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\begin{document}
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\put(143,66){\makebox(0,0)[tl]{\footnotesize Proceedings of the Ninth Prague Topological Symposium}}
\put(143,50){\makebox(0,0)[tl]{\footnotesize Contributed papers from the symposium held in}}
\put(143,34){\makebox(0,0)[tl]{\footnotesize Prague, Czech Republic, August 19--25, 2001}}
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\title[Characterization of $p$-symmetric Heegaard splittings]{An intrinsic
characterization of $p$-symmetric Heegaard splittings}
\thanks{This contribution is extracted from \cite{M}
M. Mulazzani, {\em On $p$-symmetric Heegaard splittings},
J. Knot Theory Ramifications {\bf 9} (2000), no.~8, 1059--1067.
Reprinted with permission from World Scientific Publishing Co.}
\author{Michele Mulazzani}
\address{Department of Mathematics, University of Bologna\\ 
I-40127 Bologna, Italy\\
and C.I.R.A.M., Bologna, Italy}
\email{mulazza@dm.unibo.it}
\thanks{Michele Mulazzani,
{\em An intrinsic characterization of $p$-symmetric Heegaard splittings},
Proceedings of the Ninth Prague Topological Symposium, (Prague, 2001),
pp.~217--222, Topology Atlas, Toronto, 2002; {\tt arXiv:math.GT/0112231}}
\begin{abstract}
We show that every $p$-fold strictly-cyclic branched covering of a
$b$-bridge link in $\S^3$ admits a $p$-symmetric Heegaard splitting of
genus $g=(b-1)(p-1)$. 
This gives a complete converse to a result of Birman and Hilden, and gives
an intrinsic characterization of $p$-symmetric Heegaard splittings as
$p$-fold strictly-cyclic branched coverings of links.
\end{abstract}
\subjclass[2000]{Primary 57M12, 57R65; Secondary 20F05, 57M05, 57M25}
\keywords{3-manifolds, Heegaard splittings, cyclic branched coverings,
links, plats, bridge number, braid number}
\maketitle

\section{Introduction}

The concept of $p$-symmetric Heegard splittings has been introduced by
Birman and Hilden (see \cite{BH}) in an extrinsic way, depending on a
particular embedding of the handlebodies of the splitting in the ambient
space $\E^3$. The definition of such particular splittings was motivated
by the aim to prove that every closed, orientable 3-manifold of Heegaard
genus $g\le 2$ is a 2-fold covering of $\S^3$ branched over a link of
bridge number $g+1$ and that, conversely, the 2-fold covering of $\S^3$
branched over a link of bridge number $b\le 3$ is a closed, orientable
3-manifold of Heegaard genus $b-1$ (compare also \cite{Vi}).

A genus $g$ Heegaard splitting $M=Y_g\cup_{\f}Y'_g$ is called {\it
$p$-symmetric\/}, with $p>1$, if there exist a disjoint embedding of $Y_g$
and $Y'_g$ into $\E^3$ such that $Y'_g=\tau(Y_g)$, for a translation $\tau$ 
of $\E^3$, and an orientation-preserving homeomorphism $\P:\E^3\to\E^3$ of
period $p$, such that $\P(Y_g)=Y_g$ and, if $\GG$ denotes the cyclic group
of order $p$ generated by $\P$ and $\F:\partial Y_g\to\partial Y_g$ is the
orientation-preserving homeomorphism 
$\F=\tau^{-1}_{\vert\partial Y'_g}\f$,
the following conditions are fulfilled:
\begin{itemize}
\item[i)] $Y_g/\GG$ is homeomorphic to a 3-ball;
\item[ii)] $\mbox{Fix}(\P_{\vert Y_g}^h)=\mbox{Fix}(\P_{\vert Y_g})$,
for each $1\le h\le p-1$;
\item[iii)] 
$\mbox{Fix}(\P_{\vert Y_g})/\GG$ is an unknotted set of arcs\footnote{A
set of mutually disjoint arcs $\{t_1,\ldots,t_n\}$ properly embedded in a
handlebody $Y$ is {\it unknotted\/} if there is a set of mutually disjoint
discs $D=\{D_1,\ldots,D_n\}$ properly embedded in $Y$ such that $t_i\cap
D_i=t_i\cap\partial D_i=t_i$, $t_i\cap D_j=\emptyset$ and $\partial
D_i-t_i\subset\partial Y$ for $1\le i,j\le n$ and $i\neq j$.} in the ball
$Y_g/\mathcal{G}$;
\item[iv)] there exists an integer $p_0$ such that
$\F\P_{\vert\partial Y_g}\F^{-1}=(\P_{\vert\partial Y_g})^{p_0}$.
\end{itemize}

\begin{remark} 
By the positive solution of the Smith Conjecture \cite{MB} it is easy to
see that necessarily $p_0\equiv\pm 1$ mod $p$.
\end{remark}

The map $\P'=\tau\P\tau^{-1}$ is obviously an orientation-preserving
homeomorphism of period $p$ of $\E^3$ with the same properties as $\P$,
with respect to $Y'_g$, and the relation $\f\P_{\vert\partial
Y_g}\f^{-1}=(\P'_{\vert\partial Y'_g})^{p_0}$ easily holds.

The {\it $p$-symmetric Heegaard genus\/} $g_p(M)$ of a 3-manifold $M$ is
the smallest integer $g$ such that $M$ admits a $p$-symmetric Heegaard
splitting of genus $g$.

The following results have been established in \cite{BH}:
\begin{enumerate}
\item 
Every closed, orientable 3-manifold of $p$-symmetric Heegaard genus $g$
admits a representation as a $p$-fold cyclic covering of $\S^3$, branched
over a link which admits a $b$-bridge presentation, where $g=(b-1)(p-1)$.
\item 
The $p$-fold cyclic covering of $\S^3$ branched over a knot of braid
number $b$ is a closed, orientable 3-manifold $M$ which admits a
$p$-symmetric Heegaard splitting of genus $g=(b-1)(p-1)$.
\end{enumerate}

Note that statement 2 is not a complete converse of 1, since it only
concerns knots and, moreover, $b$ denotes the braid number, which is
greater than or equal to (often greater than) the bridge number. In this
paper we fill this gap, giving a complete converse to statement 1. Since
the coverings involved in 1 are strictly-cyclic (see next section for
details on strictly-cyclic branched coverings of links), our statement
will concern this kind of coverings. More precisely, we shall prove in
Theorem \ref{Theorem 3} that a $p$-fold strictly-cyclic covering of
$\S^3$, branched over a link of bridge number $b$, is a closed, orientable
3-manifold $M$ which admits a $p$-symmetric Heegaard splitting of genus
$g=(b-1)(p-1)$, and therefore has $p$-symmetric Heegaard genus $g_p(M)\le
(b-1)(p-1)$. This result gives an intrinsic interpretation of
$p$-symmetric Heegaard splittings as $p$-fold strictly-cyclic branched
coverings of links.

\section{Main results}

Let 
$$
\b=\{(p_k(t),t)\,\vert\, 1\le k\le
2n\,,\,t\in[0,1]\}\subset\E^2\times[0,1]
$$
be a geometric
$2n$-string braid of $\E^3$ \cite{Bi}, where
$p_1,\ldots,p_{2n}:[0,1]\to\E^2$ are continuous maps such that
$p_{k}(t)\neq p_{k'}(t)$, for every $k\neq k'$ and $t\in[0,1]$,
and such that
$\{p_1(0),\ldots,p_{2n}(0)\}=\{p_1(1),\ldots,p_{2n}(1)\}$. We set
$P_k=p_k(0)$, for each $k=1,\ldots,2n$, and
$$
A_i=(P_{2i-1},0),
B_i=(P_{2i},0),
A'_i=(P_{2i-1},1),
B'_i=(P_{2i},1),
$$
for each $i=1,\ldots,n$ (see Figure 1). Moreover, we set
$\FF=\{P_1,\ldots,P_{2n}\}$, $\FF_1=\{P_1,P_3\ldots,P_{2n-1}\}$
and $\FF_2=\{P_2,P_4,\ldots,P_{2n}\}$.

The braid $\b$ is realized through an ambient isotopy
$$
{\wh\b}:\E^2\times[0,1]\to\E^2\times[0,1],\
{\wh\b}(x,t)=(\b_t(x),t),
$$
where $\b_t$ is an homeomorphism of
$\E^2$ such that $\b_0=\mbox{Id}_{\E^2}$ and $\b_t(P_i)=p_i(t)$,
for every $t\in[0,1]$. Therefore, the braid $\b$ naturally defines
an orientation-preserving homeomorphism
${\wti\b}=\b_1:\E^2\to\E^2$, which fixes the set $\FF$. Note that
$\b$ uniquely defines ${\wti\b}$, up to isotopy of $\E^2$ mod
$\FF$.

Connecting the point $A_i$ with $B_i$ by a circular arc $\a_i$
(called {\it top arc\/}) and the point $A'_i$ with $B'_i$ by a
circular arc $\a'_i$ (called {\it bottom arc\/}), as in Figure 1,
for each $i=1,\ldots,n$, we obtain a $2n$-plat presentation of a
link $L$ in $\E^3$, or equivalently in $\S^3$. As is well known,
every link admits plat presentations and, moreover, a $2n$-plat
presentation corresponds to an $n$-bridge presentation of the
link. So, the bridge number $b(L)$ of a link $L$ is the smallest
positive integer $n$ such that $L$ admits a representation by a
$2n$-plat. For further details on braid, plat and bridge
presentations of links we refer to \cite{Bi}.

\begin{figure}[bht]
\begin{center}
\includegraphics*[totalheight=5cm]{figure1.eps}
\end{center}
\caption{A $2n$-plat presentation of a link.}
\label{Fig. 1}
\end{figure}

\begin{remark} 
A $2n$-plat presentation of a link 
$L\subset\E^3\subset\S^3=\E^3\cup\{\infty\}$ furnishes a
$(0,n)$-decomposition \cite{MS}
$(\S^3,L)=(D,A_n)\cup_{\f'}(D',A'_n)$ of the link, where $D$ and
$D'$ are the 3-balls 
$$D=(\E^2\times]-\infty,0])\cup\{\infty\} \mbox{ and } 
D'=(\E^2\times[1,+\infty[)\cup\{\infty\},$$
$$A_n=\a_1\cup\cdots\cup\a_n, \ A'_n=\a'_1\cup\cdots\cup\a'_n$$ 
and
$\f':\partial D\to\partial D'$ is defined by $\f'(\infty)=\infty$
and $\f'(x,0)=({\wti\b}(x),1)$, for each $x\in\E^2$.
\end{remark}

If a $2n$-plat presentation of a $\m$-component link 
$L=\bigcup_{j=1}^{\m}L_j$ is given, each component $L_j$ of $L$ contains
$n_j$ top arcs and $n_j$ bottom arcs. 
Obviously, $\sum_{j=1}^{\m}n_j=n$. A $2n$-plat presentation of a link $L$ 
will be called {\it special \/} if:
\begin{itemize}
\item[(1)] 
the top arcs and the bottom arcs belonging to $L_1$ are
$\a_1,\ldots,\a_{n_1}$ and $\a'_1,\ldots,\a'_{n_1}$ respectively, the top
arcs and the bottom arcs belonging to $L_2$ are
$\a_{n_1+1},\ldots,\a_{n_1+n_2}$ and $\a'_{n_1+1},\ldots,\a'_{n_1+n_2}$
respectively, $\ldots$, the top arcs and the bottom arcs belonging on
$L_\m$ are 
$$\a_{n_1+\cdots+n_{\m-1}+1},\ldots,\a_{n_1+\cdots+n_{\m}}=\a_{n}$$
and
$$\a'_{n_1+\cdots+n_{\m-1}+1},\ldots,\a'_{n_1+\cdots+n_{\m}}=\a'_{n}$$
respectively;
\item[(2)] 
$p_{2i-1}(1)\in\FF_1$ and $p_{2i}(1)\in\FF_2$, for each $i=1,\ldots,n$.
\end{itemize}

It is clear that, because of (2), the homeomorphism $\wti\b$, associated
to a $2n$-string braid $\b$ defining a special plat presentation, keeps
fixed both the sets $\FF_1$ and $\FF_2$. Although a special plat
presentation of a link is a very particular case, we shall prove that
every link admits such kind of presentation.

\begin{proposition}\label{Proposition special} 
Every link $L$ admits a special $2n$-plat presentation, for each 
$n\ge b(L)$.
\end{proposition}

\begin{proof}
Let $L$ be presented by a $2n$-plat. We show that this presentation is
equivalent to a special one, by using a finite sequence of moves on the
plat presentation which changes neither the link type nor the number of
plats. The moves are of the four types $I$, $I'$, $II$ and $II'$ depicted
in Figure 2. First of all, it is straightforward that condition (1) can be
satisfied by applying a suitable sequence of moves of type $I$ and $I'$.
Furthermore, condition (2) is equivalent to the following: $(2')$ there
exists an orientation of $L$ such that, for each $i=1,\ldots,n$, the top
arc $\a_i$ is oriented from $A_i$ to $B_i$ and the bottom arc $\a'_i$ is
oriented from $B'_i$ to $A'_i$. Therefore, choose any orientation on $L$
and apply moves of type $II$ (resp. moves of type $II'$) to the top arcs
(resp. bottom arcs) which are oriented from $B_i$ to $A_i$ (resp. from
$A'_i$ to $B'_i$). 
\end{proof}

\begin{figure}[bht]
\begin{center}
\includegraphics*[totalheight=10cm]{figure2.eps}
\end{center}
\caption{Moves on plat presentations.}
\label{Fig. 2}
\end{figure}

A $p$-fold branched cyclic covering of an oriented $\m$-component link
$L=\bigcup_{j=1}^{\m}L_j\subset\S^3$ is completely determined (up to
equivalence) by assigning to each component $L_j$ an integer $c_j\in{\bf
Z}_p-\{0\}$, such that the set $\{c_1,\ldots,c_{\m}\}$ generates the group
${\bf Z}_p$. The monodromy associated to the covering sends each meridian
of $L_j$, coherently oriented with the chosen orientations of $L$ and
$\S^3$, to the permutation $(1\,2\,\cdots\,p)^{c_j}\in\Sigma_p$. 
Multiplying
each $c_j$ by the same invertible element of ${\bf Z}_p$, we obtain an
equivalent covering.

Following \cite{MM} we shall call a branched cyclic covering:
\begin{itemize}
\item[a)] 
{\it strictly-cyclic\/} if $c_{j'}=c_{j''}$, for every 
$j',j''\in\{1,\ldots,\m\}$,
\item[b)] 
{\it almost-strictly-cyclic\/} if $c_{j'}=\pm c_{j''}$, for every
$j',j''\in\{1,\ldots,\m\}$, 
\item[c)] 
{\it meridian-cyclic\/} if $\gcd(b,c_j)=1$, for every
$j\in\{1,\ldots,\m\}$,
\item[d)] 
{\it singly-cyclic\/} if $\gcd(b,c_j)=1$, for some $j\in\{1,\ldots,\m\}$,
\item [e)] 
{\it monodromy-cyclic\/} if it is cyclic.
\end{itemize}

The following implications are straightforward: 
$$\text{ a)} 
\Rightarrow \text{ b)} 
\Rightarrow \text{ c)}
\Rightarrow \text{ d)}
\Rightarrow\text{ e)}.$$ 
Moreover, the five definitions are equivalent when $L$ is a knot. 
Similar definitions and properties also hold for a $p$-fold cyclic
covering of a 3-ball, branched over a set of properly embedded
(oriented) arcs.

It is easy to see that, by a suitable reorientation of the link, an
almost-strictly-cyclic covering becomes a strictly-cyclic one. As a
consequence, it follows from Remark 1 that every branched cyclic covering
of a link arising from a $p$-symmetric Heegaard splitting -- according to
Birman-Hilden construction -- is strictly-cyclic.

Now we show that, conversely, every $p$-fold branched strictly-cyclic
covering of a link admits a $p$-symmetric Heegaard splitting.

\begin{theorem}\label{Theorem 3} 
A $p$-fold strictly-cyclic covering of $\S^3$ branched over a link $L$ of
bridge number $b$ is a closed, orientable 3-manifold $M$ which admits a
$p$-symmetric Heegaard splitting of genus $g=(b-1)(p-1)$. So the
$p$-symmetric Heegaard genus of $M$ is $$g_p(M)\le(b-1)(p-1).$$
\end{theorem}

\begin{proof}
Let $L$ be presented by a special $2b$-plat arising from a braid $\b$, and
let $(\S^3,L)=(D,A_b)\cup_{\f'}(D',A'_b)$ be the $(0,b)$-decomposition
described in Remark 2. Now, all arguments of the proofs of Theorem 3 of
\cite{BH} entirely apply and the condition of Lemma 4 of \cite{BH} is
satisfied, since the homeomorphism $\wti\b$ associated to $\b$ fixes both
the sets $\FF_1$ and $\FF_2$. 
\end{proof}

As a consequence of Theorem \ref{Theorem 3} and Birman-Hilden results,
there is a natural one-to-one correspondence between $p$-symmetric
Heegaard splittings and $p$-fold strictly-cyclic branched coverings of
links.

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\bibitem{BH}
Joan~S. Birman and Hugh~M. Hilden, \emph{Heegaard splittings of branched
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\bibitem{MM}
John~P. Mayberry and Kunio Murasugi, \emph{Torsion-groups of abelian coverings
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\bibitem{MB}
John~W. Morgan and Hyman Bass (eds.), \emph{The {S}mith conjecture}, Academic
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\bibitem{MS}
Kanji Morimoto and Makoto Sakuma, \emph{On unknotting tunnels for knots}, Math.
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\bibitem{M}
Michele Mulazzani, \emph{On $p$-symmetric {H}eegaard splittings}, J. Knot
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\bibitem{Vi}
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\end{thebibliography}

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