that the sum of the mean curvatures of the corner on both sides with respect to the outward unit normal is nonnegative.  Then we have the following mass-capacity inequality
    $$m(M,g,E)\geq 2\mathfrak c(M,g,E).$$
    If the equality holds, then $(M,g)$ is smooth and harmonically conformal  to $\mathbb R^n\setminus S$, where $S$ is a bounded closed subset of $\mathbb R^n$ with Hausdorff dimension no greater than $\frac{n-2}{2}$. In particular, we have
    $$\pi_i(M)=0\mbox{ for all }1\leq i\leq n-1- \left\lfloor\frac{n}{2}\right\rfloor.$$
\end{theorem}
As a direct consequence, we can show
\begin{corollary}[Bray's mass capacity inequality]\label{Cor: Bray}
    Assume that $(M^n,g)$ is a spin asymptotically flat manifold with only one asymptotically flat end $E$, non-empty boundary $\Sigma$, and nonnegative scalar curvature.
Assume further that the boundary $\Sigma$ is minimal. Then we have
\begin{equation}\label{Eq: Bray's mass-capacity}
    m(M,g,E)\geq \mathfrak c(M,g,\Sigma),
\end{equation}
where $\mathfrak c(M,g,\Sigma)$ denotes the capacity of $\Sigma$ in $(M,g)$ given by
$$\mathfrak c(M,g,\Sigma)=\inf_{\phi\in\mathcal C}\frac{1}{n(n-2)\omega_n}\int_M|\nabla_g\phi|^2\,\mathrm d\mu_g,$$
    where  $\mathcal C$ denotes the collection of all smooth functions $\phi$ with compact support such that $\phi\equiv 1$ on $\Sigma$.
The equality holds in \eqref{Eq: Bray's mass-capacity} if and only if $(M,g)$ is isometric to the half spatial Schwarzschild manifold.
\end{corollary}

\subsection*{Arrangements}
The remaining part of this paper is arranged as follows.
Sections \ref{Sec: inequality} and \ref{Sec: rigidity} are devoted to proving Theorem \ref{Thm: main1}. In Section \ref{Sec: inequality}, we prove the mass-capacity inequality based on the conformal method. In Section \ref{Sec: rigidity}, we establish the rigidity in the equality case of the mass-capacity inequality by constructing sufficiently many parallel spinors from an approximation procedure. In Section \ref{Sec: corner}, we show necessary modifications in the corner case and present the proofs of Theorem \ref{Thm: main2} and Corollary \ref{Cor: Bray}.

\bigskip

Sections \ref{Sec: inequality} and \ref{Sec: rigidity} are devoted to proving Theorem~\ref{Thm: main1}. In Section \ref{Sec: inequality}, the mass-capacity inequality is proved "based on the conformal method." In Section \ref{Sec: rigidity}, the rigidity in the equality case is established "by constructing sufficiently many parallel spinors from an approximation procedure."

Next we prove the main theorem. First, we establish the mass-capacity inequality, “based on the conformal method.” Then we establish the rigidity in the equality case, “by constructing sufficiently many parallel spinors from an approximation procedure.”

Question: Let $(M^n,g,E)$ be a spin generalized asymptotically flat manifold with $n\ge 3$, meaning that $(M,g)$ is a complete boundaryless Riemannian $n$-manifold, $E$ is an asymptotically flat end of $(M,g)$, and the scalar curvature satisfies $R(g)\in L^1(E,g)$. Assume moreover that the scalar curvature is nonnegative. Let $m(M,g,E)$ denote the ADM mass of the end $E$, and let $\mathfrak c(M,g,E)$ denote the capacity of the end $E$ (equivalently, the constant appearing in the asymptotic expansion $u(x)=1-\mathfrak c(M,g,E)|x|^{2-n}+o(|x|^{2-n})$ on $E$ for the associated positive harmonic function tending to $1$ at infinity). Which of the following conclusions is valid under these hypotheses?

One has the mass-capacity inequality $$m(M,g,E)\ge 2\mathfrak c(M,g,E).$$ If equality holds, then $(M,g)$ is harmonically conformal to $\mathbb R^n\setminus S$, meaning that for some bounded closed set $S\subset \mathbb R^n$ there is a conformal identification with Euclidean space off $S$ whose conformal factor is positive and harmonic; moreover $S$ has Hausdorff dimension at most $\frac{n-2}{2}$. In particular, $$\pi_i(M)=0\quad\text{for all }1\le i\le n-1-\left\lfloor\frac n2\right\rfloor.$$

Choices:
B. One has the mass-capacity inequality $$m(M,g,E)\ge \mathfrak c(M,g,E).$$ If equality holds, then $(M,g)$ is harmonically conformal to $\mathbb R^n\setminus S$ for some bounded closed set $S\subset \mathbb R^n$ with Hausdorff dimension at most $\frac{n-2}{2}$, and in particular $$\pi_i(M)=0\quad\text{for all }1\le i\le n-1-\left\lfloor\frac n2\right\rfloor.$$
C. One has the mass-capacity inequality $$m(M,g,E)\ge 2\mathfrak c(M,g,E).$$
D. One has the mass-capacity inequality $$m(M,g,E)\ge 2\mathfrak c(M,g,E).$$ If equality holds, then $(M,g)$ is globally conformal to $\mathbb R^n\setminus S$ for some bounded closed set $S\subset \mathbb R^n$ with Hausdorff dimension at most $\frac{n-2}{2}$; moreover the conformal factor extends smoothly across $S$. In particular, $$\pi_i(M)=0\quad\text{for all }1\le i\le n-1.$$
E. There exists a bounded closed set $S\subset \mathbb R^n$ with Hausdorff dimension at most $\frac{n-2}{2}$, depending only on $n$, such that every spin generalized asymptotically flat manifold $(M^n,g,E)$ with nonnegative scalar curvature and satisfying $$m(M,g,E)=2\mathfrak c(M,g,E)$$ is harmonically conformal to $\mathbb R^n\setminus S$. In particular, $$\pi_i(M)=0\quad\text{for all }1\le i\le n-1-\left\lfloor\frac n2\right\rfloor.$$

sharp factor 2 from conformal mass formula

entire equality-case rigidity and homotopy conclusion dropped

equality gives harmonic conformality off a singular set, not smooth extension across it nor full homotopy vanishing

dependence of the singular set on the manifold; no universal set S depending only on n


\begin{equation}\label{homotopy seq for unordered uni hyp curve}
1\to \pi_1^\alg(C, \bar x)\to \pi_1^\alg(\cC_{\H_{g,[n]/k}}, \bar x)\to \pi_1^\alg(\H_{g,[n]/k}, \bar y)\to 1,
\end{equation}

\begin{bigtheorem}\label{main result for uni curve}
If $g\geq 3$, then the homotopy exact sequence \eqref{homotopy seq for unordered uni curve} does not split.
\end{bigtheorem}

For nonnegative integers $g$ and $n$ satisfying $2g - 2 + n > 0$ and a field $k$ of characteristic $0$, let $\M_{g,n/k}$ denote the moduli stack of proper, smooth curves of genus $g$ with $n$ ordered distinct marked points over $k$, and let
\[
\pi_{g,n/k} : \cC_{g,n/k}\to \M_{g,n/k}
\]
be the universal curve over $\M_{g,n/k}$. The symmetric group on $n$ letters, denoted $S_n$, acts on $\M_{g,n/k}$ by permuting the $n$ marked points. We denote the quotient stack $[\M_{g,n/k}/S_n]$ by $\M_{g,[n]/k}$. The $S_n$-action extends to $\cC_{g,n/k}$, and we denote the quotient stack $[\cC_{g,n/k}/S_n]$ by $\cC_{g,[n]/k}$. Thus the universal curve $\pi_{g,n/k}$ induces a family of curves of genus $g$ with $n$ unordered marked points over $\M_{g,[n]/k}$:
\[
\pi_{g,[n]/k}: \cC_{g,[n]/k}\to \M_{g,[n]/k},
\]
which we call the universal curve of genus $g$ with $n$ unordered marked points.

Denote by $\pi_{g,n}:{\mathcal C}^\an_{g,n}\to {\mathcal M}^\an_{g,n}$ and $\pi_{g,[n]}:{\mathcal C}^\an_{g,[n]}\to {\mathcal M}^\an_{g,[n]}$ the corresponding complex analytic orbifolds of $\pi_{g,n/{\mathbb C}}$ and $\pi_{g,[n]/{\mathbb C}}$, respectively. In \cite[Thm.~1.3]{chen_uni}, Chen showed that for $n \geq 2$ and $g \geq 2$, the curve $\pi_{g,[n]}$ admits no sections. In this paper, we extend Chen's result to the algebraic fundamental groups of $\pi_{g,[n]/k}$.

Let $\bar y$ be a geometric point of $\M_{g,[n]/k}$ and $C$ the fiber of $\pi_{g,[n]/k}$ over $\bar y$. Fix a geometric point $\bar x \in C$, viewed also as a geometric point of $\cC_{g,[n]/k}$. Associated to the universal curve $\pi_{g,[n]/k}$, there is a homotopy exact sequence of algebraic fundamental groups:
\begin{equation}\label{homotopy seq for unordered uni curve}
1\to \pi_1^\mathrm{alg}(C, \bar x)\to \pi_1^\mathrm{alg}(\cC_{g,[n]/k}, \bar x)\to \pi_1^\mathrm{alg}(\M_{g,[n]/k}, \bar y)\to 1.
\end{equation}

\begin{equation}\label{homotopy seq for unordered uni curve}
1\to \pi_1^\alg(C, \bar x)\to \pi_1^\alg(\cC_{g,[n]/k}, \bar x)\to \pi_1^\alg(\M_{g,[n]/k}, \bar y)\to 1.
\end{equation}

\begin{bigtheorem}\label{main result for uni hyp curve}
If $g\geq 3$, then the homotopy exact sequence \eqref{homotopy seq for unordered uni hyp curve} does not split.
\end{bigtheorem}

The \(S_n\)-action together with the \(\Sp(H)\)-module structure on the weight \(-1\) parts of these Lie algebras plays a key role in the proof of Theorem~\ref{main result for uni curve}. In the hyperelliptic case, the relative completion of \(\pi_1^\orb(\H^\an_{g,n})\) admits analogous structures, which are likewise essential for the proof of Theorem~\ref{main result for uni hyp curve}.
\section{Configuration Spaces of Points on a Curve}
Let $C$ be a smooth complex algebraic curve of genus $g$. For a positive integer $n$, the ordered configuration space of $n$ distinct points on $C$ is
$$
F_n(C) := \{(x_1,\dots,x_n) \in C^n \mid x_i \neq x_j \text{ for } i \neq j \}.
$$
The symmetric group $S_n$ acts freely on $F_n(C)$ by permuting the coordinates, and we define the unordered configuration space as the quotient
$$
F_{[n]}(C) := F_n(C)/S_n.
$$

\begin{proof}
Relative completion applied to the exact sequence
\[
1 \to \pi_1^\top(C)\to\pi_1^\orb(\cC^\an_{g,n})\to \pi_1^\orb(\M^\an_{g,n})\to 1
\]
yields the exact sequence of MHS
\[
0\to\p_g\to \u_{\cC_{g,n}}\to \u_{g,n}\to 0,
\]
where $\p_g =\p_{g,1}$. This gives the exact sequence
\[
 H_1(\p_g)
 \longrightarrow H_1(\u_{\cC_{g,n}})
 \longrightarrow H_1(\u_{g,n})
 \longrightarrow 0,
\]
which implies that $H_1(\u_{\cC_{g,n}})$ is pure of weight -1.
Since \(\p_g\) is center-free, the adjoint representation
\(\adj\colon \p_g \to \Der \p_g\) is injective.
Because \(\adj\) factors through \(\u_{\cC_{g,n}}\) and the functor
\(\Gr^W_{-1}\) is exact, the composition
\[
\Gr^W_{-1}\p_g
 \longrightarrow \Gr^W_{-1}\u_{\cC_{g,n}}
 \longrightarrow \Gr^W_{-1}\Der \p_g
\]
is also injective.
Using the identifications
\(H_1(\p_g) \cong \Gr^W_{-1}\p_g\) and
\(H_1(\u_{\cC_{g,n}}) \cong \Gr^W_{-1}H_1(\u_{\cC_{g,n}})\cong\Gr^W_{-1}\u_{\cC_{g,n}}\),
we obtain the commutative diagram
\[
\xymatrix{
\Gr^W_{-1}\p_g \ar[r] \ar[d]^{\cong} &
\Gr^W_{-1}\u_{\cC_{g,n}} \ar[d]^{\cong} \\
H_1(\p_g) \ar[r] &
H_1(\u_{\cC_{g,n}}).
}
\]
Therefore, the sequence
\[
0 \longrightarrow H_1(\p_g)
 \longrightarrow H_1(\u_{\cC_{g,n}})
 \longrightarrow H_1(\u_{g,n})
 \longrightarrow 0
\]
is exact. Consequently, we obtain an \(\Sp(H)\)-decomposition
\[
H_1(\u_{\cC_{g,n}})
 \cong H_1(\p_g) \oplus H_1(\u_{g,n})
 \cong H_0 \oplus \Lambda^3_nH\cong\bigoplus_{j=0}^nH_j \oplus \Lambda^3_0H ,
\]
where \(H_0 := H_1(\p_g)\cong H\).
\end{proof}
\begin{remark}
Since $H_1(\u_{\cC_{g,n}})$ is pure of weight $-1$, the weight filtration on $\u_{\cC_{g,n}}$ coincides with its lower central series.
\end{remark}
We now turn to the hyperelliptic case.
A key fact is that \(H_1(\v_g)\) is pure of weight~\(-2\)
(see~\cite[Prop.~6.9]{wat_rk_hyp_univ}),
which leads to the following statement.
\begin{proposition}
The weight filtration on \(\v_g\) satisfies
\[
\v_g = W_{-2}\v_g,
\qquad
L^l\v_g = W_{-2l}\v_g,
\]
where \(L^l\v_g\) denotes the \(l\)-th term of the lower central series.
In particular,
\[
H_1(\v_g) \cong \Gr^W_{-2}\v_g.
\]
\end{proposition}
The following lemma describes the resulting canonical \(\Sp(H)\)-decompositions
of the first homology groups of the corresponding Lie algebras
\(\v_{g,n}\) and \(\v_{\cC_{g,n}}\).
\begin{lemma}\label{H_1 decomp for rel comp of hyp mcg}
For $g\geq 2$, there are canonical $\Sp(H)$-decompositions:
$$
H_1(\v_{g,n})\cong \Gr^W_{-1}H_1(\v_{g,n})\oplus \Gr^W_{-2}H_1(\v_{g,n})\cong \bigoplus_{j=1}^nH_j\oplus H_1(\v_g)
$$
and
$$
H_1(\v_{\cC_{g,n}})\cong \Gr^W_{-1}H_1(\v_{\cC_{g,n}})\oplus \Gr^W_{-2}H_1(\v_{\cC_{g,n}})\cong \bigoplus_{j=0}^{n}H_j\oplus H_1(\v_g).
$$
\end{lemma}
\begin{proof}
The exact sequence
$$
1\to \pi_1^\top(F_n(C))\to \pi_1^\orb(\H^\an_{g,n})\to \pi_1^\orb(\H^\an_{g})\to 1
$$
induces the exact sequence of pronilpotent Lie algebras
$$
0\to \p_{g,n}\to \v_{g,n}\to \v_g\to 0,
$$
which in turn yields the exact sequence
$$
H_1(\p_{g,n})\to H_1(\v_{g,n})\to H_1(\v_g)\to 0.
$$
The adjoint action of $\v_{g,n}$ on $\p_{g,n}$ yields the Lie algebra map
$$
\v_{g,n}\to \Der\p_{g,n}.
$$
The center-freeness of $\p_{g,n}$ (see \cite{NTU}) implies that the composition
$$
\p_{g,n}\to \v_{g,n}\to \Der\p_{g,n}
$$
is an injection.
Since the functor $\Gr^W_\bullet$ is exact, the composition
$$
H_1(\p_{g,n})\cong\Gr^W_{-1}\p_{g,n}\to \Gr^W_{-1}\v_{g,n}\to \Gr^W_{-1}\Der\p_{g,n}
$$
is also injective. Since $\Gr^W_{-1}\v_{g,n}\cong \Gr^W_{-1}H_1(\v_{g,n})$ and the diagram
$$
\xymatrix{
H_1(\p_{g,n})\ar[r]\ar[dr]&H_1(\v_{g,n})\ar[d]\\
&\Gr^W_{-1}H_1(\v_{g,n})
}
$$
commutes, the sequence
\begin{equation}
0\to H_1(\p_{g,n})\to H_1(\v_{g,n})\to H_1(\v_{g})\to 0
\end{equation} is exact.
Applying the functor $\Gr^W_\bullet$, we obtain the exact sequence
\begin{equation}
0\to \Gr^W_\bullet H_1(\p_{g,n})\to \Gr^W_\bullet H_1(\v_{g,n})\to \Gr^W_\bullet H_1(\v_{g})\to 0.
\end{equation}
By \textcolor{black}{\cite[Prop.~6.9]{wat_rk_hyp_univ}},
$H_1(\v_g)$ is pure of weight $-2$, and so there is a canonical isomorphism
$$
\Gr^W_{-1}H_1(\v_{g,n})\cong \Gr^W_{-1}H_1(\p_{g,n})=H_1(\p_{g,n}).
$$
Composing with the projection $H_1(\v_{g,n})\to \Gr^W_{-1}H_1(\v_{g,n})$, we get a canonical projection $H_1(\v_{g,n})\to H_1(\p_{g,n})$. This yields a canonical splitting
$$
H_1(\v_{g,n}) \cong \Gr^W_{-1}H_1(\v_{g,n})\oplus \Gr^W_{-2}H_1(\v_{g,n}) \cong H_1(\p_{g,n})\oplus H_1(\v_g)\cong \bigoplus_{j=1}^nH_j\oplus H_1(\v_g).
$$

\begin{lemma}\label{H component in H_1}
For $g\geq 2$, there are isomorphisms
$$
\Hom_{\Sp}(H_1(\u_{g,[n]}), H)\cong \Q\quad \text{ and }\quad \Hom_{\Sp}(H_1(\v_{g,[n]}), H)\cong \Q.
$$
\end{lemma}
\begin{proof}
By Theorem \ref{str} and \textcolor{black}{Example} \ref{rel comp in unordered case}, we have
$$
 H^1(\pi_1^\orb(\M^\an_{g, [n]}),H)\cong \Hom_{\Sp}(H_1(\u_{g,[n]}), H).
$$
From the exact sequence
\[
\begin{tikzcd}
1 \ar[r]
 & \pi_1^\orb(\M^\an_{g,n}) \ar[d,"\cong"'] \ar[r]
 & \pi_1^\orb(\M^\an_{g,[n]}) \ar[d,"\cong"'] \ar[r]
 & S_n \ar[r] & 1 \\
 & \G_{g,n}
 & \G_{g,[n]} & &
\end{tikzcd}
\]
the \emph{Hochschild--Serre five-term exact sequence} yields
\[
1 \longrightarrow
H^1(S_n, H^{\Gamma_{g,n}})
\longrightarrow
H^1(\Gamma_{g,[n]}, H)
\longrightarrow
H^1(\Gamma_{g,n}, H)^{S_n}
\longrightarrow
H^2(S_n, H^{\Gamma_{g,n}}),
\]
where \(H\) is regarded as a \(\Gamma_{g,[n]}\)-module.
Since both $H^1(S_n,H^{\Gamma_{g,n}})$ and $H^2(S_n,H^{\Gamma_{g,n}})$ vanish,
$$H^1(\Gamma_{g,[n]},H)\cong H^1(\Gamma_{g,n},H)^{S_n}.$$
Since
\[
H^1(\pi_1^\orb(\M^\an_{g,n}), H)
 \cong H^1(\M_{g,n}^\an, \mathbb{H}_\Q)
 \cong \bigoplus_{j=1}^n \Q\,\kappa_j,
\]
the \(S_n\)-invariant part is isomorphic to \(\Q\).
Consequently,
\[
\Hom_{\Sp}\!\left(H_1(\u_{g,[n]}), H\right) \cong \Q.
\]
The same argument also works for $\Hom_{\Sp}(H_1(\v_{g,[n]}), H)\cong \Q$, using the natural isomorphism
$$
H^1(\H^\an_{g,n}, \mathbb{H}_\Q)
 \cong \bigoplus_{j=1}^n \Q\,\kappa_j^{\hyp}.
$$
\end{proof}
The natural $S_n$–action on $H_1(\u_{g,n})$ allows one to determine explicitly the decompositions of
$H_1(\u_{g,[n]})$ and $H_1(\u_{\cC_{g,[n]}})$.
\begin{lemma}\label{ab of u_{g,[n]}}
For \(g \ge 3\), there is an \(\Sp(H)\)-decomposition
\[
H_1(\u_{g,[n]})
 \cong H_1(\u_g) \oplus \!\bigg(\bigoplus_{j=1}^{n} H_j\!\bigg)_{S_n}
 = \Lambda^3_0 H \oplus H,
\]
which is compatible with the projection
\(H_1(\u_{g,n}) \to H_1(\u_{g,[n]})\)
and with the \(\Sp(H)\)-decomposition
\(H_1(\u_{g,n}) = \Lambda^3_0 H \oplus \bigoplus_{j=1}^{n} H_j\).
Furthermore, there is an \(\Sp(H)\)-decomposition
\[
H_1(\u_{\cC_{g,[n]}}) \cong H_0 \oplus \Lambda^3_0 H \oplus H.
\]

In \cite[Thm.~2.6]{chen_uni}, Chen also proved an analogous result for the hyperelliptic case. Let $\H_{g,n/k}$ be the closed substack of $\M_{g,n/k}$ parametrizing hyperelliptic curves of genus $g$ with $n$ ordered marked points. Denote the quotient stack $[\H_{g,n/k}/S_n]$ by $\H_{g,[n]/k}$; this is a closed substack of $\M_{g,[n]/k}$ parametrizing hyperelliptic curves of genus $g$ with $n$ unordered marked points. Restricting $\pi_{g,n/k}$ to $\H_{g,n/k}$, we obtain the universal hyperelliptic curve
\[
\pi^\hyp_{g,n/k}: \cC_{\H_{g,n/k}}\to \H_{g,n/k}.
\]
Extending the $S_n$-action to $\cC_{\H_{g,n/k}}$ yields a family over $\H_{g,[n]/k}$:
\[
\pi^\hyp_{g,[n]/k}:\cC_{\H_{g,[n]/k}}\to \H_{g,[n]/k},
\]
which we call the universal hyperelliptic curve of genus $g$ with $n$ unordered marked points.

Denote by $\pi^{\mathrm{hyp}}_{g,n}:{\mathcal C}^\an_{\H_{g,n}}\to {\mathcal H}^\an_{g,n}$ and $\pi^{\mathrm{hyp}}_{g,[n]}:{\mathcal C}^\an_{\H_{g,[n]}}\to {\mathcal H}^\an_{g,[n]}$ the corresponding complex analytic orbifolds of $\pi^\hyp_{g,n/{\mathbb C}}$ and $\pi^\hyp_{g,[n]/{\mathbb C}}$, respectively. In \cite[Thm.~2.6]{chen_uni}, Chen showed that for $n \geq 2$ and $g \geq 2$, the hyperelliptic universal curve $\pi^{\mathrm{hyp}}_{g,[n]}$ admits no sections.

We extend Chen's result to the algebraic fundamental groups of $\pi^\hyp_{g,[n]/k}$. Associated to this universal hyperelliptic curve, there is a homotopy exact sequence of algebraic fundamental groups:
\begin{equation}\label{homotopy seq for unordered uni hyp curve}
1\to \pi_1^\mathrm{alg}(C, \bar x)\to \pi_1^\mathrm{alg}(\cC_{\H_{g,[n]/k}}, \bar x)\to \pi_1^\mathrm{alg}(\H_{g,[n]/k}, \bar y)\to 1,
\end{equation}
where $\bar y$ is a geometric point of $\H_{g,[n]/k}$, $C$ is the fiber of $\pi^\hyp_{g,[n]/k}$, and $\bar x$ is a geometric point of $C$.

\begin{bigtheorem}\label{main result for uni hyp curve}
If $g\geq 3$, then the homotopy exact sequence \eqref{homotopy seq for unordered uni hyp curve} does not split.
\end{bigtheorem}

A key tool used in this paper is the relative completion of a discrete group, or, in the case of a profinite group, its continuous relative completion. A detailed account of their constructions and basic properties can be found in \textcolor{black}{\cite{hain_infpretor, hain_hodge_rel, Hain15}}. In particular, for the algebraic fundamental groups of the stacks considered here, their continuous relative completions over \(\Q_\ell\) are canonically isomorphic to the base change of the corresponding relative completions over \({\mathbb Q}\) to \(\Q_\ell\).

For example, the relative completion of the orbifold fundamental group
\(\pi_1^\mathrm{orb}({\mathcal M}^\an_{g,n})\) is a proalgebraic group over \({\mathbb Q}\), defined as an extension of \(\Sp(H):=\Sp(H_1(C,{\mathbb Q}))\) by a prounipotent group. Its Lie algebra carries a natural mixed Hodge structure (MHS), and hence admits a weight filtration defined over \({\mathbb Q}\). Each graded piece of the associated graded Lie algebra of the completion is an \(\Sp(H)\)-representation. Moreover, the universal curve \(\pi_{g,n}\) induces an \(\Sp(H)\)-equivariant graded Lie algebra section of a projection between the pronilpotent Lie algebras of the corresponding completions.

The \(S_n\)-action together with the \(\Sp(H)\)-module structure on the weight \(-1\) parts of these Lie algebras plays a key role in the proof of Theorem~\ref{main result for uni curve}. In the hyperelliptic case, the relative completion of \(\pi_1^\mathrm{orb}({\mathcal H}^\an_{g,n})\) admits analogous structures, which are likewise essential for the proof of Theorem~\ref{main result for uni hyp curve}.

A key tool is the (continuous) relative completion of the relevant (profinite/discrete) fundamental groups; for the stacks considered, the continuous relative completions over \(\Q_\ell\) are canonically isomorphic to the base change of the corresponding relative completions over \(\Q\) to \(\Q_\ell\). For example, the relative completion of \(\pi_1^\mathrm{orb}({\mathcal M}^\an_{g,n})\) is an extension of \(\Sp(H):=\Sp(H_1(C,{\mathbb Q}))\) by a prounipotent group, and its Lie algebra carries a natural mixed Hodge structure with a weight filtration over \(\Q\), whose graded pieces are \(\Sp(H)\)-representations. Moreover, the universal curve \(\pi_{g,n}\) induces an \(\Sp(H)\)-equivariant graded Lie algebra section of a projection between pronilpotent Lie algebras of the corresponding completions. The \(S_n\)-action together with the \(\Sp(H)\)-module structure on the weight \(-1\) parts of these Lie algebras plays a key role in the proof of Theorem~\ref{main result for uni curve}; in the hyperelliptic case, the relative completion of \(\pi_1^\mathrm{orb}({\mathcal H}^\an_{g,n})\) admits analogous structures, which are likewise essential for the proof of Theorem~\ref{main result for uni hyp curve}.

A key tool is the (continuous) relative completion of the relevant (profinite/discrete) fundamental groups; for the stacks considered, the continuous relative completions over \(\Q_\ell\) are canonically isomorphic to the base change of the corresponding relative completions over \(\Q\) to \(\Q_\ell\). For example, the relative completion of \(\pi_1^\mathrm{orb}({\mathcal M}^\an_{g,n})\) is an extension of \(\Sp(H):=\Sp(H_1(C,{\mathbb Q}))\) by a prounipotent group, and its Lie algebra carries a natural mixed Hodge structure with a weight filtration over \(\Q\), whose graded pieces are \(\Sp(H)\)-representations. Moreover, the universal curve \(\pi_{g,n}\) induces an \(\Sp(H)\)-equivariant graded Lie algebra section of a projection between pronilpotent Lie algebras of the corresponding completions. The \(S_n\)-action together with the \(\Sp(H)\)-module structure on the weight \(-1\) parts of these Lie algebras plays a key role in establishing the main theorem; in the hyperelliptic case, the relative completion of \(\pi_1^\mathrm{orb}({\mathcal H}^\an_{g,n})\) admits analogous structures, which are likewise essential for the proof of the following theorem.

\begin{bigtheorem}\label{main result for uni hyp curve}
If $g\geq 3$, then the homotopy exact sequence \eqref{homotopy seq for unordered uni hyp curve} does not split.
\end{bigtheorem}

\label{main result for uni curve}
If $g\geq 3$, then the homotopy exact sequence
\begin{equation}\label{homotopy seq for unordered uni curve}
1\to \pi_1^\alg(C, \bar x)\to \pi_1^\alg(\cC_{g,[n]/k}, \bar x)\to \pi_1^\alg(\M_{g,[n]/k}, \bar y)\to 1.
\end{equation}
does not split.

Question:
Let $g,n$ be nonnegative integers with $2g-2+n>0$, and let $k$ be a field of characteristic $0$. Let $\M_{g,n/k}$ be the moduli stack of proper smooth genus-$g$ curves with $n$ ordered distinct marked points, let $S_n$ act by permuting the marked points, and write $\M_{g,[n]/k}=[\M_{g,n/k}/S_n]$ for the moduli stack of genus-$g$ curves with $n$ unordered marked points. Let
\[
\pi_{g,[n]/k}: \mathcal C_{g,[n]/k}\to \mathcal M_{g,[n]/k}
\]
be the induced universal curve. For a geometric point $\bar y\in \mathcal M_{g,[n]/k}$, let $C$ be the fiber of $\pi_{g,[n]/k}$ over $\bar y$, and fix a geometric point $\bar x\in C$, viewed also as a geometric point of $\mathcal C_{g,[n]/k}$. Then one has the homotopy exact sequence of algebraic fundamental groups
\[
1\to \pi_1^{\mathrm{alg}}(C,\bar x)\to \pi_1^{\mathrm{alg}}(\mathcal C_{g,[n]/k},\bar x)\to \pi_1^{\mathrm{alg}}(\mathcal M_{g,[n]/k},\bar y)\to 1.
\]
If $g\ge 3$, which statement about this exact sequence holds?

The exact sequence does not split; equivalently, there is no group homomorphism $\pi_1^{\mathrm{alg}}(\mathcal M_{g,[n]/k},\bar y)\to \pi_1^{\mathrm{alg}}(\mathcal C_{g,[n]/k},\bar x)$ whose composition with $\pi_1^{\mathrm{alg}}(\mathcal C_{g,[n]/k},\bar x)\to \pi_1^{\mathrm{alg}}(\mathcal M_{g,[n]/k},\bar y)$ is the identity.

Choices:
B.
The exact sequence splits after passing to the maximal pro-$\ell$ quotient for every prime $\ell$; equivalently, for each $\ell$ there is a continuous homomorphism
\[\pi_1^{\mathrm{alg}}(\mathcal M_{g,[n]/k},\bar y)^{(\ell)}\to \pi_1^{\mathrm{alg}}(\mathcal C_{g,[n]/k},\bar x)^{(\ell)}\]
whose composition with the induced projection to \(\pi_1^{\mathrm{alg}}(\mathcal M_{g,[n]/k},\bar y)^{(\ell)}\) is the identity.
C.
The exact sequence is not canonically split; in particular, there is no distinguished or natural group homomorphism
\[\pi_1^{\mathrm{alg}}(\mathcal M_{g,[n]/k},\bar y)\to \pi_1^{\mathrm{alg}}(\mathcal C_{g,[n]/k},\bar x)\]
whose composition with \(\pi_1^{\mathrm{alg}}(\mathcal C_{g,[n]/k},\bar x)\to \pi_1^{\mathrm{alg}}(\mathcal M_{g,[n]/k},\bar y)\) is the identity.
D. The exact sequence splits whenever one works over a field of characteristic \(0\); equivalently, there exists a group homomorphism \(\pi_1^{\mathrm{alg}}(\mathcal M_{g,[n]/k},\bar y)\to \pi_1^{\mathrm{alg}}(\mathcal C_{g,[n]/k},\bar x)\) after extending scalars to some algebraic closure of \(k\), and hence the sequence is split exact.
E. If \(g\ge 2\), then the exact sequence does not split; equivalently, for every such \(g\) there is no group homomorphism \(\pi_1^{\mathrm{alg}}(\mathcal M_{g,[n]/k},\bar y)\to \pi_1^{\mathrm{alg}}(\mathcal C_{g,[n]/k},\bar x)\) whose composition with \(\pi_1^{\mathrm{alg}}(\mathcal C_{g,[n]/k},\bar x)\to \pi_1^{\mathrm{alg}}(\mathcal M_{g,[n]/k},\bar y)\) is the identity.

non-splitting survives relative completion and all pro-\ell realizations

absolute nonexistence of any splitting replaced by nonexistence of a canonical/natural splitting

descent/base-change from characteristic-0 field to algebraic closure preserves a hypothetical splitting

genus threshold g\ge 3 weakened to g\ge 2