\section{Introduction}

We first explain about the case of orientable surfaces.
For $g\geq0$, let $\Sigma_g$ denote an orientable closed surface of genus $g$, that is, $\Sigma_g$ is a connected sum of $g$ tori.
The \textit{mapping class group} $\mathcal{M}(\Sigma_g)$ of $\Sigma_g$ is the group which consists of isotopy classes of all orientation preserving diffeomorphisms of $\Sigma_g$.
The \textit{level $2$ mapping class group} $\mathcal{M}_2(\Sigma_g)$ of $\Sigma_g$ is the subgroup of $\mathcal{M}(\Sigma_g)$ which acts on $H_1(\Sigma_g;\mathbb{Z}/2\mathbb{Z})$ trivially.
Let $\mathcal{T}^2(\Sigma_g)$ denote the subgroup of $\mathcal{M}(\Sigma_g)$ generated by squares of \textit{Dehn twists} about non-separating curves.
It is known that $\mathcal{T}^2(\Sigma_g)=\mathcal{M}_2(\Sigma_g)$ (see \cite{Hu}).
Let $\mathcal{I}(\Sigma_g)$ denote the subgroup of $\mathcal{M}(\Sigma_g)$ which acts on $H_1(\Sigma_g;\mathbb{Z})$ trivially, called the \textit{Torelli group} of $\Sigma_g$.
It is clear that $\mathcal{I}(\Sigma_g)\subset\mathcal{M}_2(\Sigma_g)$, and hence we have that $\mathcal{T}^2(\Sigma_g)\mathcal{I}(\Sigma_g)=\mathcal{M}_2(\Sigma_g)$.

In this paper, we would like to consider the similar problem for the case of non-orientable surfaces, as follows.
For $g\geq1$, let $N_g$ denote a non-orientable closed surface of genus $g$, that is, $N_g$ is a connected sum of $\Sigma_h$ and $g-2h$ real projective planes for $\displaystyle0\leq{h}\leq\frac{g-1}{2}$.
In this paper, we describe $N_g$ as a surface obtained by attaching $g-2h$ M\"obius bands to a surface obtained by removing disjoint $g-2h$ disks from $\Sigma_h$, as shown in Figure~\ref{non-ori-surf}.
We call these attached M\"obius bands \textit{crosscaps}.
The \textit{mapping class group} $\mathcal{M}(N_g)$ of $N_g$ is the group which consists of isotopy classes of all diffeomorphisms of $N_g$.
The \textit{level $2$ mapping class group} $\mathcal{M}_2(N_g)$ of $N_g$ is the subgroup of $\mathcal{M}(N_g)$ which acts on $H_1(N_g;\mathbb{Z}/2\mathbb{Z})$ trivially.
Let $\mathcal{T}^2(N_g)$ denote the subgroup of $\mathcal{M}(N_g)$ generated by squares of Dehn twists about non-separating curves.
Note that $\mathcal{T}^2(N_g)\subset\mathcal{M}_2(N_g)$.
$\mathcal{M}_2(N_g)$ can be normally generated by one \textit{crosscap slide} (see \cite{Sz2}), and a crosscap slide can not be described as any product of Dehn twists (see \cite{Li}).
Hence $\mathcal{M}_2(N_g)$ can not be generated by only Dehn twists, and so $\mathcal{T}^2(N_g)\neq\mathcal{M}_2(N_g)$.
We consider the subgroup $\mathcal{T}_2(N_g)$ of $\mathcal{M}_2(N_g)$ generated by all Dehn twists, called the \textit{twist subgroup} of $\mathcal{M}_2(N_g)$.
Note that a generating set for $\mathcal{T}_2(N_g)$ which does not consist of squares of Dehn twists about non-separating curves is known (see \cite{KO}).
In this paper, we prove that $\mathcal{T}_2(N_g)\neq\mathcal{T}^2(N_g)$ either.

\begin{figure}[htbp]
\includegraphics{non-ori-surf}
\caption{A model of a non-orientable closed surface $N_g$.}\label{non-ori-surf}
\end{figure}

More precisely, in Section~\ref{main}, we prove the following, where $\mathcal{I}(N_g)$ is the subgroup of $\mathcal{M}(N_g)$ which acts on $H_1(N_g;\mathbb{Z})$ trivially, called the \textit{Tolleri group} of $N_g$.

\begin{thm}\label{main-1}
For $g\geq3$, the quotient of $\mathcal{T}_2(N_g)$ by $\mathcal{T}^2(N_g)\mathcal{I}(N_g)$ is isomorphic to $\left(\mathbb{Z}/2\mathbb{Z}\right)^{\tbinom{g-1}{2}-1}$ if $g$ is odd and to $\left(\mathbb{Z}/2\mathbb{Z}\right)^{\tbinom{g-1}{2}}$ if $g$ is even.
\end{thm}

Moreover, in Section~\ref{T^2I}, we prove the following.

\begin{thm}\label{main-2}
$\mathcal{T}^2(N_g)\mathcal{I}(N_g)=\mathcal{T}^2(N_g)$ if $g\geq3$ is odd.
\end{thm}

In addition, as an application of Theorem~\ref{main-1}, we give a finite generating set for $\mathcal{T}^2(N_g)\mathcal{I}(N_g)$, in Section~\ref{gen-T^2I}.
Finally, we examine about actions on non-separating simple closed curves of $N_g$ by $\mathcal{M}_2(N_g)$, in Appendix~\ref{action}.

Through this paper, the product $gf$ of mapping classes $f$ and $g$ means that we apply $f$ first and then $g$, and we do not distinguish a simple closed curve from its isotopy class.










\section{Preliminaries}\label{pre}





\subsection{On Dehn twist and crosscap slide}\

First, We define a \textit{Dehn twist} and a \textit{crosscap slide} which are elements of $\mathcal{M}(N_g)$.
For a simple closed curve $c$ of $N_g$, its regular neighborhood is either an annulus  or a M\"obius band.
We call $c$ a \textit{two sided} or a \textit{one sided} simple closed curve respectively.
For a two sided simple closed curve $c$, the \textit{Dehn twist} $t_c$ about $c$ is the isotopy class of the map described as shown in Figure~\ref{dehn-slide}~(a).
The direction of the twist is indicated by an arrow written beside $c$ as shown in Figure~\ref{dehn-slide}~(a).
For a one sided simple closed curve $\mu$ of $N_g$ and an oriented two sided simple closed curve $\alpha$ of $N_g$ such that $N_g\setminus\{\alpha\}$ is non-orientable when $g\geq3$ and that $\mu$ and $\alpha$ intersect transversely at only one point, the \textit{crosscap slide} $Y_{\mu,\alpha}$ about $\mu$ and $\alpha$ is the isotopy class of the map described by pushing the crosscap which is a regular neighborhood of $\mu$ once along $\alpha$, as shown in Figure~\ref{dehn-slide}~(b).

\begin{figure}[htbp]
\subfigure[The Dehn twist $t_c$ about $c$.]{\includegraphics{dehn}}~\subfigure[The crosscap slide $Y_{\mu,\alpha}$ about $\mu$ and $\alpha$.]{\includegraphics{crosscap-slide}}
\caption{Descriptions of a Dehn twist and a crosscap slide.}\label{dehn-slide}
\end{figure}

Next, we explain about some relations on Dehn twists and crosscap slides.
For details, for instance see \cite{Sz1}.
For a simple closed curve $c$ which bounds a disk or a crosscap, we have that
$$t_c=1.$$
For any Dehn twist $t_c$, crosscap slide $Y_{\mu,\alpha}$ and $f\in\mathcal{M}(N_g)$, we have that
$$ft_cf^{-1}=t_{f(c)},~fY_{\mu,\alpha}f^{-1}=Y_{f(\mu),f(\alpha)}.$$
Let $\mu_1$, $\mu_2$ and $\alpha$ be simple closed curves as shown in Figure~\ref{crosscap-slide-rel}~(a).
Then we have that
$$Y_{\mu_2,\alpha}^{-1}Y_{\mu_1,\alpha}=Y_{\mu_2,\alpha}Y_{\mu_1,\alpha}^{-1}=t_\alpha^2.$$
For two crosscap slides $Y_{\mu,\alpha}$ and $Y_{\mu,\beta}$ such that $\overline{\alpha}\overline{\beta}$ is simple, where $\overline{\alpha}$ and $\overline{\beta}$ are oriented loops of $N_{g-1}$ obtained by collapsing a regular neighborhood of $\mu$ and $\overline{\alpha}\overline{\beta}$ is a composition loop of $\overline{\alpha}$ and $\overline{\beta}$ based at the collapsing point, let $\delta_1(\mu,\alpha,\beta)$ and $\delta_2(\mu,\alpha,\beta)$ are simple closed curves determined by $\mu$, $\alpha$ and $\beta$ as shown in Figure~\ref{crosscap-slide-rel}~(b).
Then we have that
$$Y_{\mu,\beta}Y_{\mu,\alpha}=t_{\delta_1(\mu,\alpha,\beta)}t_{\delta_2(\mu,\alpha,\beta)}.$$
In addition, for any crosscap slide $Y_{\mu,\alpha}$, we have that
$$Y_{\mu,\alpha}^2=t_{\delta(\mu,\alpha)},~Y_{\mu,\alpha}=Y_{\mu,\alpha^{-1}}^{-1},$$
where $\delta(\mu,\alpha)$ is a simple closed curve determined by $\mu$ and $\alpha$ as shown in Figure~\ref{dehn-slide}~(b).

\begin{figure}[htbp]
\subfigure[]{\includegraphics{crosscap-slide-rel-1}}\hspace{10mm}
\subfigure[]{\includegraphics{crosscap-slide-rel-2}}
\caption{Relations on crosscap slides.}\label{crosscap-slide-rel}
\end{figure}





\subsection{On $\mathcal{M}_2(N_g)$ and $\mathcal{I}(N_g)$}\label{M_2-I}\

For $1\leq{i_1}<i_2<\dots<i_k\leq{g}$, let $\alpha_{i_1,i_2,\dots,i_k}$ be a simple closed curve of $N_g$ as shown in Figure~\ref{alpha}.
For $g\geq4$, Szepietowski \cite{Sz2} gave a finite generating set for $\mathcal{M}_2(N_g)$, and then Hirose-Sato \cite{HS} gave a minimal generating set for $\mathcal{M}_2(N_g)$ as follows.

\begin{figure}[htbp]
\includegraphics{alpha}
\caption{Simple closed curves $\alpha_{i_1,i_2,\dots,i_k}$ and $\alpha^\prime_{i_1,i_2,\dots,i_k}$ for $1\leq{i_1}<i_2<\dots<i_k\leq{g}$.}\label{alpha}
\end{figure}

\begin{thm}[\cite{HS}]\label{level2-gen}
For $g\geq4$, $\mathcal{M}_2(N_g)$ is minimally generated by
\begin{itemize}
\item	$Y_{\alpha_i,\alpha_{i,j}}$ for $1\leq{i<j}\leq{g}$,
\item	$Y_{\alpha_j,\alpha_{i,j}}$ for $1\leq{i<j}\leq{g-1}$, 
\item	$t_{\alpha_{1,j,k,l}}^2$ for $1<j<k<l\leq{g}$.
\end{itemize}
\end{thm}

For $g\leq3$, $\mathcal{I}(N_g)$ is trivial.
For $g\geq4$, a normal generating set for $\mathcal{I}(N_g)$ in $\mathcal{M}(N_g)$ is known as follows.

\begin{thm}[\cite{HK, Ko2}]\label{torelli-gen}
Let $\alpha$, $\beta_1$, $\beta_2$ and $\gamma$ be simple closed curves as shown in Figure~\ref{Torelli-gen}.
For $g\geq5$, $\mathcal{I}(N_g)$ is normally generated by $t_\alpha$ and $t_{\beta_1}t_{\beta_2}^{-1}$ in $\mathcal{M}(N_g)$.
$\mathcal{I}(N_4)$ is normally generated by $t_\alpha$, $t_{\beta_1}t_{\beta_2}^{-1}$ and $t_\gamma$ in $\mathcal{M}(N_4)$.

\begin{figure}[htbp]
\includegraphics{Torelli-gen}
\caption{Simple closed curves $\alpha$, $\beta_1$, $\beta_2$ and $\gamma$.}\label{Torelli-gen}
\end{figure}

\end{thm}

For $n\geq1$, let $\Gamma_2(n)$  denote the kernel of the natural homomorphism $GL(n,\mathbb{Z})\to{}GL(n,\mathbb{Z}/2\mathbb{Z})$.
A finite presentation for $\Gamma_2(n)$ was given by Fullarton \cite{Fu} and the second author \cite{Ko1} independently, and then Hirose and the second author \cite{HK} gave another presentation as follows.
Put $n=g-1$.
For $1\leq{i,j}\leq{g-1}$ with $i\neq{j}$, let $Y_{i,j}$ be the matrix whose $(i,j)$ entry is $2$, $(i,i)$ entry is $-1$, the other diagonal entries are $1$ and the other entries are $0$, and $Y_{i,g}$ the matrix whose $(i,i)$ entry is $-1$, the other diagonal entries are $1$ and the other entries are $0$.
$\Gamma_2(g-1)$ has the following presentation.

\begin{prop}[\cite{HK}]\label{G_2-pre}
For $g\geq2$, $\Gamma_2(g-1)$ admits a presentation with generators $Y_{i,j}$ for $1\leq{i}\leq{g-1}$ and $1\leq{j}\leq{g}$ with $i\neq{j}$.
The defining relators are
\begin{enumerate}
\item	$Y_{i,j}^2$,
\item	$(Y_{i,j}Y_{k,j})^2$, $(Y_{i,j}Y_{k,l})^2$,
\item	$(Y_{i,j}Y_{i,k}Y_{j,k})^2$, $(Y_{i,j}Y_{i,k}Y_{i,l})^2$,
\item	$(Y_{j,i}Y_{i,j}Y_{k,j}Y_{j,k}Y_{i,k}Y_{k,i})^2$,
\end{enumerate}
where indices $i$, $j$, $k$ and $l$ are distinct.
\end{prop}

We regard $\alpha_{i_1,i_2,\dots,i_k}$ as an element of $H_1(N_g;\mathbb{Z})$.
Note that $\alpha_{i_1,i_2,\dots,i_k}=\alpha_{i_1}+\alpha_{i_2}+\cdots+\alpha_{i_k}$.
Then, as a $\mathbb{Z}$-module, we have presentations
\begin{eqnarray*}
H_1(N_g;\mathbb{Z})/{\langle{\alpha_{1,2,\dots,g}}\rangle}&=&\langle{\alpha_1,\alpha_2,\dots,\alpha_g\mid\alpha_{1,2,\dots,g}=0}\rangle\\
&=&\langle{\alpha_1,\alpha_2,\dots,\alpha_{g-1}}\rangle.
\end{eqnarray*}
We can regard an automorphism of $H_1(N_g;\mathbb{Z})/{\langle{\alpha_{1,2,\dots,g}}\rangle}$ as a matrix in $GL(g-1,\mathbb{Z})$.
For $f\in\mathcal{M}(N_g)$, let $f_\ast\in{}GL(g-1,\mathbb{Z})$ be the natural automorphism of $H_1(N_g;\mathbb{Z})/{\langle{\alpha_{1,2,\dots,g}}\rangle}$ induced from $f$.
We have that $(Y_{\alpha_i,\alpha_{i,j}})_\ast=Y_{i,j}$ and $(Y_{\alpha_j,\alpha_{i,j}})_\ast=Y_{j,i}$.
It is known that the quotient $\mathcal{M}_2(N_g)/{\mathcal{I}(N_g)}$ of $\mathcal{M}_2(N_g)$ by $\mathcal{I}(N_g)$ is isomorphic to $\Gamma_2(g-1)$ for $g\geq3$ (see \cite{MP, HK}).










\section{Proof of Theorem~\ref{main-1}}\label{main}

In this section, we prove Theorem~\ref{main-1}.
First we prepare some things.

For $1\leq{i}\leq{g-1}$, let $Y_{g,i}=(Y_{\alpha_g,\alpha_{i,g}})_\ast\in\Gamma_2(g-1)$, and we can check that
$$Y_{g,i}=(Y_{1,i}Y_{1,g})(Y_{2,i}Y_{2,g})\cdots(Y_{i-1,i}Y_{i-1,g})\cdot(Y_{i+1,i}Y_{i+1,g})\cdots(Y_{g-1,i}Y_{g-1,g})Y_{i,g}.$$
Hence, from Proposition~\ref{G_2-pre} we have the following.

\begin{cor}\label{G_2-pre2}
For $g\geq3$, $\Gamma_2(g-1)$ admits a presentation with generators $Y_{i,j}$ for $1\leq{i,j}\leq{g}$ with $i\neq{j}$.
The defining relators are $(1)$-$(4)$ of Proposition~\ref{G_2-pre} and
\begin{enumerate}
\item[(5)]	$(Y_{1,i}Y_{1,g})(Y_{2,i}Y_{2,g})\cdots(Y_{i-1,i}Y_{i-1,g})\cdot(Y_{i+1,i}Y_{i+1,g})\cdots(Y_{g-1,i}Y_{g-1,g})Y_{i,g}Y_{g,i}^{-1}$ for $1\leq{i}\leq{g-1}$.
\end{enumerate}
\end{cor}

\begin{lem}\label{G_2-rel}
For $g\geq3$, in $\Gamma_2(g-1)$, the relation $Y_{i,j}Y_{g,j}=Y_{g,j}Y_{i,j}$ holds for $1\leq{i}\leq{g-1}$.
\end{lem}

\begin{proof}
By relators of $\Gamma_2(g-1)$, we have relations
\begin{eqnarray*}
Y_{i,j}(Y_{k,j}Y_{k,g})&=&(Y_{k,j}Y_{k,g})Y_{i,j},\\
Y_{i,j}(Y_{i,j}Y_{i,g})&=&(Y_{i,j}Y_{i,g})(Y_{i,g}Y_{i,j}Y_{i,g}),\\
(Y_{i,g}Y_{i,j}Y_{i,g})(Y_{k,j}Y_{k,g})&=&(Y_{k,j}Y_{k,g})(Y_{i,g}Y_{i,j}Y_{i,g})
\end{eqnarray*}
for $k\neq{i}$.
Hence we calculate
\begin{eqnarray*}
Y_{i,j}Y_{g,j}
&=&Y_{i,j}(Y_{1,j}Y_{1,g})\cdots(Y_{j-1,j}Y_{j-1,g})\cdot(Y_{j+1,j}Y_{j+1,g})\cdots(Y_{g-1,j}Y_{g-1,g})Y_{j,g}\\
&=&(Y_{1,j}Y_{1,g})\cdots(Y_{j-1,j}Y_{j-1,g})\cdot(Y_{j+1,j}Y_{j+1,g})\cdots(Y_{g-1,j}Y_{g-1,g})(Y_{i,g}Y_{i,j}\underset{(2)}{\underline{Y_{i,g})Y_{j,g}}}\\
&=&(Y_{1,j}Y_{1,g})\cdots(Y_{j-1,j}Y_{j-1,g})\cdot(Y_{j+1,j}Y_{j+1,g})\cdots(Y_{g-1,j}Y_{g-1,g})\underset{(3)}{\underline{Y_{i,g}Y_{i,j}Y_{j,g}}}Y_{i,g}\\
&=&(Y_{1,j}Y_{1,g})\cdots(Y_{j-1,j}Y_{j-1,g})\cdot(Y_{j+1,j}Y_{j+1,g})\cdots(Y_{g-1,j}Y_{g-1,g})Y_{j,g}Y_{i,j}\underset{(1)}{\underline{Y_{i,g}Y_{i,g}}}\\
&=&(Y_{1,j}Y_{1,g})\cdots(Y_{j-1,j}Y_{j-1,g})\cdot(Y_{j+1,j}Y_{j+1,g})\cdots(Y_{g-1,j}Y_{g-1,g})Y_{j,g}Y_{i,j}\\
&=&Y_{g,j}Y_{i,j}.
\end{eqnarray*}
Thus we obtain the claim.
\end{proof}

Let $N$ be the normal subgroup of $\Gamma_2(g-1)$ normally generated by $Y_{j,i}^{-1}Y_{i,j}$ for $1\leq{i,j}\leq{g}$.

\begin{lem}\label{G/N-1}
For $g\geq3$, the quotient $\Gamma_2(g-1)/{N}$ of $\Gamma_2(g-1)$ by $N$ admits a presentation with generators $Y_{i,j}$ for $1\leq{i<j}\leq{g}$.
The defining relators are
\begin{enumerate}
\item[$\overline{(1)}$]	$Y_{i,j}^2$,
\item[$\overline{(2)}$]	$(Y_{i,j}Y_{k,j})^2$, $(Y_{i,j}Y_{i,l})^2$, $(Y_{i,j}Y_{j,l})^2$, $(Y_{i,j}Y_{k,l})^2$,
\item[$\overline{(5)}$]	$(Y_{1,i}Y_{1,g})\cdots(Y_{i-1,i}Y_{i-1,g})\cdot(Y_{i,i+1}Y_{i+1,g})\cdots(Y_{i,g-1}Y_{g-1,g})$ for $2\leq{i}\leq{g-1}$, and $Y_{1,g}Y_{2,g}\cdots{}Y_{g-1,g}$ if $g$ is odd.
\end{enumerate}
\end{lem}

\begin{proof}
Since we have that $Y_{j,i}=Y_{i,j}$ for $1\leq{i<j}\leq{g}$ in $\Gamma_2(g-1)/{N}$, by the Tietze transformation, $\Gamma_2(g-1)/{N}$ is generated by $Y_{i,j}$ for $1\leq{i<j}\leq{g}$.

By the relator~(1) of $\Gamma_2(g-1)$, we have the relator~$\overline{(1)}$.
By the relator $(Y_{i,j}Y_{k,j})^2$ of $\Gamma_2(g-1)$, we have relators
\begin{itemize}
\item	$(Y_{i,j}Y_{k,j})^2$ if $1\leq{i,k}<j\leq{g}$,
\item	$(Y_{j,i}Y_{j,k})^2$ if $1\leq{j}<{i,k}\leq{g-1}$,
\item	$(Y_{i,j}Y_{j,k})^2$ if $1\leq{i<j<k}\leq{g-1}$.
\end{itemize}
In addition, by Lemma~\ref{G_2-rel}, we have relators
\begin{itemize}
\item	$(Y_{j,i}Y_{j,g})^2$ if $i>j$,
\item	$(Y_{i,j}Y_{j,g})^2$ if $i<j$.
\end{itemize}
Moreover, by the relator $(Y_{i,j}Y_{k,l})^2$ of $\Gamma_2(g-1)$, we have the relator $(Y_{i,j}Y_{k,l})^2$.
Hence we have the relator~$\overline{(2)}$.
By the relators~$\overline{(1)}$ and $\overline{(2)}$, the relators~(3) and (4) are trivial relators.
By the relator~(5) of $\Gamma_2(g-1)$, we have the relator
$$(Y_{1,i}Y_{1,g})\cdots(Y_{i-1,i}Y_{i-1,g})\cdot(Y_{i,i+1}Y_{i+1,g})\cdots(Y_{i,g-1}Y_{g-1,g}),$$
and hence by the relators~$\overline{(1)}$ and $\overline{(2)}$,
\begin{eqnarray*}
Y_{1,i}&=&Y_{1,g}(Y_{2,i}Y_{2,g})\cdots(Y_{i-1,i}Y_{i-1,g})\cdot(Y_{i,i+1}Y_{i+1,g})\cdots(Y_{i,g-1}Y_{g-1,g})\\
&=&(Y_{1,g}\cdots{}Y_{i-1,g}Y_{i+1,g}\cdots{}Y_{g-1,g})(Y_{2,i}\cdots{}Y_{i-1,i}Y_{i,i+1}\cdots{}Y_{i,g-1}).
\end{eqnarray*}
for $2\leq{i}\leq{g-1}$.
By the relators~$\overline{(1)}$ and $\overline{(2)}$, the relator~(5) of $\Gamma_2(g-1)$ for $i=1$ can be transformed as
\begin{eqnarray*}
(Y_{1,2}Y_{2,g})\cdots(Y_{1,g-1}Y_{g-1,g})
&=&(Y_{1,2}\cdots{}Y_{1,g-1})(Y_{2,g}\cdots{}Y_{g-1,g})\\
&=&(Y_{1,g}\cdot{}Y_{3,g}\cdots{}Y_{g-1,g})(Y_{2,3}\cdots{}Y_{2,g-1})\\
&&(Y_{1,g}Y_{2,g}\cdot{}Y_{4,g}\cdots{}Y_{g-1,g})(Y_{2,3}\cdot{}Y_{3,4}\cdots{}Y_{3,g-1})\\
&&(Y_{1,g}Y_{2,g}Y_{3,g}\cdot{}Y_{5,g}\cdots{}Y_{g-1,g})(Y_{2,4}Y_{3,4}\cdot{}Y_{4,5}\cdots{}Y_{4,g-1})\\
&&\vdots\\
&&(Y_{1,g}\cdots{}Y_{g-3,g}\cdot{}Y_{g-1,g})(Y_{2,g-2}\cdots{}Y_{g-3,g-2}\cdot{}Y_{g-2,g-1})\\
&&(Y_{1,g}\cdots{}Y_{g-2,g})(Y_{2,g-1}\cdots{}Y_{g-2,g-1})\\
&&(Y_{2,g}\cdots{}Y_{g-1,g})\\
&=&(Y_{1,g}\cdots{}Y_{g-1,g})^{g-2}\prod_{2\leq{i<j}\leq{g-1}}Y_{i,j}^2\\
&=&
\left\{
\begin{array}{ll}
Y_{1,g}\cdots{}Y_{g-1,g}&\textrm{if}~g~\textrm{is odd},\\
1&\textrm{if}~g~\textrm{is even}.\\
\end{array}
\right.
\end{eqnarray*}
Hence we have the relator~$\overline{(5)}$.

Thus we obtain the claim.
\end{proof}

By the relators~$\overline{(1)}$ and $\overline{(2)}$, it follows that $\Gamma_2(g-1)/{N}$ is a $\mathbb{Z}/2\mathbb{Z}$-module.
By the relator~$\overline{(5)}$, generators $Y_{1,i}$ are not needed, for $2\leq{i}\leq{g}$ if $g$ is odd and for $2\leq{i}\leq{g-1}$ if $g$ is even.
Therefore we have the following.

\begin{cor}\label{G/N-2}
For $g\geq3$, the quotient $\Gamma_2(g-1)/{N}$ of $\Gamma_2(g-1)$ by $N$ is isomorphic to $\left(\mathbb{Z}/2\mathbb{Z}\right)^{\tbinom{g-1}{2}}$ if $g$ is odd and to $\left(\mathbb{Z}/2\mathbb{Z}\right)^{\tbinom{g-1}{2}+1}$ if $g$ is even.
\end{cor}

\begin{lem}\label{norm-gen-T^2}
In $\mathcal{M}_2(N_g)$, $\mathcal{T}^2(N_g)$ is normally generated by $t_{\alpha_{i_1,i_2,\dots,i_k}}^2$ for even indices $1\leq{i_1<i_2<\cdots<i_k\leq{g}}$.
\end{lem}

\begin{proof}
For any non-separating simple closed curve $\alpha$ of $N_g$, there are indices $1\leq{i_1<i_2<\cdots<i_k\leq{g}}$ such that the $\mathbb{Z}/2\mathbb{Z}$ coefficient homology classes of $\alpha$ and $\alpha_{i_1,i_2,\dots,i_k}$ are equal, and then there is $f\in\mathcal{M}_2(N_g)$ such that $f(\alpha_{i_1,i_2,\dots,i_k})=\alpha$ (see Theorem~\ref{SCC/M_2}).
Hence for any $t_\alpha^2\in\mathcal{T}^2(N_g)$, there is $f\in\mathcal{M}_2(N_g)$ such that $t_\alpha^2=t_{f(\alpha_{i_1,i_2,\dots,i_k})}^2=ft_{\alpha_{i_1,i_2,\dots,i_k}}^2f^{-1}$.
Thus we obtain the claim.
\end{proof}

We now prove Theorem~\ref{main-1}.

\begin{proof}[Proof of Theorem~\ref{main-1}]
It is known that $\mathcal{T}_2(N_g)$ is an index $2$ subgroup of $\mathcal{M}_2(N_2)$ (see \cite{Li, KO}).
Since $\mathcal{T}_2(N_g)\supset\mathcal{T}^2(N_g)\mathcal{I}(N_g)$, by Corollary~\ref{G/N-2}, it suffices to show that the quotient of $\mathcal{M}_2(N_2)$ by $\mathcal{T}^2(N_g)\mathcal{I}(N_g)$ is isomorphic to $\Gamma_2(g-1)/{N}$.
Let $\phi:\mathcal{M}_2(N_g)\to\Gamma_2(g-1)$ be the homomorphism defined in Subsection~\ref{M_2-I}, that is, $\phi(f)=f_\ast$.
Since $\ker\phi$ is $\mathcal{I}(N_g)$, we show that $\phi(\mathcal{T}^2(N_g))$ is $N$.

For $1\leq{i<j}\leq{g}$, we have that $(Y_{\alpha_j,\alpha_{i,j}}^{-1})_\ast(Y_{\alpha_i,\alpha_{i,j}})_\ast=Y_{j,i}^{-1}Y_{i,j}\in{N}$.
In addition, since the equality $t_{\alpha_{i,j}}^2=Y_{\alpha_j,\alpha_{i,j}}^{-1}Y_{\alpha_i,\alpha_{i,j}}$ holds, we have that $\phi(t_{\alpha_{i,j}}^2)=Y_{j,i}^{-1}Y_{i,j}$.
Therefore $\phi(\mathcal{T}^2(N_g))\supset{N}$.

In order to prove $\phi(\mathcal{T}^2(N_g))\subset{N}$, it suffices to show that $\phi(t_{\alpha_{i_1,i_2,\dots,i_k}}^2)$ is in $N$, by Lemma~\ref{norm-gen-T^2}.
Let $\alpha^\prime_{i_1,i_2,\dots,i_k}$ be a simple closed curve as shown in Figure~\ref{alpha}.
Since $t_{\alpha_{i_1,i_2,\dots,i_k}}^{-1}t_{\alpha^\prime_{i_1,i_2,\dots,i_k}}$ is in $\mathcal{I}(N_g)$ and hence its image by $\phi$ is $1$, we have that
$$\phi(t_{\alpha_{i_1,i_2,\dots,i_k}}^2)=\phi(t_{\alpha_{i_1,i_2,\dots,i_k}}^2\cdot{}t_{\alpha_{i_1,i_2,\dots,i_k}}^{-1}t_{\alpha^\prime_{i_1,i_2,\dots,i_k}})=\phi(t_{\alpha_{i_1,i_2,\dots,i_k}}t_{\alpha^\prime_{i_1,i_2,\dots,i_k}}).$$
By the proof of Lemma~\ref{chain-T^2}, $t_{\alpha_{i_1,i_2,\dots,i_k}}t_{\alpha^\prime_{i_1,i_2,\dots,i_k}}$ is a product of $T_{s,t}$ for $1\leq{s<t}\leq{k}$, where
$$T_{s,t}=(t_{\alpha_{i_{s+1},i_{s+2}}}t_{\alpha_{i_{s+2},i_{s+3}}}\cdots{}t_{\alpha_{i_{t-1},i_t}})^{-1}t_{\alpha_{i_s,i_{s+1}}}^2(t_{\alpha_{i_{s+1},i_{s+2}}}t_{\alpha_{i_{s+2},i_{s+3}}}\cdots{}t_{\alpha_{i_{t-1},i_t}}).$$
Note that $T_{s,t}$ is the square of the Dehn twist about a simple closed curve as shown in Figure~\ref{st}.
Similar to the proof of Lemma~\ref{norm-gen-T^2}, there is $f\in\mathcal{M}_2(N_g)$ such that $T_{s,t}=ft_{\alpha_{i_s,i_t}}^2f^{-1}$.
Since $\phi(t_{\alpha_{i_s,i_t}}^2)=Y_{i_t,i_s}^{-1}Y_{i_s,i_t}$, it follows that $\phi(t_{\alpha_{i_1,i_2,\dots,i_k}}t_{\alpha^\prime_{i_1,i_2,\dots,i_k}})$ is in $N$.
Therefore	 $\phi(\mathcal{T}^2(N_g))\subset{N}$.

\begin{figure}[htbp]
\includegraphics{st}
\caption{The loop $(t_{\alpha_{i_{s+1},i_{s+2}}}t_{\alpha_{i_{s+2},i_{s+3}}}\cdots{}t_{\alpha_{i_{t-1},i_t}})^{-1}(\alpha_{i_s,i_{s+1}})$.}\label{st}
\end{figure}

Thus we complete the proof.
\end{proof}










\section{Proof of Theorem~\ref{main-2}}\label{T^2I}

In this section, we prove Theorem~\ref{main-2}.
At first, we show the following.

\begin{lem}\label{chain-T^2}
Let $d_1$, $d_2$ and $d_3$ be simple closed curves of a surface $S$ as shown in Figure~\ref{chain}.
Then $t_{d_1}t_{d_2}$ and $t_{d_3}$ are in $\mathcal{T}^2(S)$.

\begin{figure}[htbp]
\includegraphics{chain}
\caption{Simple closed curves $c_1$, $c_2$, $\dots$, $c_k$, $d_1$, $d_2$ and $d_3$.}\label{chain}
\end{figure}

\end{lem}

\begin{proof}
Let $c_i$ be a simple closed curve of $S$ as shown in Figure~\ref{chain}, for $1\leq{i}\leq{k}$.
It is known that
\begin{itemize}
\item	$t_{d_1}t_{d_2}=(t_{c_1}t_{c_2}\cdots{}t_{c_k})^{k+1}$,
\item	$t_{d_3}=(t_{c_1}t_{c_2}\cdots{}t_{c_k})^{2(k+1)}$,
\end{itemize}
called the $k$-chain relation.
When $k=1$, it is clear that $t_{d_1}t_{d_2}$ is in $\mathcal{T}^2(S)$.
When $k\geq2$, by the commutativity relation $t_{c_i}t_{c_j}=t_{c_j}t_{c_i}$ with $|i-j|\geq2$ and the braid relation $t_{c_i}t_{c_{i+1}}t_{c_i}=t_{c_{i+1}}t_{c_i}t_{c_{i+1}}$, we calculate
\begin{eqnarray*}
(t_{c_1}t_{c_2}\cdots{}t_{c_k})^{k+1}
&=&(t_{c_1}t_{c_2}\cdots{}t_{c_{k-1}})^kt_{c_k}t_{c_{k-1}}\cdots{}t_{c_2}t_{c_1}^2t_{c_2}\cdots{}t_{c_{k-1}}t_{c_k}\\
&=&(t_{c_1}t_{c_2}\cdots{}t_{c_{k-1}})^kt_{c_k}^2\cdot(t_{c_k}^{-1}t_{c_{k-1}}^2t_{c_k})\cdots(t_{c_k}^{-1}t_{c_{k-1}}^{-1}\cdots{}t_{c_2}^{-1}t_{c_1}^2t_{c_2}\cdots{}t_{c_{k-1}}t_{c_k}).
\end{eqnarray*}
By induction on $k$, we have that $(t_{c_1}t_{c_2}\cdots{}t_{c_k})^{k+1}$ is in $\mathcal{T}^2(S)$, and $t_{d_1}t_{d_2}$ and $t_{d_3}$ are so.
Thus we obtain the claim.
\end{proof}

We next prove Theorem~\ref{main-2}.

\begin{proof}[Proof of Theorem~\ref{main-2}]
Recall Subsection~\ref{M_2-I}, especially Theorem~\ref{torelli-gen}.
Since $\mathcal{I}(N_3)$ is trivial, it is clear that $\mathcal{T}^2(N_3)\mathcal{I}(N_3)=\mathcal{T}^2(N_3)$.
Hence suppose that $g\geq5$.
By Lemma~\ref{chain-T^2}, it follows that $t_{\beta_1}t_{\beta_2}^{-1}(=t_{\beta_1}t_{\beta_2}\cdot{}t_{\beta_2}^{-2})$ is in $\mathcal{T}^2(N_g)$.
Let $\delta$ be a simple closed curve as shown in Figure~\ref{alpha-delta}.
By Lemma~\ref{chain-T^2}, it follows that $t_\alpha{}t_\delta$ is in $\mathcal{T}^2(N_g)$.
Since $t_\delta=1$, we have that $t_\alpha$ is in $\mathcal{T}^2(N_g)$.
Therefore $\mathcal{I}(N_g)\subset\mathcal{T}^2(N_g)$ and hence $\mathcal{T}^2(N_g)\mathcal{I}(N_g)=\mathcal{T}^2(N_g)$.
Thus we finish the proof.

\begin{figure}[htbp]
\includegraphics{alpha-delta}
\caption{Simple closed curves $\alpha$ and $\delta$.}\label{alpha-delta}
\end{figure}

\end{proof}

\begin{rem}
Any Dehn twist about a separating simple closed curve is in $\mathcal{I}(N_g)$ for any genus $g$, and so in $\mathcal{T}^2(N_g)$ if $g$ is odd.
\end{rem}

\begin{rem}
We do not know whether or not $\mathcal{T}^2(N_g)\mathcal{I}(N_g)=\mathcal{T}^2(N_g)$  if $g$ is even.
\end{rem}










\section{A finite generating set for $\mathcal{T}^2(N_g)\mathcal{I}(N_g)$}\label{gen-T^2I}

By Theorem~\ref{main-1}, $\mathcal{T}^2(N_g)\mathcal{I}(N_g)$ is a finite index subgroup of $\mathcal{M}_2(N_g)$.
In addition, by Theorem~\ref{level2-gen}, $\mathcal{M}_2(N_g)$ is finitely generated.
Hence $\mathcal{T}^2(N_g)\mathcal{I}(N_g)$ can be finitely generated.
In this section, we give a finite generating set for $\mathcal{T}^2(N_g)\mathcal{I}(N_g)$ by the Reidemeister-Schreier method.

For $1\leq{i<j}\leq{g}$, let $\beta_{i,j}$ be a simple closed curve as shown in Figure~\ref{beta_ij}.
For $(i,j)$ with $1\leq{i<j}\leq{g}$, we consider the lexicographic order $<$, that is, $(i,j)<(k,l)$ if and only if $i<k$, or $i=k$ and $j<l$.
Let $\mathcal{Y}$ be the set consisting of
$$Y_{\alpha_{i_1},\alpha_{i_1,j_1}}Y_{\alpha_{i_2},\alpha_{i_2,j_2}}\cdots{}Y_{\alpha_{i_m},\alpha_{i_m,j_m}}$$
for $2\leq{i_s<j_s}\leq{g}$ if $g$ is odd and $(i_s,j_s)=(1,g)$ or $2\leq{i_s<j_s}\leq{g}$ if $g$ is even, where $s\neq{}t\iff(i_s,j_s)\neq(i_t,j_t)$, and $\mathcal{Y}_{\mathrm{LO}}$ the subset of $\mathcal{Y}$ consisting of
$$Y_{\alpha_{i_1},\alpha_{i_1,j_1}}Y_{\alpha_{i_2},\alpha_{i_2,j_2}}\cdots{}Y_{\alpha_{i_m},\alpha_{i_m,j_m}}\in\mathcal{Y},$$
where $s<t\iff(i_s,j_s)<(i_t,j_t)$.

\begin{figure}[htbp]
\includegraphics{beta_ij}
\caption{A simple closed curve $\beta_{i,j}$ for $1\leq{i<j}\leq{g}$.}\label{beta_ij}
\end{figure}

The main result of this section is as follows.

\begin{thm}\label{gen-T^2I-2}
$\mathcal{T}^2(N_g)\mathcal{I}(N_g)$ is generated by
\begin{enumerate}
\item	$ft_{\alpha_{i,j}}^2f^{-1}$ for $1\leq{i<j}\leq{g}$ and $f\in\mathcal{Y}_{\mathrm{LO}}$,
\item	$ft_{\beta_{i,j}}f^{-1}$ for $1\leq{i<j}\leq{g}$ and $f\in\mathcal{Y}_{\mathrm{LO}}$,
\item	$ft_{\alpha_{1,j,k,l}}^2f^{-1}$ for $1<j<k<l\leq{g}$ and $f\in\mathcal{Y}_{\mathrm{LO}}$.
\end{enumerate}
\end{thm}

In order to prove this theorem, we first show the following lemma.

\begin{lem}\label{gen-T^2I-1}
$\mathcal{T}^2(N_g)\mathcal{I}(N_g)$ is generated by $(1)$-$(3)$ of Theorem~\ref{gen-T^2I-2} and
\begin{enumerate}
\item[(4)]	$f[Y_{\alpha_{i},\alpha_{i,j}},Y_{\alpha_{k},\alpha_{k,l}}]f^{-1}$ for $1\leq{i<j}\leq{g}$, $1\leq{k<l}\leq{g}$ and $f\in\mathcal{Y}_{\mathrm{LO}}$.
\end{enumerate}
\end{lem}

In addition, in order to prove this lemma, we use the following lemma.

\begin{lem}\label{[a,b]}
The following holds.
$$[a,b_1b_2\cdots{}b_n]=[a,b_1]\left(b_1[a,b_2]b_1^{-1}\right)\left(b_1b_2[a,b_3]b_2^{-1}b_1^{-1}\right)\cdots\left(b_1b_2\cdots{}b_{n-1}[a,b_n]b_{n-1}^{-1}\cdots{}b_2^{-1}b_1^{-1}\right)$$
\end{lem}

\begin{proof}
We calculate
\begin{eqnarray*}
&&[a,b_1]\left(b_1[a,b_2]b_1^{-1}\right)\\
&=&\left(ab_1a^{-1}b_1^{-1}\right)\left(b_1ab_2a^{-1}b_2^{-1}b_1^{-1}\right)\\
&=&ab_1b_2a^{-1}b_2^{-1}b_1^{-1}\\
&=&[a,b_1b_2],\\
&&[a,b_1b_2\cdots{}b_{n-1}]\left(b_1b_2\cdots{}b_{n-1}[a,b_n]b_{n-1}^{-1}\cdots{}b_2^{-1}b_1^{-1}\right)\\
&=&\left(ab_1b_2\cdots{}b_{n-1}a^{-1}b_{n-1}^{-1}\cdots{}b_2^{-1}b_1^{-1}\right)\left(b_1b_2\cdots{}b_{n-1}ab_na^{-1}b_n^{-1}b_{n-1}^{-1}\cdots{}b_2^{-1}b_1^{-1}\right)\\
&=&ab_1b_2\cdots{}b_na^{-1}b_n^{-1}\cdots{}b_2^{-1}b_1^{-1}\\
&=&[a,b_1b_2\cdots{}b_n].
\end{eqnarray*}
By induction on $n$, we obtain the claim.
\end{proof}

\begin{proof}[Proof of Lemma~\ref{gen-T^2I-1}]
By Theorem~\ref{main-1}, we have the short exact sequence
$$1\to\mathcal{T}^2(N_g)\mathcal{I}(N_g)\to\mathcal{M}_2(N_g)\to\left(\mathbb{Z}/2\mathbb{Z}\right)^{\tbinom{g-1}{2}+\epsilon}\to1,$$
where $\epsilon=0$ if $g$ is odd and  $\epsilon=1$ if $g$ is even.
In order to give a generating set for $\mathcal{T}^2(N_g)\mathcal{I}(N_g)$, we use the Reidemeister-Schreier method (for details, for instance see \cite{Jo}).

By Lemma~\ref{G/N-1}, it follows that $\left(\mathbb{Z}/2\mathbb{Z}\right)^{\tbinom{g-1}{2}+\epsilon}$ is equal to $\mathcal{Y}_{\mathrm{LO}}$ as a set.
We note that $\mathcal{Y}_{\mathrm{LO}}$ is a Schreier transversal for $\mathcal{T}^2(N_g)\mathcal{I}(N_g)$ in $\mathcal{M}_2(N_g)$, that is, for any $Y_{\alpha_{i_1},\alpha_{i_1,j_1}}Y_{\alpha_{i_2},\alpha_{i_2,j_2}}\cdots{}Y_{\alpha_{i_m},\alpha_{i_m,j_m}}\in\mathcal{Y}_{\mathrm{LO}}$ with $m\geq1$, $Y_{\alpha_{i_1},\alpha_{i_1,j_1}}Y_{\alpha_{i_2},\alpha_{i_2,j_2}}\cdots{}Y_{\alpha_{i_{m-1}},\alpha_{i_{m-1},j_{m-1}}}$ is in $\mathcal{Y}_{\mathrm{LO}}$.
For any $x\in\mathcal{M}_2(N_g)$, let $\overline{x}$ be the element of $\mathcal{Y}_{\mathrm{LO}}=\left(\mathbb{Z}/2\mathbb{Z}\right)^{\tbinom{g-1}{2}+\epsilon}$ which is sent from $x$ by the natural projection.
Then $\mathcal{T}^2(N_g)\mathcal{I}(N_g)$ is generated by
$$G=\left\{\left.fx^{\pm1}\overline{fx^{\pm1}}^{-1}\right|x\in{X},f\in\mathcal{Y}_{\mathrm{LO}},fx^{\pm1}\neq\overline{fx^{\pm1}}\right\},$$
where $X$ is the generating set for $\mathcal{M}_2(N_g)$ in Theorem~\ref{level2-gen}.
For any $fx^{\pm1}\overline{fx^{\pm1}}^{-1}\in{G}$, we check that $fx^{\pm1}\overline{fx^{\pm1}}^{-1}$ is a product of elements in (1)-(4) of Lemma~\ref{gen-T^2I-1} and their inverses.
Note that $Y_{\alpha_j,\alpha_{i,j}}^{-1}Y_{\alpha_i,\alpha_{i,j}}=Y_{\alpha_j,\alpha_{i,j}}Y_{\alpha_i,\alpha_{i,j}}^{-1}=t_{\alpha_{i,j}}^2$ and $Y_{\alpha_i,\alpha_{i,j}}^2=Y_{\alpha_j,\alpha_{i,j}}^2=t_{\beta_{i,j}}$.

Let $x=Y_{\alpha_i,\alpha_{i,j}}$.
If $f$ includes $x$, there are $f_1$, $f_2\in\mathcal{Y}_{\mathrm{LO}}$ such that $f=f_1xf_2$.
Then we see
\begin{eqnarray*}
fx\overline{fx}^{-1}
&=&f_1xf_2x\overline{f_1xf_2x}^{-1}\\
&=&f_1xf_2x(f_1f_2)^{-1}\\
&=&f_1[x,f_2]f_1^{-1}\cdot{}f_1f_2x^2f_2^{-1}f_1^{-1},\\
fx^{-1}\overline{fx^{-1}}^{-1}
&=&f_1xf_2x^{-1}\overline{f_1xf_2x^{-1}}^{-1}\\
&=&f_1xf_2x^{-1}(f_1f_2)^{-1}\\
&=&f_1[x,f_2]f_1^{-1}.
\end{eqnarray*}
If $f$ does not include $x$, there are $f_1$, $f_2\in\mathcal{Y}_{\mathrm{LO}}$ such that $f=f_1f_2$ and $f_1xf_2\in\mathcal{Y}_{\mathrm{LO}}$.
Then we see
\begin{eqnarray*}
fx\overline{fx}^{-1}
&=&f_1f_2x\overline{f_1f_2x}^{-1}\\
&=&f_1f_2x(f_1xf_2)^{-1}\\
&=&f_1[x,f_2]^{-1}f_1^{-1},\\
fx^{-1}\overline{fx^{-1}}^{-1}
&=&f_1f_2x^{-1}\overline{f_1f_2x^{-1}}^{-1}\\
&=&f_1f_2x^{-1}(f_1xf_2)^{-1}\\
&=&fx^{-2}f^{-1}\cdot{}f_1[x,f_2]^{-1}f_1^{-1}.
\end{eqnarray*}
By Lemma~\ref{[a,b]}, we have that $fx^{\pm1}\overline{fx^{\pm1}}^{-1}\in{G}$.
Let $x=Y_{\alpha_j,\alpha_{i,j}}$ and $x^\prime=Y_{\alpha_i,\alpha_{i,j}}$.
We see
\begin{eqnarray*}
fx\overline{fx}^{-1}&=&fx(x^\prime)^{-1}f^{-1}\cdot{}fx^\prime\overline{fx^\prime}^{-1},\\
fx^{-1}\overline{fx^{-1}}^{-1}&=&fx^{-2}f^{-1}\cdot{}fx\overline{fx}^{-1}.
\end{eqnarray*}
Hence we have that $fx^{\pm1}\overline{fx^{\pm1}}^{-1}\in{G}$.
Let $x=t_{\alpha_{1,j,k,l}}^2$.
We see
$$fx^{\pm1}\overline{fx^{\pm1}}^{-1}=fx^{\pm1}f^{-1}.$$
Hence we have that $fx^{\pm1}\overline{fx^{\pm1}}^{-1}\in{G}$.

Thus we finish the proof.
\end{proof}

\begin{rem}\label{reduced-gen}
Put $f=Y_{\alpha_{i_1},\alpha_{i_1,j_1}}Y_{\alpha_{i_2},\alpha_{i_2,j_2}}\cdots{}Y_{\alpha_{i_m},\alpha_{i_m,j_m}}\in\mathcal{Y}_{\mathrm{LO}}$.
We notice that the generator (4) in Lemma~\ref{gen-T^2I-1} can be reduced a form satisfying $(i_m,j_m)<(i,j)<(k,l)$, since in the above proof $f_1[x,f_2]^{-1}f_1^{-1}$ is a product of these reduced generators.
\end{rem}

\begin{proof}[Proof of Theorem~\ref{gen-T^2I-2}]
It is clear that the generators (1), (2) and (3) in Lemma~\ref{gen-T^2I-1} are equal to the generators (1), (2) and (3) in Theorem~\ref{gen-T^2I-2} respectively.
We show that the reduced generator (4) in Remark~\ref{reduced-gen} is obtained from the generators (1) and (2) in Theorem~\ref{gen-T^2I-2}.

For any reduced generator $f[Y_{\alpha_{i},\alpha_{i,j}},Y_{\alpha_{k},\alpha_{k,l}}]f^{-1}$, we see
\begin{eqnarray*}
f[Y_{\alpha_{i},\alpha_{i,j}},Y_{\alpha_{k},\alpha_{k,l}}]f^{-1}
&=&
f(Y_{\alpha_{i},\alpha_{i,j}}Y_{\alpha_{k},\alpha_{k,l}})^2Y_{\alpha_{k},\alpha_{k,l}}^{-2}(Y_{\alpha_{k},\alpha_{k,l}}Y_{\alpha_{i},\alpha_{i,j}}^{-2}Y_{\alpha_{k},\alpha_{k,l}}^{-1})f^{-1}\\
&=&
f(Y_{\alpha_{i},\alpha_{i,j}}Y_{\alpha_{k},\alpha_{k,l}})^2f^{-1}\cdot{}ft_{\beta_{k,l}}^{-1}f^{-1}\cdot{}f(Y_{\alpha_{k},\alpha_{k,l}}t_{\beta_{i,j}}^{-1}Y_{\alpha_{k},\alpha_{k,l}}^{-1})f^{-1}.
\end{eqnarray*}
Since $f$ and $fY_{\alpha_{k},\alpha_{k,l}}$ are in $\mathcal{Y}_{\mathrm{LO}}$, $ft_{\beta_{k,l}}f^{-1}$ and $f(Y_{\alpha_{k},\alpha_{k,l}}t_{\beta_{i,j}}Y_{\alpha_{k},\alpha_{k,l}}^{-1})f^{-1}$ are generators (2) in Theorem~\ref{gen-T^2I-2} respectively.
Hence we check that either $f[Y_{\alpha_{i},\alpha_{i,j}},Y_{\alpha_{k},\alpha_{k,l}}]f^{-1}$ or $f(Y_{\alpha_{i},\alpha_{i,j}}Y_{\alpha_{k},\alpha_{k,l}})^2f^{-1}$ is obtained from the generators (1) and (2) in Theorem~\ref{gen-T^2I-2}.

Let $i=k$.
Then we calculate
\begin{eqnarray*}
(Y_{\alpha_{i},\alpha_{i,j}}Y_{\alpha_{k},\alpha_{k,l}})^2
&=&
\left(t_{\alpha_{j,l}}t_{Y_{\alpha_i,\alpha_{i,j}}^{-1}(\alpha_{j,l})}^{-1}\right)^2\\
&=&
t_{\alpha_{j,l}}^2(Y_{\alpha_i,\alpha_{i,j}}^{-1}t_{\alpha_{j,l}}^{-2}Y_{\alpha_i,\alpha_{i,j}})\\
&=&
t_{\alpha_{j,l}}^2Y_{\alpha_i,\alpha_{i,j}}^{-2}(Y_{\alpha_i,\alpha_{i,j}}t_{\alpha_{j,l}}^{-2}Y_{\alpha_i,\alpha_{i,j}}^{-1})Y_{\alpha_i,\alpha_{i,j}}^2\\
&=&
t_{\alpha_{j,l}}^2t_{\beta_{i,j}}^{-1}(Y_{\alpha_i,\alpha_{i,j}}t_{\alpha_{j,l}}^{-2}Y_{\alpha_i,\alpha_{i,j}}^{-1})t_{\beta_{i,j}}
\end{eqnarray*}
(see Figure~\ref{crosscap-slide-rel-3}~(a)).
Since $f$ and $fY_{\alpha_i,\alpha_{i,j}}$ are in $\mathcal{Y}_{\mathrm{LO}}$, $f(Y_{\alpha_{i},\alpha_{i,j}}Y_{\alpha_{k},\alpha_{k,l}})^2f^{-1}$ is obtained from the generators (1) and (2) in Theorem~\ref{gen-T^2I-2}.
Let $j=l$.
Then we calculate
\begin{eqnarray*}
[Y_{\alpha_{i},\alpha_{i,j}},Y_{\alpha_{k},\alpha_{k,l}}]
&=&
Y_{\alpha_{i},\alpha_{i,j}}Y_{\alpha_{k},\alpha_{k,j}}Y_{\alpha_{i},\alpha_{i,j}}^{-1}Y_{\alpha_{k},\alpha_{k,j}}^{-1}\\
&=&
Y_{Y_{\alpha_{i},\alpha_{i,j}}(\alpha_{k}),Y_{\alpha_{i},\alpha_{i,j}}(\alpha_{k,j})}Y_{\alpha_{k},\alpha_{k,j}}^{-1}\\
&=&
Y_{\alpha_{k},\alpha_{k,j}}Y_{\alpha_{k},\alpha_{i,k}}^2Y_{\alpha_{k},\alpha_{k,j}}^{-1}\\
&=&
Y_{\alpha_{k},\alpha_{k,j}}t_{\beta_{i,k}}Y_{\alpha_{k},\alpha_{k,j}}^{-1}
\end{eqnarray*}
(see Figure~\ref{crosscap-slide-rel-3}~(b)).
Since $fY_{\alpha_k,\alpha_{k,j}}$ is in $\mathcal{Y}_{\mathrm{LO}}$, $f[Y_{\alpha_{i},\alpha_{i,j}},Y_{\alpha_{k},\alpha_{k,l}}]f^{-1}$ is obtained from the generators (1) and (2) in Theorem~\ref{gen-T^2I-2}.
Let $j=k$.
Then we calculate
\begin{eqnarray*}
(Y_{\alpha_{i},\alpha_{i,j}}Y_{\alpha_{k},\alpha_{k,l}})^2
&=&
Y_{\alpha_{i},\alpha_{i,j}}Y_{\alpha_{j}\alpha_{j,l}}Y_{\alpha_{i},\alpha_{i,j}}Y_{\alpha_{j},\alpha_{i,j}}^{-1}Y_{\alpha_{j}\alpha_{j,l}}^{-1}Y_{\alpha_{i},\alpha_{i,j}}^{-1}\cdot{}Y_{\alpha_{i},\alpha_{i,j}}Y_{\alpha_{j},\alpha_{i,j}}^{-1}(Y_{\alpha_{j},\alpha_{i,j}}Y_{\alpha_{j},\alpha_{j,l}})^2\\
&=&
Y_{\alpha_{i},\alpha_{i,j}}Y_{\alpha_{j}\alpha_{j,l}}t_{\alpha_{i,j}}^2Y_{\alpha_{j}\alpha_{j,l}}^{-1}Y_{\alpha_{i},\alpha_{i,j}}^{-1}\cdot{}t_{\alpha_{i,j}}^2\left(t_{Y_{\alpha_j,\alpha_{j,l}}^{-2}(\alpha_{i,l})}t_{Y_{\alpha_j,\alpha_{j,l}}^{-1}(\alpha_{i,l})}^{-1}\right)^2\\
&=&
Y_{\alpha_{i},\alpha_{i,j}}Y_{\alpha_{j}\alpha_{j,l}}t_{\alpha_{i,j}}^2Y_{\alpha_{j}\alpha_{j,l}}^{-1}Y_{\alpha_{i},\alpha_{i,j}}^{-1}\cdot{}t_{\alpha_{i,j}}^2(Y_{\alpha_j,\alpha_{j,l}}^{-2}t_{\alpha_{i,l}}^2Y_{\alpha_j,\alpha_{j,l}}^2\cdot{}Y_{\alpha_j,\alpha_{j,l}}^{-1}t_{\alpha_{i,l}}^{-2}Y_{\alpha_j,\alpha_{j,l}})\\
&=&
Y_{\alpha_{i},\alpha_{i,j}}Y_{\alpha_{j}\alpha_{j,l}}t_{\alpha_{i,j}}^2Y_{\alpha_{j}\alpha_{j,l}}^{-1}Y_{\alpha_{i},\alpha_{i,j}}^{-1}\cdot{}t_{\alpha_{i,j}}^2t_{\beta_{j,l}}^{-1}t_{\alpha_{i,l}}^2t_{\beta_{j,l}}\cdot{}Y_{\alpha_j,\alpha_{j,l}}t_{\beta_{j,l}}^{-1}t_{\alpha_{i,l}}^{-2}t_{\beta_{j,l}}Y_{\alpha_j,\alpha_{j,l}}^{-1}
\end{eqnarray*}
(see Figure~\ref{crosscap-slide-rel-3}~(c)).
Since $f$, $fY_{\alpha_j,\alpha_{j,l}}$ and $fY_{\alpha_i,\alpha_{i,j}}Y_{\alpha_j,\alpha_{j,l}}$ are in $\mathcal{Y}_{\mathrm{LO}}$, $f(Y_{\alpha_{i},\alpha_{i,j}}Y_{\alpha_{k},\alpha_{k,l}})^2f^{-1}$ is obtained from the generators (1) and (2) in Theorem~\ref{gen-T^2I-2}.
Let $i<k<j<l$.
Then we calculate
\begin{eqnarray*}
[Y_{\alpha_{i},\alpha_{i,j}},Y_{\alpha_{k},\alpha_{k,l}}]
&=&
Y_{\alpha_{i},\alpha_{i,j}}Y_{\alpha_{k},\alpha_{k,l}}Y_{\alpha_{i},\alpha_{i,j}}^{-1}Y_{\alpha_{k},\alpha_{k,l}}^{-1}\\
&=&
Y_{\alpha_{i},\alpha_{i,j}}Y_{Y_{\alpha_{k},\alpha_{k,l}}(\alpha_{i}),Y_{\alpha_{k},\alpha_{k,l}}(\alpha_{i,j})}^{-1}\\
&=&
Y_{\alpha_{i},\alpha_{i,j}}Y_{\alpha_{i},\alpha_{i,l}}^{-1}Y_{\alpha_{i},\alpha_{i,k}}^2Y_{\alpha_{i},\alpha_{i,l}}Y_{\alpha_{i},\alpha_{i,k}}^2Y_{\alpha_{i},\alpha_{i,j}}^{-1}Y_{\alpha_{i},\alpha_{i,k}}^{-2}Y_{\alpha_{i},\alpha_{i,l}}^{-1}Y_{\alpha_{i},\alpha_{i,k}}^{-2}Y_{\alpha_{i},\alpha_{i,l}}\\
&=&
Y_{\alpha_{i},\alpha_{i,j}}(t_{\beta_{i,l}}^{-1}Y_{\alpha_{i},\alpha_{i,l}}t_{\beta_{i,k}}Y_{\alpha_{i},\alpha_{i,l}}^{-1}t_{\beta_{i,l}}t_{\beta_{i,k}})Y_{\alpha_{i},\alpha_{i,j}}^{-1}\cdot{}t_{\beta_{i,k}}^{-1}t_{\beta_{i,l}}^{-1}\cdot{}Y_{\alpha_{i},\alpha_{i,l}}t_{\beta_{i,k}}^{-1}Y_{\alpha_{i},\alpha_{i,l}}^{-1}\cdot{}t_{\beta_{i,l}}
\end{eqnarray*}
(see Figure~\ref{crosscap-slide-rel-3}~(d)).
Since $f$, $fY_{\alpha_i,\alpha_{i,j}}$, $fY_{\alpha_i,\alpha_{i,l}}$ and $fY_{\alpha_i,\alpha_{i,j}}Y_{\alpha_i,\alpha_{i,l}}$ are in $\mathcal{Y}_{\mathrm{LO}}$, $f[Y_{\alpha_{i},\alpha_{i,j}},Y_{\alpha_{k},\alpha_{k,l}}]f^{-1}$ is obtained from the generators (1) and (2) in Theorem~\ref{gen-T^2I-2}.
For the other cases, we have that $f[Y_{\alpha_{i},\alpha_{i,j}},Y_{\alpha_{k},\alpha_{k,l}}]f^{-1}=1$.

\begin{figure}[htbp]
\subfigure[$Y_{\alpha_{i},\alpha_{i,j}}Y_{\alpha_{i},\alpha_{i,l}}=t_{\alpha_{j,l}}t_{Y_{\alpha_i,\alpha_{i,j}}^{-1}(\alpha_{j,l})}^{-1}$.]{\includegraphics{crosscap-slide-rel-3-1}}
\subfigure[$Y_{Y_{\alpha_{i},\alpha_{i,j}}(\alpha_{k}),Y_{\alpha_{i},\alpha_{i,j}}(\alpha_{k,j})}=Y_{\alpha_{k},\alpha_{k,j}}Y_{\alpha_{k},\alpha_{i,k}}^2$.]{\includegraphics{crosscap-slide-rel-3-2}}
\subfigure[$Y_{\alpha_{j},\alpha_{i,j}}Y_{\alpha_{j},\alpha_{j,l}}=t_{Y_{\alpha_j,\alpha_{j,l}}^{-2}(\alpha_{i,l})}t_{Y_{\alpha_j,\alpha_{j,l}}^{-1}(\alpha_{i,l})}^{-1}$.]{\includegraphics{crosscap-slide-rel-3-3}}
\subfigure[$Y_{Y_{\alpha_{k},\alpha_{k,l}}(\alpha_{i}),Y_{\alpha_{k},\alpha_{k,l}}(\alpha_{i,j})}^{-1}=Y_{\alpha_{i},\alpha_{i,l}}^{-1}Y_{\alpha_{i},\alpha_{i,k}}^2Y_{\alpha_{i},\alpha_{i,l}}Y_{\alpha_{i},\alpha_{i,k}}^2Y_{\alpha_{i},\alpha_{i,j}}^{-1}Y_{\alpha_{i},\alpha_{i,k}}^{-2}Y_{\alpha_{i},\alpha_{i,l}}^{-1}Y_{\alpha_{i},\alpha_{i,k}}^{-2}Y_{\alpha_{i},\alpha_{i,l}}$.]{\includegraphics{crosscap-slide-rel-3-4}}
\caption{}\label{crosscap-slide-rel-3}
\end{figure}

Thus we complete the proof.
\end{proof}










