Bihomogeneity of solenoids
Alex Clark
Robbert Fokkink
Solenoids are inverse limit spaces over regular covering maps of closed manifolds. M.C. McCord has shown that solenoids are topologically homogeneous and that they are principal bundles with a profinite structure group. We show that if a solenoid is bihomogeneous, then its structure group contains an open abelian subgroup. This leads to new examples of homogeneous continua that are not bihomogeneous.
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\volumeyear{2002}
\papernumber{1}
\published{12 January 2002}
\pagenumbers{1}{9}
\received{22 August 2001}
\revised{8 January 2002}
\accepted{10 January 2002}


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\title{Bihomogeneity of solenoids}

\author{Alex Clark\\Robbert Fokkink}

\address{University of North Texas, Department of Mathematics\\Denton
TX 76203-1430, U.S.A.}
\secondaddress{Technische Universiteit Delft,
Faculty of Information Technology and Systems\\Division Mediamatica,
P.O. Box 5031, 2600 GA Delft, Netherlands}

\asciiaddress{University of North Texas, Department of Mathematics\\Denton
TX 76203-1430, USA\\and\\Technische Universiteit Delft,
Faculty of Information Technology and Systems\\Division Mediamatica,
P.O. Box 5031, 2600 GA Delft, Netherlands}



\email{alexc@unt.edu, r.j.fokkink@its.tudelft.nl}

\begin{document}

\begin{abstract}
Solenoids are inverse limit spaces over regular covering maps of closed
manifolds.
M.C. McCord has shown that solenoids are topologically homogeneous and
that they are principal bundles with a profinite structure group.
We show that if a solenoid is bihomogeneous,
then its structure group
contains an open abelian subgroup. This leads
to new
examples of homogeneous continua that are not bihomogeneous.
\end{abstract}

\primaryclass{54F15}
\secondaryclass{55R10}
\keywords{Homogeneous continuum, covering space, profinite group,\break
principal bundle}
\asciikeywords{Homogeneous continuum, covering space, profinite group,
principal bundle}

\makeshorttitle

A topological space $X$ is \emph{homogeneous\/} if for every pair of points
$x,y\in X$ there is a homeomorphism $h:X\rightarrow X$ satisfying
$h\left(
x\right) =y$. The space is \emph{bihomogeneous\/} if for each
such pair there is a homeomorphism satisfying $h\left( x\right) =y$ and
$
h\left( y\right) =x.$
A compact and connected space is called a \emph{continuum}.
Knaster and Van Dantzig asked whether a
homogeneous
continuum is necessarily bihomogeneous. This was settled in the negative
by
Krystyna Kuperberg \cite{Kuperberg}. Subsequent counterexamples were
given
by Minc, Kawamura and Greg Kuperberg \cite{Minc, Kawamura, GKuperberg}.
The
counterexamples in \cite{Kuperberg,GKuperberg} are locally connected.
Ungar \cite{Ungar} has studied stronger types of homogeneity
conditions and showed that these conditions imply local connectivity.

A solenoid $M_{\infty }$ is an inverse limit space
over closed connected manifolds with bonding maps that are covering maps.
We shall silently
assume that the bonding maps are not $1-1$, so that
$M_{\infty }$ is not locally connected.
McCord~\cite{McCord} has shown that solenoids are homogeneous
provided that compositions of the bonding covering maps are regular. Minc~\cite{Minc}
presented an example of a homogeneous but not bihomogeneous infinite-dimensional 
continuum similar to a solenoid, and Krystyna Kuperberg~\cite{K2} 
observed that a similar construction could be used to construct a 
finite-dimensional solenoid which is homogeneous but not bihomogeneous.
We shall show
that $M_{\infty }$ is bihomogeneous only if a certain
condition
related to commutativity (or lack thereof) of $\pi _{1}\left(
M_{i}\right) $
is met. In case the solenoid
is $2$-dimensional, the condition is both necessary and sufficient.

\section{Path-components of solenoids as left-cosets of the structure
group}

A {\it (strong) solenoid} $M_\infty$ is an inverse-limit space of closed
manifolds
$M_i$ with bonding maps $p_i\colon M_{i+1}\to M_i$ for $i\in\N$ which
are covering maps, such that any composition $p_{i+k}\circ \ldots\circ
p_i$
is regular.
Solenoids are homogeneous spaces
and they have dense path-components.
\smallbreak
A $G$-bundle $(E,B,p,F)$ is {\it principal} if the
structure group $G$ acts effectively on the fibers.
As a consequence, the fiber
$F$ is homeomorphic to $G$, and
$G$ is naturally equivalent to
the group of deck-transformations.
\begin{theorem}[McCord, \protect\cite{McCord}]
Suppose that $M_\infty=\lim_{\leftarrow}(M_i,p_i)$ is a solenoid.
Let $\pi_0\colon M_\infty\to M_0$ be the projection onto the first
coordinate
and let $\Gamma_0=\pi_0^{-1}(m_0)$ be a fiber.
Then $(M_{\infty },M_{0},\pi_0,\Gamma _{0})$ is a principal-bundle.
\end{theorem}
The projection $\pi_0$ is not to be confused with a homotopy group.
Note that a solenoid $\lim_{\leftarrow }(M_{i},f_{i})$ is a principal
bundle over any $M_i$ and we have singled out $M_0$.
The spaces $M_i$ are called the {\it factor spaces} of the solenoid.
We think of the fundamental groups $\pi_1(M_i)$ as (normal) subgroups
of $\pi_1(M_0)$.
The structure group $\Gamma_0$ is isomorphic to the profinite group
$\lim_{\leftarrow}\pi_1(M_0)/\pi_1(M_i)$.
\medbreak
Choose base-points $m_i\in M_i$ such that $p_i(m_i)=m_{i-1}$, so
$m_\infty=(m_i)$ is an element of $M_\infty$.
We identify the structure group $\Gamma_0$ with the fiber of $m_0$
and we identify $m_\infty$ with the unit element of $\Gamma_0$.
The fundamental group $\pi_1(M_0)$ acts on the base-point fiber
$\Gamma_0$ by path lifting: for $g\in\Gamma_0$ and $\gamma\in\pi_1(M_0,m_0)$,
define $g\circ\gamma$ as the end-point of the lifted path $\tilde\gamma$
starting from the initial-point $g$.
One verifies that this right action of $\pi_1(M_0)$
commutes with left multiplication of $\Gamma_0$.
More precisely,
suppose that $h$ is a deck-transformation and that $\tilde\gamma$ is a
lifted path with initial-point $g$.
Then $h(\tilde\gamma)$ has initial point $h(g)$ and end-point
$h(g\circ\gamma)$.
Identify the structure group with the group of deck-transformations,
so we get that $(hg)\circ\gamma=h(g\circ\gamma)$.

\begin{definition}
Suppose that $(M_{\infty },M_0,\pi_0,\Gamma_0)$ is a solenoid.
We shall call the $\pi_1(M_0)$-orbit of $e\in\Gamma_0$
the characteristic group and we shall denote it by $\gamma_0$.
Let $K_\infty\subset\pi_1(M_0)$ be the intersection of all
$\pi_1(M_i)$.
Then $\gamma_0$ is isomorphic to $\pi_1(M_0)/K_\infty$ and we shall
refer to $K_\infty$ as the kernel of $\pi_1(M_0)$.
\end{definition}

Our definition deviates from the common terminology, as in \cite{Steenrod},
where the equivalence class of $\gamma_0$ under inner automorphisms of
$\Gamma_0$ is called the characteristic class.
Note that $\gamma_0$ inherits a topology from $\Gamma_0$.

\begin{lemma}\label{leftcosets}
The path components of a solenoid are naturally equivalent to the left
cosets
$\Gamma_0/\gamma_0$.
\end{lemma}
\begin{proof}
Suppose that $x,y\in\Gamma_0$ are elements of the base-point fiber.
Then $x\circ\gamma=y$ for some $\gamma\in\pi_1(M_0)$ if and only
if there exists a path $\tilde\gamma\subset M_\infty$ that connects
$x$ to $y$.
\end{proof}

If we replace the base space $M_0$ by $M_i$ for some index $i$,
then we get a principal bundle $(M_\infty,M_i,\pi_i,\Gamma_i)$,
where $\Gamma_i\subset \Gamma_0$ is the subgroup of transformations that
leave $M_i$ invariant.
The topology of $\Gamma_0$ is induced by taking the $\Gamma_i$ as
an open neighborhood base of the identity.
One verifies that the charateristic group of the bundle, denoted
$\gamma_i$,
is equal to $\gamma_0\cap\Gamma_i$.
Hence the $\gamma_i$ are open subgroups of $\gamma_0$.

\begin{lemma}
For $j>i$ the inclusion $\Gamma _{j}\subset \Gamma _{i}$ induces a
natural isomorphism between $\Gamma _{j}/\gamma _{j}$
and $\Gamma _{i}/\gamma _{i}.$
\end{lemma}

Since path components are dense in $M_{\infty },$ the
characteristic subgroups $\gamma _{i}$ are dense in $\Gamma _{i}$.

\section{The permutation of path-components by self-\break hom\-eo\-morphisms}

A solenoid $M_\infty$ can be represented as a subspace of $\prod M_i$,
the Cartesian product of its factor spaces.
We identify
$M_i$ with the subspace of $\prod M_i$ defined by:
$$M_i=\{(x_j) : x_j\in M_j,\ x_j=p^i_j(x_i) \text{ if } j\leq i,\
x_j=m_j \text{ if }j>i\}$$
where $p^i_j\colon M_i\to M_j$ is a composition of bonding maps.
In this representation,
the factor spaces $M_i$ and $M_\infty$ all have the same base-point.
\medbreak
A \emph{morphism} between fiber bundles
can be represented by a commutative diagram:
\begin{equation*}
\begin{array}{rcl}
E_{1}& \overset{h}{\longrightarrow } & E_{2}\  \\
p_1 \downarrow &  &\downarrow p_2 \\
B_{1} & \overset{f}{\longrightarrow } & B_{2}\
\end{array}
\end{equation*}
We shall say that $h$ is the {\it lifted map} and that $f$ is the {\it
base-map}.
We say that morphisms are homotopic if their base-maps are.
By the unique path-lifting property, a morphism between bundles
with a totally disconnected fiber is determined by the base-map
$f\colon B_1\to B_2$
and the image under $h$ of a single element of $E_1$.
For pointed spaces, therefore, a bundle-morphism is determined by the
base-map only.
This implies that,
for principal bundles with a totally disconnected fiber,
bundle-morphisms
commute with deck-transformations; i.e., for
a lifted map $h$ and a deck-transformation $\varphi\colon E_1\to E_1$,
we have that $h\circ\varphi=\psi\circ h$ for some
deck-transformation $\psi\colon E_2\to E_2$.

\begin{lemma}\label{induceshomomorphism}
Suppose that $(E_{i},B_{i},p_{i},\Gamma_{i})$ are principal
$\Gamma_i$-bundles with a totally disconnected fiber
(for $i=1,2$).
Then a base-point preserving
bundle-morphism induces a homomorphism of the structure group.
Furthermore, homotopic morphisms induce the same homomorphism.
\end{lemma}
\begin{proof}
First note that the lifted map $h$ maps $\Gamma_1$ to $\Gamma_2$.
Deck-transformations are (left) translations
$x\to ax$ of the
base-point fiber $\Gamma_i$ ($i=1,2$).
Since a bundle-morphism commutes with deck-transformations,
$h\colon \Gamma_1\to \Gamma_2$ satisfies $h(ax)=f(a)h(x)$ for some
$f\colon \Gamma_1\to \Gamma_2$.
Substitute $x=e$ to find that $h(ax)=h(a)h(x)$.
Now homotopic bundle-morphisms give homotopic homomorphisms $h\colon
\Gamma_1\to\Gamma_2$.
Since the groups are totally disconnected, the homomorphisms are
necessarily
the same.
\end{proof}

We shall say that a bundle morphism of a solenoid is an
\textit{automorphism}
if the commutative diagram can be extended on the right-hand side
\begin{equation*}
\begin{array}{rclrcl}
M_{\infty}& \overset{h_1}{\longrightarrow } & M_{\infty}&
\overset{h_2}{\longrightarrow } & M_{\infty}\\
\pi_j \downarrow &  &\downarrow \pi_i &  &\downarrow \pi_k\\
M_{j} & \overset{f_1}{\longrightarrow } & M_{i}&
\overset{f_2}{\longrightarrow } & M_{k}
\end{array}
\end{equation*}

such that $f_2\circ f_1$ is homotopic to $p^j_k$. We shall say that
$h_1$ is the
inverse of $h_2$.
For instance, the covering
projection $p^j_i\colon M_j\to M_i$ with
lifted map $id_{M_\infty}$
yields an automorphism.
We show that for every self-homeomorphism of a
solenoid,
there is an automorphism that acts in the same way on the space of
path-components.

\begin{theorem}\label{homotopicmorphism}
Suppose that $h$ is a base-point preserving
self-homeomorphism of a solenoid $M_{\infty }$.
Then $h$ is homotopic to the lifted map of
an automorphism of $M_\infty$.
\end{theorem}

\begin{proof}
Since $M_0$ is an ANR, the composition
$\pi_0\circ h\colon M_\infty\to M_0$
extends to $H\colon U\to M_0$ for
a neighborhood of $M_\infty\subset U$ in $\prod M_i$.
The restriction $H\colon M_i\to M_0$ is well-defined for
sufficiently large $i$.
Note that $H$ preserves the base-point of $M_i$.
For sufficiently large $i$, the maps $H\circ \pi_i$ and $\pi_0\circ h$
are homotopic.
By the homotopy lifting property, $H\circ \pi_i$ can then be lifted to
$\tilde H\colon M_\infty\to M_\infty$, which is homotopic to $h$.

Now apply the same argument to $\pi_i\circ h^{-1}$ to find
a map $G\colon M_j\to M_i$ for sufficiently large $j$ which
can be lifted to $\tilde G\colon M_\infty\to M_\infty$.
By choosing $j$ and $i$ sufficiently large, the composition
$H\circ G\colon M_j\to M_0$ gets arbitrarily close
to and hence homotopic to the covering map $p^j_0$.
\end{proof}

Theorem \ref{homotopicmorphism} and Lemma \ref{induceshomomorphism}
describe
how a self-homeomorphism acts on path-components of
a solenoid (provided that it preserves the
base-point).

\begin{lemma}
Suppose that $h$ is the lifted map of an automorphism of a solenoid
$M_\infty$.
For some index $i$, $h$ induces a monomorphism
$\hat h\colon\Gamma_i\to \Gamma_0$ such that $\hat
{h}^{-1}(\gamma_0)=\gamma_i$
and $\hat h(\Gamma_i)$ is an open subgroup of $\Gamma_0$.
\end{lemma}
\begin{proof}
By Lemma \ref{induceshomomorphism} we know that $h$ induces
a homomorphism $\hat h\colon \Gamma_i\to\Gamma_0$.
Since homeomorphisms preserve path-components,
Lemma \ref{leftcosets} implies that $h$ induces a homomorphism
$\Gamma_i/\gamma_i\to \Gamma_0/\gamma_0$.
Since $h$ is the lifted map of an automorphism, it has an inverse $g$
which induces a homomorphism $\hat{g}\colon\Gamma_j\to \Gamma_0$.
The composition $\hat g\circ\hat h$, which is defined on an open
subgroup,
is equal to
the identity.
By Lemma \ref{induceshomomorphism},
$\hat g\circ\hat h$ is equal to the homomorphism induced
by $p^j_i$, which is the identity.
\end{proof}

\section{An algebraic condition for bihomogeneity}

\begin{definition}
Suppose that $\Gamma_0$ is the structure group of a solenoid with
characteristic group $\gamma_0$.
We define $\Mon(\Gamma_0,\gamma_0)$ as the set of monomorphisms
$f\colon \Gamma_i\to \Gamma_0$, such that $f(\gamma_i)=\gamma_0\cap
f(\Gamma_i)$.
\end{definition}

We say that an element of $\Mon(\Gamma_0,\gamma_0)$ is a
{\emph{characteristic automorphism}}.
A self-homeomorphism $H$ of $M_\infty$ need not preserve
the base-point.
It can however be represented as a composition
of a homeomorphism $h$ that preserves the path-component of
the base-point and a deck-transformation.
Since $h$ is homotopic to a base-point preserving homeomorphism,
$H$ permutes the path-components in the same way as a composition of
a base-point preserving homeomorphism and a deck-transformation.
In terms of $\Gamma_0/\gamma_0$,
this is a composition of a characteristic automorphism
$\varphi$ and a left translation $z\to wz$ of $\Gamma_0$.

\begin{definition}
We say that a solenoid is \emph{algebraically bihomogeneous\/} if
it satisfies the following condition.
For every $x,y\in \Gamma _{0}$ there are elements $w\in\Gamma_0$
and $\varphi\in\Mon(\Gamma_0,\gamma_0)$ such that
$z\to w\varphi(z)$ switches the residue classes
$x\mathop{\mathrm{mod}}\gamma_0$ and
$y\mathop{\mathrm{mod}}\gamma_0$.
\end{definition}

Obviously, bihomogeneity implies algebraic bihomogeneity.
The condition of algebraic bihomogeneity may seem awkward, but
fortunately there is a simpler characterization as we shall
see below.
We denote $x\sim y$ if $x,y$ are in the same residue class of
$\gamma_{0}$.

\begin{lemma}\label{inverse}
A solenoid $M_{\infty }$ is algebraically bihomogeneous if and if only for
every $z\in \Gamma _{0}$ there is a characteristic automorphism
$\varphi$ such that $\varphi (z)\sim z^{-1}$.
\end{lemma}

\begin{proof}
Suppose that $M_{\infty }$ is algebraically bihomogeneous.
For every $z\in
\Gamma _{0}$, we can switch the cosets of $z$ and $e$.
More precisely, there exists a
$w\in \Gamma_0$ and a $\varphi\in\Mon(\Gamma_0,\gamma_0)$
such that $zg=w\varphi (e)$ and $eg^{\prime }=w\varphi (z)$
for $g,g^{\prime }\in \gamma _{0}$.
Since $\varphi (e)=e,$ it follows that
$w=zg$ and $\varphi (z)=g^{-1}z^{-1}g^{\prime }$. Compose $
\varphi $ with the inner automorphism $x\rightarrow gxg^{-1}$ to obtain
$
\psi \in \Mon(\Gamma _{0},\gamma_0)$ satisfying $\psi
\left( z\right) \sim z^{-1}.$

If $\varphi (z)=z^{-1}g$ for some $g\in \gamma _{0}$, then
compose $\varphi $ with the inner automorphism $x\rightarrow gxg^{-1}$
to
get $\psi \in \Mon(\Gamma _{0},\gamma_0)$ satisfying
$\psi (z)=gz^{-1}$. Then $x\rightarrow zg^{-1}\psi (x)$ switches the
cosets of~$e$ and~$z$.
This implies algebraic bihomogeneity.
\end{proof}

Since $
z\rightarrow z^{-1}$ is a homomorphism if and only if the group is abelian, we have the
following corollary.

\begin{corollary}
A solenoid with an abelian structure group $\Gamma _{i}$ is
algebraically
bihomogeneous.
\end{corollary}

This condition is automatically met if $\pi _{1}\left(
M_{i}\right) $ is abelian.

\begin{lemma}
Suppose that $\gamma _{0}$ is a characteristic group. Then
$\Mon(\Gamma _{0},\gamma_0)$ is countable.
\end{lemma}

\begin{proof}
There are only countably many subgroups $\gamma_i$ and each of these
is finitely generated.
Hence, there are only finitely many homomorphisms
$f\colon\gamma_i\to\gamma_0$.
Since characteristic automorphisms are determined by their action on
some $\gamma_i$, the result follows.
\end{proof}

\begin{theorem}
\label{main}Let $M_{\infty }$ be a bihomogeneous solenoid with structure
group $\Gamma _{0}$. Then $\Gamma _{0}$ contains an open abelian
subgroup.
\end{theorem}

\begin{proof}
Suppose that $\varphi \colon \Gamma _{j}\rightarrow \Gamma _{0}$ is a
characteristic automorphism. For $g\in \gamma _{0}$ define the subset $
V(\varphi ,g)=\{z\in \Gamma _{j}\colon z\varphi (z)=g\}\subset \Gamma
_{0}.$
As $\varphi $ ranges over $\Mon(\Gamma_{0},\gamma_0)$
and $g$ ranges over $\gamma _{0}$, the countable family of all
$V(\varphi ,g)
$ covers $\Gamma _{0}$ by Lemma \ref{inverse}.
Hence one of
these sets, say $V(\varphi _{0},g_{0})$, is of second category in
$\Gamma
_{0}$.
It follows that $K=\{z\in \Gamma _{0}\colon
z\varphi
_{0}(z)=g_{0}\}$ is closed with non-empty interior in $\Gamma _{0}.$
Since $K$ has non-empty interior,
there exist a $z_{0}\in K$ and a neighborhood $V$ of $e$ such that
$\varphi
(z_{0}\delta )=\delta ^{-1}z_{0}^{-1}g_{0}$ for all $\delta \in V$. It
follows that $\varphi (\delta )=g_{0}^{-1}z_{0}\delta
^{-1}z_{0}^{-1}g_{0}$.
By composition with the inner automorphism $x\rightarrow
z_{0}^{-1}g_{0}xg_{0}^{-1}z_{0},$ we get a homomorphism $\psi $ such
that $
\psi (\delta )=\delta ^{-1}$ for $\delta \in V$. The group generated by
$V$
is an open abelian subgroup of $\Gamma _{0}$.
\end{proof}

For any neighborhood $V$ of the identity,
$\gamma _{j}\subset V$ for large enough~$j$.
Hence there
exists an open abelian subgroup of $\Gamma _{0}$
if and only if $\gamma _{j}$ is abelian for some $j.$

\begin{corollary}\label{condition}
Let $M_{\infty }$ be a solenoid and let $K_\infty$ be the kernel
of $\pi_1(M_0)$.
Then $M_\infty$ is algebraically bihomogeneous if and only if
$\pi_1(M_j)/K_\infty$ is abelian for sufficiently large index~$j$, or,
equivalently, $\Gamma_j$ is abelian for sufficiently large index~$j$.
\end{corollary}

\section{An application}

Our algebraic condition for (topological)
bihomogeneity in Corollary \ref{condition}
is necessary but not sufficient.
For this, there should exist a homeomorphism $h\colon M_i\to M_i$
which induces an isomorphism $h_*\colon\pi_1(M_i)\to\pi_1(M_i)$
such that $h_*(x)=x^{-1}$ (modulo $K_\infty$).
The problem whether homomorphisms between fundamental groups
are realized by continuous maps is known as the geometric
realization problem.
It is a classical result of Nielsen \cite{Nielsen} that closed surfaces
admit geometric realizations.
This can be extended to certain three-dimensional
manifolds~\cite{Waldhausen}.
The following result now follows from Nielsen's theorem.

\begin{theorem}
A two-dimensional solenoid $S_\infty$
with kernel $K_\infty\subset \pi_1(S_0) $ is
bihomogeneous if and only if $\pi_1(S_i)/K_\infty$
is abelian for sufficiently large index $i$.
\end{theorem}

One easily constructs two-dimensional solenoids that are
not bihomogeneous, using results from geometric group theory.
The fundamental group $\pi_1(S)$ of a closed surface is subgroup separable,
see~\cite{Scott};
i.e., for every subgroup $H\subset \pi_1(S)$
there is a descending chain of subgroups of finite index with kernel $H$.
Hence, there
exists a solenoid with base-space $S$ and kernel $H$.
For a closed surface $S$ of genus greater than~$1$,
the fundamental group contains no abelian subgroup of finite index.
Therefore, a solenoid with base-space $S$
and kernel~$\{e\}$ is a (simply-connected) continuum which is
not bihomogeneous.

\section{Final remarks}

One-dimensional solenoids
are indecomposable continua.
It is not difficult to show that
higher-dimensional solenoids are not.
Rogers~\cite{Rogers2} has
shown
that a homogeneous, hereditarily indecomposable continuum is at most
one-dimensional. His question whether there exists a
homogeneous, indecomposable continuum of dimension greater than one
remains open.
\medbreak
Our example of a non-bihomogeneous space
is based on obstructions of the fundamental
group, which seems to be
characteristic for all examples so far.
So it is natural
to ask whether there exists a
simply-connected Peano
continuum that is homogeneous but not bihomogeneous. More generally, 
it is natural to ask whether there exists a continuum with trivial first \v{C}ech cohomology
that is homogeneous but not bihomogeneous.

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{\bf G\,P Scott},
{\it Subgroups of surface groups are almost geometric},
J. London Math. Soc. (2) 17 (1978) 555--565.

\bibitem{Steenrod} {\bf Norman Steenrod}, {\it The topology of fibre bundels},
Princeton (1951)

\bibitem{Ungar} {\bf G\,S Ungar}, {\it On all kinds of homogeneous
spaces},
Trans. Amer. Math. Soc. 212 (1975) 393--400

\bibitem{Waldhausen}
{\bf F Waldhausen},
{\it  On irreducible $3$-manifolds which are sufficiently large},
Ann. of Math. (2) 87 (1968) 56--88
\end{thebibliography}

\Addresses\recd

\end{document}

agtout.tex
%%%
%%%  This is agtout.tex.  
%%%
%%%  This the version of  gtoutput.tex  intended to finish formatting
%%%  papers published in Algebriac & Geometric Topology and stored in the
%%%  arXiv.   All versions of  gtoutput.tex  are copyright 
%%%  GT Publications and are to be used _only_ for formatting
%%%  the officially published version of ABT or G&T papers.
%%%
%%%
%%%                                             Colin Rourke  27.102000
%%%
%%%  To create header file  head.xxx  comment out the first \endinput

%  test for latex or plain tex
\def\ifplaintex{\expandafter\ifx\csname documentclass\endcsname\relax}

\def\gtp{{\mathsurround=0pt\it $\cal G\mskip-2mu$eometry \&\ 
$\cal T\!\!$opology $\cal P\!$ublications}}  % GT publications

\def\Addressesr{\bigskip
{\small \parskip 0pt \leftskip 0pt \rightskip 0pt plus 1fil \def\\{\par}
\sl\theaddress\par
\medskip
\rm Email:\stdspace\tt\theemail\hfill\rm Received:\qua\receiveddate \par}}


\def\recd{{\small Received:\qua\receiveddate\ifx\reviseddate\relax
\else\qquad Revised:\qua\reviseddate\fi\par}} 

%  define the various new ingredients of the title page and the data
%  output files

\def\lognumber#1{\def\thelognumber{#1}}
\def\volumenumber#1{\def\thevolumenumber{#1}}
\def\volumeyear#1{\def\thevolumeyear{#1}}
\def\papernumber#1{\def\thepapernumber{#1}}
\def\pagenumbers#1#2{\def\startpage{#1}\def\finishpage{#2}}
\def\published#1{\def\publishdate{#1}}
\def\proposed#1{\def\theproposer{#1}}
\def\seconded#1{\def\theseconders{#1}}
\def\received#1{\def\receiveddate{#1}}
\def\revised#1{\def\reviseddate{#1}}
\def\accepted#1{\def\accepteddate{#1}}
\def\asciititle#1{\def\theasciititle{#1}}
\def\covertitle#1{\def\thecovertitle{#1}}
\def\asciiauthors#1{\def\theasciiauthors{#1}}
\def\asciiaddress#1{\def\theasciiaddress{#1}}
\def\asciiemail#1{\def\theasciiemail{#1}}
\def\coverauthors#1{\def\thecoverauthors{#1}}
\long\def\asciiabstract#1{\long\def\theasciiabstract{#1}}
\def\asciikeywords#1{\def\theasciikeywords{#1}}
\def\isbn#1{\def\theisbn{#1}}
\def\erratum#1{\def\theerratum{non-empty}\def\erratumnumber{#1}}

%  initialise

\let\\\par\let\thelognumber\relax\let\thevolumenumber\relax
\let\thepapernumber\relax\let\thevolumeyear\relax\let\startpage\relax
\let\finishpage\relax\let\publishdate\relax\let\receiveddate\relax
\let\reviseddate\relax\let\accepteddate\relax\let\theasciititle\relax
\let\thecovertitle\relax\let\theasciiauthors\relax\let\theasciiaddress\relax
\let\theasciiabstract\relax\let\theasciikeywords\relax
\let\theproposer\relax\let\theseconders\relax\let\theisbn\relax
\let\thecoverauthors\relax\let\theerratum\relax\let\theasciiemail\relax



%%%% fonts for AGT logo:

\ifplaintex
\font\logobig=cmssbx10 scaled 3836
\font\logomed=cmssbx10 scaled 2557
\else
\font\logobig=cmssbx10 scaled 4200
\font\logomed=cmssbx10 scaled 2800
\fi

\long\def\makeagttitle{   %%% start of definition of \makeagttitle
\count0=\startpage
\agt\hfill      %   Journal title (top left) 
%   logo (top right)
\hbox to 45truept{\vbox to 0pt{\vglue -13truept{\logomed A\kern -.37em{\logobig 
T}\kern -.38em G}\vss}\hss}
%
\break
{\small Volume \thevolumenumber\ (\thevolumeyear)
\startpage--\finishpage\nl
Published: \publishdate}

\vglue .25truein

% title
{\parskip=0pt\leftskip 0pt plus
1fil\def\\{\par\smallskip}{\Large\bf\thetitle}\par\medskip} \vglue
0.05truein

% authors :
%
{\parskip=0pt\leftskip 0pt plus 1fil\def\\{\par}{\sc\theauthors}
\par\medskip}%
 
\vglue 0.03truein 

%  abstract and classification numbers:

{\small\leftskip 25truept\rightskip 25truept{\bf Abstract}\stdspace\theabstract

{\bf AMS Classification}\stdspace\theprimaryclass
\ifx\thesecondaryclass\relax\else; \thesecondaryclass\fi\par
{\bf Keywords}\stdspace \thekeywords\par}\vglue 7truept

}   %%%% end of definition of \makeagttitle

\ifplaintex
%  get print centerpage:
\hoffset 14truemm
\voffset 31truemm
%  fonts for headline and footline
\font\phead=cmsl9 scaled 950
\font\pnum=cmbx10 scaled 913
\font\pfoot=cmsl9 scaled 950
%  headline and footline
\headline{\vbox to 0pt{\vskip -4.5mm\line{\small\phead\ifnum
\count0=\startpage ISSN 1472-2739 (on-line) 1472-2747 (printed)
\hfill {\pnum\folio}\else\ifodd\count0\def\\{ }% 
\ifx\theshorttitle\relax\thetitle\else\theshorttitle\fi\hfill{\pnum\folio}
\else\def\\{ and }{\pnum\folio}\hfill\ifx\theshortauthors\relax\theauthors
\else\theshortauthors\fi\fi\fi}\vss}}
\footline{\vbox to 0pt{\vglue 0mm\line{\small\pfoot\ifnum\count0=\startpage
\copyright\ \gtp\hfill\else
\agt, Volume \thevolumenumber\ (\thevolumeyear)\hfill\fi}\vss}}
\else
%  get print centerpage:
\headsep 23pt
\footskip 35pt
\hoffset -4truemm
\voffset 12.5truemm
%  fonts for headline and footline
\font\lhead=cmsl9 scaled 1050
\font\lnum=cmbx10 
\font\lfoot=cmsl9 scaled 1050
\makeatletter
%  headline and footline
\def\@oddhead{{\small\lhead\ifnum\count0=\startpage ISSN 1472-2739 
(on-line) 1472-2747 (printed)\hfill {\lnum\number\count0}\else\ifodd\count0
\def\\{ }\ifx\theshorttitle\relax \thetitle \else\theshorttitle\fi\hfill
{\lnum\number\count0}\else\def\\{ and }{\lnum\number\count0}
\hfill\ifx\theshortauthors\relax 
\theauthors\else\theshortauthors\fi\fi\fi}}\def\@evenhead{\@oddhead}
\def\@oddfoot{\small\lfoot\ifnum\count0=\startpage\copyright\ \gtp\hfill\else
\agt, Volume \thevolumenumber\ (\thevolumeyear)\hfill\fi}
\def\@evenfoot{\@oddfoot}
\makeatother
\fi
%  force \makeagttitle
\let\maketitlepage\makeagttitle
\let\makeshorttitle\maketitlepage
\let\maketitle\maketitlepage

\endinput  %%%comment out to create xxx header file


\newwrite\gtoutfile
\long\gdef\makeheadfile{  %%% start of definition of \makeheadfile
{\def\\{, }\def\s{ }
\immediate\openout\gtoutfile head.xxx
\immediate\write\gtoutfile{To: math@arxiv.org}
\immediate\write\gtoutfile{Subject: put OR rep NNNNN:ppppp}
\immediate\write\gtoutfile{--text follows this line--}
\immediate\write\gtoutfile{Proxy-for: \ifx\theasciiauthors\relax
\theauthors\else\theasciiauthors\fi\s<\ifx\theasciiemail\relax\theemail\else\theasciiemail\fi>}
\immediate\write\gtoutfile{\noexpand\\}
\immediate\write\gtoutfile{Authors: \ifx\theasciiauthors\relax
\theauthors\else\theasciiauthors\fi}
{\def\\{ }\immediate\write\gtoutfile{Title: \ifx\theasciititle\relax
\thetitle\else\theasciititle\fi}}
\immediate\write\gtoutfile{Subj-class: GT or SG, GR etc}
\immediate\write\gtoutfile{MSC-class: \theprimaryclass\ifx\thesecondaryclass\relax\else, \thesecondaryclass\fi}
\immediate\write\gtoutfile{Journal-ref: Algebr. Geom. Topol. \thevolumenumber\s
(\thevolumeyear) \startpage-\finishpage}
\immediate\write\gtoutfile{Comments: Published by Algebraic and
Geometric Topology at}
\immediate\write\gtoutfile{\s\s\s  http://www.maths.warwick.ac.uk/agt/AGTVol\thevolumenumber/agt-\thevolumenumber-\thepapernumber.abs.html}
\immediate\write\gtoutfile{\noexpand\\}
\immediate\write\gtoutfile{}
\ifx\theasciiabstract\relax
\immediate\write\gtoutfile{\theabstract}\else
\immediate\write\gtoutfile{\theasciiabstract}\fi
\immediate\write\gtoutfile{}
\immediate\write\gtoutfile{\noexpand\\}
\immediate\write\gtoutfile{}
\immediate\closeout\gtoutfile}}  %%% end of definition of \makeheadfile


\def\maketitlepage{\makeagttitle\makeheadfile}
\let\makeshorttitle\maketitlepage
\let\maketitle\maketitlepage

\endinput   
%%%
%%%  This is agtout.tex.  
%%%
%%%  This the version of  gtoutput.tex  intended to finish formatting
%%%  papers published in Algebriac & Geometric Topology and stored in the
%%%  arXiv.   All versions of  gtoutput.tex  are copyright 
%%%  GT Publications and are to be used _only_ for formatting
%%%  the officially published version of ABT or G&T papers.
%%%
%%%
%%%                                             Colin Rourke  27.102000
%%%
%%%  To create header file  head.xxx  comment out the first \endinput

%  test for latex or plain tex
\def\ifplaintex{\expandafter\ifx\csname documentclass\endcsname\relax}

\def\gtp{{\mathsurround=0pt\it $\cal G\mskip-2mu$eometry \&\ 
$\cal T\!\!$opology $\cal P\!$ublications}}  % GT publications

\def\Addressesr{\bigskip
{\small \parskip 0pt \leftskip 0pt \rightskip 0pt plus 1fil \def\\{\par}
\sl\theaddress\par
\medskip
\rm Email:\stdspace\tt\theemail\hfill\rm Received:\qua\receiveddate \par}}


\def\recd{{\small Received:\qua\receiveddate\ifx\reviseddate\relax
\else\qquad Revised:\qua\reviseddate\fi\par}} 

%  define the various new ingredients of the title page and the data
%  output files

\def\lognumber#1{\def\thelognumber{#1}}
\def\volumenumber#1{\def\thevolumenumber{#1}}
\def\volumeyear#1{\def\thevolumeyear{#1}}
\def\papernumber#1{\def\thepapernumber{#1}}
\def\pagenumbers#1#2{\def\startpage{#1}\def\finishpage{#2}}
\def\published#1{\def\publishdate{#1}}
\def\proposed#1{\def\theproposer{#1}}
\def\seconded#1{\def\theseconders{#1}}
\def\received#1{\def\receiveddate{#1}}
\def\revised#1{\def\reviseddate{#1}}
\def\accepted#1{\def\accepteddate{#1}}
\def\asciititle#1{\def\theasciititle{#1}}
\def\covertitle#1{\def\thecovertitle{#1}}
\def\asciiauthors#1{\def\theasciiauthors{#1}}
\def\asciiaddress#1{\def\theasciiaddress{#1}}
\def\asciiemail#1{\def\theasciiemail{#1}}
\def\coverauthors#1{\def\thecoverauthors{#1}}
\long\def\asciiabstract#1{\long\def\theasciiabstract{#1}}
\def\asciikeywords#1{\def\theasciikeywords{#1}}
\def\isbn#1{\def\theisbn{#1}}
\def\erratum#1{\def\theerratum{non-empty}\def\erratumnumber{#1}}

%  initialise

\let\\\par\let\thelognumber\relax\let\thevolumenumber\relax
\let\thepapernumber\relax\let\thevolumeyear\relax\let\startpage\relax
\let\finishpage\relax\let\publishdate\relax\let\receiveddate\relax
\let\reviseddate\relax\let\accepteddate\relax\let\theasciititle\relax
\let\thecovertitle\relax\let\theasciiauthors\relax\let\theasciiaddress\relax
\let\theasciiabstract\relax\let\theasciikeywords\relax
\let\theproposer\relax\let\theseconders\relax\let\theisbn\relax
\let\thecoverauthors\relax\let\theerratum\relax\let\theasciiemail\relax



%%%% fonts for AGT logo:

\ifplaintex
\font\logobig=cmssbx10 scaled 3836
\font\logomed=cmssbx10 scaled 2557
\else
\font\logobig=cmssbx10 scaled 4200
\font\logomed=cmssbx10 scaled 2800
\fi

\long\def\makeagttitle{   %%% start of definition of \makeagttitle
\count0=\startpage
\agt\hfill      %   Journal title (top left) 
%   logo (top right)
\hbox to 45truept{\vbox to 0pt{\vglue -13truept{\logomed A\kern -.37em{\logobig 
T}\kern -.38em G}\vss}\hss}
%
\break
{\small Volume \thevolumenumber\ (\thevolumeyear)
\startpage--\finishpage\nl
Published: \publishdate}

\vglue .25truein

% title
{\parskip=0pt\leftskip 0pt plus
1fil\def\\{\par\smallskip}{\Large\bf\thetitle}\par\medskip} \vglue
0.05truein

% authors :
%
{\parskip=0pt\leftskip 0pt plus 1fil\def\\{\par}{\sc\theauthors}
\par\medskip}%
 
\vglue 0.03truein 

%  abstract and classification numbers:

{\small\leftskip 25truept\rightskip 25truept{\bf Abstract}\stdspace\theabstract

{\bf AMS Classification}\stdspace\theprimaryclass
\ifx\thesecondaryclass\relax\else; \thesecondaryclass\fi\par
{\bf Keywords}\stdspace \thekeywords\par}\vglue 7truept

}   %%%% end of definition of \makeagttitle

\ifplaintex
%  get print centerpage:
\hoffset 14truemm
\voffset 31truemm
%  fonts for headline and footline
\font\phead=cmsl9 scaled 950
\font\pnum=cmbx10 scaled 913
\font\pfoot=cmsl9 scaled 950
%  headline and footline
\headline{\vbox to 0pt{\vskip -4.5mm\line{\small\phead\ifnum
\count0=\startpage ISSN 1472-2739 (on-line) 1472-2747 (printed)
\hfill {\pnum\folio}\else\ifodd\count0\def\\{ }% 
\ifx\theshorttitle\relax\thetitle\else\theshorttitle\fi\hfill{\pnum\folio}
\else\def\\{ and }{\pnum\folio}\hfill\ifx\theshortauthors\relax\theauthors
\else\theshortauthors\fi\fi\fi}\vss}}
\footline{\vbox to 0pt{\vglue 0mm\line{\small\pfoot\ifnum\count0=\startpage
\copyright\ \gtp\hfill\else
\agt, Volume \thevolumenumber\ (\thevolumeyear)\hfill\fi}\vss}}
\else
%  get print centerpage:
\headsep 23pt
\footskip 35pt
\hoffset -4truemm
\voffset 12.5truemm
%  fonts for headline and footline
\font\lhead=cmsl9 scaled 1050
\font\lnum=cmbx10 
\font\lfoot=cmsl9 scaled 1050
\makeatletter
%  headline and footline
\def\@oddhead{{\small\lhead\ifnum\count0=\startpage ISSN 1472-2739 
(on-line) 1472-2747 (printed)\hfill {\lnum\number\count0}\else\ifodd\count0
\def\\{ }\ifx\theshorttitle\relax \thetitle \else\theshorttitle\fi\hfill
{\lnum\number\count0}\else\def\\{ and }{\lnum\number\count0}
\hfill\ifx\theshortauthors\relax 
\theauthors\else\theshortauthors\fi\fi\fi}}\def\@evenhead{\@oddhead}
\def\@oddfoot{\small\lfoot\ifnum\count0=\startpage\copyright\ \gtp\hfill\else
\agt, Volume \thevolumenumber\ (\thevolumeyear)\hfill\fi}
\def\@evenfoot{\@oddfoot}
\makeatother
\fi
%  force \makeagttitle
\let\maketitlepage\makeagttitle
\let\makeshorttitle\maketitlepage
\let\maketitle\maketitlepage

\endinput  %%%comment out to create xxx header file


\newwrite\gtoutfile
\long\gdef\makeheadfile{  %%% start of definition of \makeheadfile
{\def\\{, }\def\s{ }
\immediate\openout\gtoutfile head.xxx
\immediate\write\gtoutfile{To: math@arxiv.org}
\immediate\write\gtoutfile{Subject: put OR rep NNNNN:ppppp}
\immediate\write\gtoutfile{--text follows this line--}
\immediate\write\gtoutfile{Proxy-for: \ifx\theasciiauthors\relax
\theauthors\else\theasciiauthors\fi\s<\ifx\theasciiemail\relax\theemail\else\theasciiemail\fi>}
\immediate\write\gtoutfile{\noexpand\\}
\immediate\write\gtoutfile{Authors: \ifx\theasciiauthors\relax
\theauthors\else\theasciiauthors\fi}
{\def\\{ }\immediate\write\gtoutfile{Title: \ifx\theasciititle\relax
\thetitle\else\theasciititle\fi}}
\immediate\write\gtoutfile{Subj-class: GT or SG, GR etc}
\immediate\write\gtoutfile{MSC-class: \theprimaryclass\ifx\thesecondaryclass\relax\else, \thesecondaryclass\fi}
\immediate\write\gtoutfile{Journal-ref: Algebr. Geom. Topol. \thevolumenumber\s
(\thevolumeyear) \startpage-\finishpage}
\immediate\write\gtoutfile{Comments: Published by Algebraic and
Geometric Topology at}
\immediate\write\gtoutfile{\s\s\s  http://www.maths.warwick.ac.uk/agt/AGTVol\thevolumenumber/agt-\thevolumenumber-\thepapernumber.abs.html}
\immediate\write\gtoutfile{\noexpand\\}
\immediate\write\gtoutfile{}
\ifx\theasciiabstract\relax
\immediate\write\gtoutfile{\theabstract}\else
\immediate\write\gtoutfile{\theasciiabstract}\fi
\immediate\write\gtoutfile{}
\immediate\write\gtoutfile{\noexpand\\}
\immediate\write\gtoutfile{}
\immediate\closeout\gtoutfile}}  %%% end of definition of \makeheadfile


\def\maketitlepage{\makeagttitle\makeheadfile}
\let\makeshorttitle\maketitlepage
\let\maketitle\maketitlepage

\endinput   
%%%
%%%  This is agtout.tex.  
%%%
%%%  This the version of  gtoutput.tex  intended to finish formatting
%%%  papers published in Algebriac & Geometric Topology and stored in the
%%%  arXiv.   All versions of  gtoutput.tex  are copyright 
%%%  GT Publications and are to be used _only_ for formatting
%%%  the officially published version of ABT or G&T papers.
%%%
%%%
%%%                                             Colin Rourke  27.102000
%%%
%%%  To create header file  head.xxx  comment out the first \endinput

%  test for latex or plain tex
\def\ifplaintex{\expandafter\ifx\csname documentclass\endcsname\relax}

\def\gtp{{\mathsurround=0pt\it $\cal G\mskip-2mu$eometry \&\ 
$\cal T\!\!$opology $\cal P\!$ublications}}  % GT publications

\def\Addressesr{\bigskip
{\small \parskip 0pt \leftskip 0pt \rightskip 0pt plus 1fil \def\\{\par}
\sl\theaddress\par
\medskip
\rm Email:\stdspace\tt\theemail\hfill\rm Received:\qua\receiveddate \par}}


\def\recd{{\small Received:\qua\receiveddate\ifx\reviseddate\relax
\else\qquad Revised:\qua\reviseddate\fi\par}} 

%  define the various new ingredients of the title page and the data
%  output files

\def\lognumber#1{\def\thelognumber{#1}}
\def\volumenumber#1{\def\thevolumenumber{#1}}
\def\volumeyear#1{\def\thevolumeyear{#1}}
\def\papernumber#1{\def\thepapernumber{#1}}
\def\pagenumbers#1#2{\def\startpage{#1}\def\finishpage{#2}}
\def\published#1{\def\publishdate{#1}}
\def\proposed#1{\def\theproposer{#1}}
\def\seconded#1{\def\theseconders{#1}}
\def\received#1{\def\receiveddate{#1}}
\def\revised#1{\def\reviseddate{#1}}
\def\accepted#1{\def\accepteddate{#1}}
\def\asciititle#1{\def\theasciititle{#1}}
\def\covertitle#1{\def\thecovertitle{#1}}
\def\asciiauthors#1{\def\theasciiauthors{#1}}
\def\asciiaddress#1{\def\theasciiaddress{#1}}
\def\asciiemail#1{\def\theasciiemail{#1}}
\def\coverauthors#1{\def\thecoverauthors{#1}}
\long\def\asciiabstract#1{\long\def\theasciiabstract{#1}}
\def\asciikeywords#1{\def\theasciikeywords{#1}}
\def\isbn#1{\def\theisbn{#1}}
\def\erratum#1{\def\theerratum{non-empty}\def\erratumnumber{#1}}

%  initialise

\let\\\par\let\thelognumber\relax\let\thevolumenumber\relax
\let\thepapernumber\relax\let\thevolumeyear\relax\let\startpage\relax
\let\finishpage\relax\let\publishdate\relax\let\receiveddate\relax
\let\reviseddate\relax\let\accepteddate\relax\let\theasciititle\relax
\let\thecovertitle\relax\let\theasciiauthors\relax\let\theasciiaddress\relax
\let\theasciiabstract\relax\let\theasciikeywords\relax
\let\theproposer\relax\let\theseconders\relax\let\theisbn\relax
\let\thecoverauthors\relax\let\theerratum\relax\let\theasciiemail\relax



%%%% fonts for AGT logo:

\ifplaintex
\font\logobig=cmssbx10 scaled 3836
\font\logomed=cmssbx10 scaled 2557
\else
\font\logobig=cmssbx10 scaled 4200
\font\logomed=cmssbx10 scaled 2800
\fi

\long\def\makeagttitle{   %%% start of definition of \makeagttitle
\count0=\startpage
\agt\hfill      %   Journal title (top left) 
%   logo (top right)
\hbox to 45truept{\vbox to 0pt{\vglue -13truept{\logomed A\kern -.37em{\logobig 
T}\kern -.38em G}\vss}\hss}
%
\break
{\small Volume \thevolumenumber\ (\thevolumeyear)
\startpage--\finishpage\nl
Published: \publishdate}

\vglue .25truein

% title
{\parskip=0pt\leftskip 0pt plus
1fil\def\\{\par\smallskip}{\Large\bf\thetitle}\par\medskip} \vglue
0.05truein

% authors :
%
{\parskip=0pt\leftskip 0pt plus 1fil\def\\{\par}{\sc\theauthors}
\par\medskip}%
 
\vglue 0.03truein 

%  abstract and classification numbers:

{\small\leftskip 25truept\rightskip 25truept{\bf Abstract}\stdspace\theabstract

{\bf AMS Classification}\stdspace\theprimaryclass
\ifx\thesecondaryclass\relax\else; \thesecondaryclass\fi\par
{\bf Keywords}\stdspace \thekeywords\par}\vglue 7truept

}   %%%% end of definition of \makeagttitle

\ifplaintex
%  get print centerpage:
\hoffset 14truemm
\voffset 31truemm
%  fonts for headline and footline
\font\phead=cmsl9 scaled 950
\font\pnum=cmbx10 scaled 913
\font\pfoot=cmsl9 scaled 950
%  headline and footline
\headline{\vbox to 0pt{\vskip -4.5mm\line{\small\phead\ifnum
\count0=\startpage ISSN 1472-2739 (on-line) 1472-2747 (printed)
\hfill {\pnum\folio}\else\ifodd\count0\def\\{ }% 
\ifx\theshorttitle\relax\thetitle\else\theshorttitle\fi\hfill{\pnum\folio}
\else\def\\{ and }{\pnum\folio}\hfill\ifx\theshortauthors\relax\theauthors
\else\theshortauthors\fi\fi\fi}\vss}}
\footline{\vbox to 0pt{\vglue 0mm\line{\small\pfoot\ifnum\count0=\startpage
\copyright\ \gtp\hfill\else
\agt, Volume \thevolumenumber\ (\thevolumeyear)\hfill\fi}\vss}}
\else
%  get print centerpage:
\headsep 23pt
\footskip 35pt
\hoffset -4truemm
\voffset 12.5truemm
%  fonts for headline and footline
\font\lhead=cmsl9 scaled 1050
\font\lnum=cmbx10 
\font\lfoot=cmsl9 scaled 1050
\makeatletter
%  headline and footline
\def\@oddhead{{\small\lhead\ifnum\count0=\startpage ISSN 1472-2739 
(on-line) 1472-2747 (printed)\hfill {\lnum\number\count0}\else\ifodd\count0
\def\\{ }\ifx\theshorttitle\relax \thetitle \else\theshorttitle\fi\hfill
{\lnum\number\count0}\else\def\\{ and }{\lnum\number\count0}
\hfill\ifx\theshortauthors\relax 
\theauthors\else\theshortauthors\fi\fi\fi}}\def\@evenhead{\@oddhead}
\def\@oddfoot{\small\lfoot\ifnum\count0=\startpage\copyright\ \gtp\hfill\else
\agt, Volume \thevolumenumber\ (\thevolumeyear)\hfill\fi}
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\makeatother
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%  force \makeagttitle
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\let\makeshorttitle\maketitlepage
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\endinput  %%%comment out to create xxx header file


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\immediate\closeout\gtoutfile}}  %%% end of definition of \makeheadfile


\def\maketitlepage{\makeagttitle\makeheadfile}
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\let\maketitle\maketitlepage

\endinput   
%%%
%%%  This is agtout.tex.  
%%%
%%%  This the version of  gtoutput.tex  intended to finish formatting
%%%  papers published in Algebriac & Geometric Topology and stored in the
%%%  arXiv.   All versions of  gtoutput.tex  are copyright 
%%%  GT Publications and are to be used _only_ for formatting
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%%%
%%%
%%%                                             Colin Rourke  27.102000
%%%
%%%  To create header file  head.xxx  comment out the first \endinput

%  test for latex or plain tex
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%  define the various new ingredients of the title page and the data
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\def\lognumber#1{\def\thelognumber{#1}}
\def\volumenumber#1{\def\thevolumenumber{#1}}
\def\volumeyear#1{\def\thevolumeyear{#1}}
\def\papernumber#1{\def\thepapernumber{#1}}
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\def\published#1{\def\publishdate{#1}}
\def\proposed#1{\def\theproposer{#1}}
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\def\asciiaddress#1{\def\theasciiaddress{#1}}
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\def\isbn#1{\def\theisbn{#1}}
\def\erratum#1{\def\theerratum{non-empty}\def\erratumnumber{#1}}

%  initialise

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\hfill {\pnum\folio}\else\ifodd\count0\def\\{ }% 
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\def\\{ }\ifx\theshorttitle\relax \thetitle \else\theshorttitle\fi\hfill
{\lnum\number\count0}\else\def\\{ and }{\lnum\number\count0}
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\def\@oddfoot{\small\lfoot\ifnum\count0=\startpage\copyright\ \gtp\hfill\else
\agt, Volume \thevolumenumber\ (\thevolumeyear)\hfill\fi}
\def\@evenfoot{\@oddfoot}
\makeatother
\fi
%  force \makeagttitle
\let\maketitlepage\makeagttitle
\let\makeshorttitle\maketitlepage
\let\maketitle\maketitlepage

\endinput  %%%comment out to create xxx header file


\newwrite\gtoutfile
\long\gdef\makeheadfile{  %%% start of definition of \makeheadfile
{\def\\{, }\def\s{ }
\immediate\openout\gtoutfile head.xxx
\immediate\write\gtoutfile{To: math@arxiv.org}
\immediate\write\gtoutfile{Subject: put OR rep NNNNN:ppppp}
\immediate\write\gtoutfile{--text follows this line--}
\immediate\write\gtoutfile{Proxy-for: \ifx\theasciiauthors\relax
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\immediate\write\gtoutfile{Subj-class: GT or SG, GR etc}
\immediate\write\gtoutfile{MSC-class: \theprimaryclass\ifx\thesecondaryclass\relax\else, \thesecondaryclass\fi}
\immediate\write\gtoutfile{Journal-ref: Algebr. Geom. Topol. \thevolumenumber\s
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\immediate\write\gtoutfile{Comments: Published by Algebraic and
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\immediate\write\gtoutfile{\s\s\s  http://www.maths.warwick.ac.uk/agt/AGTVol\thevolumenumber/agt-\thevolumenumber-\thepapernumber.abs.html}
\immediate\write\gtoutfile{\noexpand\\}
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\immediate\write\gtoutfile{\theabstract}\else
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\immediate\write\gtoutfile{}
\immediate\write\gtoutfile{\noexpand\\}
\immediate\write\gtoutfile{}
\immediate\closeout\gtoutfile}}  %%% end of definition of \makeheadfile


\def\maketitlepage{\makeagttitle\makeheadfile}
\let\makeshorttitle\maketitlepage
\let\maketitle\maketitlepage

\endinput   

math0201287.extracted_bib
\begin{thebibliography}


\bibitem{Fokkinkoversteegen} {\bf R\,J Fokkink}, {\bf L\,G Oversteegen}, 
{\it Homogeneous weak solenoids}, preprint, 2001

\bibitem{Kawamura} {\bf K Kawamura}, {\it On a construction of
homogeneous,
non-bihomogeneous continua of P. Minc}, Topology Proc. 19 (1994)
121--127

\bibitem{Keesling} {\bf J Keesling}, {\it The group of homeomorphisms of
a
solenoid}, Trans. Amer. Math. Soc. 172 (1972) 119--131

\bibitem{GKuperberg} {\bf G Kuperberg}, {\it Another homogeneous,
non-bihomogeneous Peano continuum}, Bull. Polish Acad. Sci. Math. 44
(1996) 457--460

\bibitem{Kuperberg} {\bf K Kuperberg}, {\it On the bihomogeneity problem
of
Knaster}, Trans. Amer. Math. Soc. 321 (1990) 129--143

\bibitem{K2} {\bf K Kuperberg}, {\it Bihomogeneity and Menger manifolds}, 
Topology Appl. 84 (1998) 175--184

\bibitem{McCord} {\bf M\,C McCord}, {\it Inverse limit sequences with
covering
maps}, Trans. Amer. Math. Soc. 114 (1965) 197--209

\bibitem{Minc} {\bf P Minc}, {\it Solenoids and bihomogeneity}, Continua,
Lecture Notes in Pure and Appl. Math. 170, Marcel Dekker (1995)
297--304

\bibitem{Nielsen} {\bf J Nielsen}, {\it Untersuchungen zur Topologie der
geschlossenen zweiseitigen Fl\"{a}ch\-en I}, Acta Math. 50 (1927)
189--358

\bibitem{Rogers2} {\bf J\,T Rogers Jr}, {\it Homogeneous hereditarily
indecomposable continua are tree-like}, Houston J. Math. 8 (1982)
421--428

\bibitem{RT1} {\bf J\,T Rogers Jr}, {\bf J\,L Tolefson}, {\it Maps
between
weak solenoidal spaces,} Colloquium Math. 23 (1971) 245--249

\bibitem{RT2} {\bf J\,T Rogers Jr}, {\bf J\,L Tolefson}, {\it 
Homeomorphisms homotopic to induced homeomorphisms of weak solenoids}, 
Colloquium Math. 25 (1972) 81--87

\bibitem{Scott}
{\bf G\,P Scott},
{\it Subgroups of surface groups are almost geometric},
J. London Math. Soc. (2) 17 (1978) 555--565.

\bibitem{Steenrod} {\bf Norman Steenrod}, {\it The topology of fibre bundels},
Princeton (1951)

\bibitem{Ungar} {\bf G\,S Ungar}, {\it On all kinds of homogeneous
spaces},
Trans. Amer. Math. Soc. 212 (1975) 393--400

\bibitem{Waldhausen}
{\bf F Waldhausen},
{\it  On irreducible $3$-manifolds which are sufficiently large},
Ann. of Math. (2) 87 (1968) 56--88
\end{thebibliography}