The surjectivity problem for one-generator,\\ one-relator extensions of torsion-free groups
Marshall M Cohen
Colin Rourke
% We use Klyachko's methods to prove that the natural map G to G-hat, where G is a torsion-free group and G-hat is obtained by adding a new generator t and a new relator w, is surjective only if w is conjugate to gt or gt^{-1} for some g in G. This solves a special case of the surjectivity problem for group extensions, raised by Cohen [Whitehead torsion, group extensions, and Zeeman's conjecture in high dimensions, Topology, 16 (1977) 79--88].
2001-5.tex
%%%%%%%%%%%%%%   Geometry and Topology: 2001-5.tex  %%%%%%%%%%%%%
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%%%%     The surjectivity problem for one-generator, one-relator ....
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%%%%              Marshall M Cohen and Colin Rourke  
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%%%%                Published in Volume 1(2001) pages 127-142
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%%%%                   Publication date 6 March 2001
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%%%%                     This is a plain TeX file
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\input gtmacros
\input pictex
\input rlepsf

\input gtoutput

\lognumber{136}
\volumenumber{5}\papernumber{5}\volumeyear{2001}
\pagenumbers{127}{142}
\accepted{6 March 2001}
\proposed{Cameron Gordon}
\seconded{Joan Birman, Wolfgang Metzler}
\received{12 September 2000}
\published{6 March 2001}

%%% abbreviations etc

\def\gen#1{\langle #1 \rangle}
\def\ngen#1{\langle \langle #1 \rangle \rangle}
\let\la\langle
\let\ra\rangle
\let\wat\widehat
\let\bar\overline
\font\spec=cmtex10 scaled 1095 
\def\d{\hbox{\spec \char'017\kern 0.05em}} 
\def\a{\longrightarrow}
\def\implies{\Longrightarrow}
\def\ex{\mathop{\rm ex}}
\def\inv{^{-1}}
\def\co{\mskip 1.5mu\colon\thinspace}
\def\H{{\cal H}}
\def\o{{\cal O}}
\def\Z{\Bbb Z}




\reflist

\key{Co} 
{\bf M\,M Cohen}, {\it A Course in Simple-Homotopy Theory},
Graduate Texts in Mathenmatics 10, Springer--Verlag, New York (1973)

\key{Coh}
{\bf M\,M Cohen}, {\it Whitehead torsion, group extensions, and
Zeeman's conjecture in high dimensions}, Topology, 16 (1977) 79--88

\key{CL}
{\bf Marshall M Cohen}, {\bf Martin Lustig}, {\it The conjugacy
problem for Dehn twist automorphisms of free groups}, Comment.
Math. Helv. 74 (1999) 179--200

\key{FR}
{\bf Roger Fenn}, {\bf Colin Rourke}, {\it Klyachko's methods and the
solution of equations over torsion-free groups}, L'Enseignment
Math\'ematique, 42 (1996) 49--74

\key{FRo}
{\bf Roger Fenn}, {\bf Colin Rourke}, {\it Characterisation of a class
of equations with solutions over torsion-free groups}, from: ``The
Epstein Birthday Schrift'', (I Rivin, C Rourke and C Series, editors),
Geometry and Topology Monographs, Volume 1 (1998) 163--17

\key{Me}
{\bf W Metzler}, {\it \"Uber den Homotopietyp zweidimensionaler
$CW$--Komplexe und Elementartransformationen bei Darstellungen von
Gruppen durch Erzeugende und definierende Relationen}, J. Reine
Angew. Math. 285 (1976) 7--23

\key{Ke}
{\bf M Kervaire}, {\it On higher dimensional knots}, from: ``Differential
and combinatorial top\-ology -- a symposium in honour of Marston Morse'',
Princeton Math. Series, 27 (1965)

\key{Kl}
{\bf A Klyachko}, {\it Funny property of sphere and equations over
groups}, Comm. in Alg. 21 (1993) 2555--2575

\key{Rot}
{\bf O\,S Rothaus}, {\it On the non-trivialty of some group extensions
given by generators and relations}, Ann. of Math. 106 (1977) 599--612

\key{RS}
{\bf C\,P Rourke}, {\bf B\,J Sanderson}, {\it Introduction to
piecewise-linear topology}, Sprin\-ger study edition, Springer--Verlag,
Berlin (1982)



\endreflist


\title{The surjectivity problem for one-generator,\\
one-relator extensions of torsion-free groups}
\shorttitle{Surjectivity problem for extensions of torsion-free groups}
\authors{Marshall M Cohen\\Colin Rourke}
\asciiaddress{Cornell University, Ithaca NY, 14853-4102, 
USA\\Warwick University,
Coventry, CV4 7AL, UK}
\address{Cornell University, Ithaca NY, 14853-4102, 
USA\\{\rm and}\\Warwick University,
Coventry, CV4 7AL, UK}
\email{marshall@math.cornell.edu\\cpr@maths.warwick.ac.uk}

\abstract
We prove that the natural map $G\to\wat G$, where $G$ is a
torsion-free group and $\wat G$ is obtained by adding a new generator
$t$ and a new relator $w$, is surjective only if $w$ is conjugate to
$gt$ or $gt\inv$ where $g\in G$.  This solves a special case of the
surjectivity problem for group extensions, raised by Cohen [\Coh].
\endabstract

\asciiabstract{%
We use Klyachko's methods to prove that the natural map G to G-hat,
where G is a torsion-free group and G-hat is obtained by adding a new
generator t and a new relator w, is surjective only if w is conjugate
to gt or gt^{-1} for some g in G.  This solves a special case of the
surjectivity problem for group extensions, raised by Cohen [Whitehead
torsion, group extensions, and Zeeman's conjecture in high dimensions,
Topology, 16 (1977) 79--88].}


\primaryclass{20E22, 20F05}\secondaryclass{57M20, 57Q10}
\keywords{Surjectivity problem, torsion-free groups, Whitehead torsion,\break
Kervaire conjecture}
\asciikeywords{Surjectivity problem, torsion-free groups, Whitehead torsion,
Kervaire conjecture}

\maketitle



\section{Introduction}

In this paper we prove the following theorem.

\proclaim{Main Theorem}
Suppose that $G$ is a torsion-free group and that $\gen{t}$ is an
infinite cyclic group with generator t. Let $w$ be an
element of the free product  $G*\gen{t} $  and let $\ngen{w}$
be the normal subgroup of $G*\gen{t}$ generated by w.  View $G$
as a subgroup of $G*\gen{t}$ and let $i$ be the inclusion $G \a G*\gen{t}$.
Consider the natural homomorphism
$$q = \pi i\co \ G \ {\buildrel i \over \a} \ G*\gen{t}\ \
{\buildrel\pi\over \a}\ \ \wat{G} = {G*\gen{t}\over \ngen{w}}.$$ If
$q$ is onto then $w$ is conjugate to $gt$ or $gt\inv$ for some $g \in
G$.
\endproc

There are standard ways in which this algebraic situation may be
realized topologically. These lead to the following results.

\proclaim{Corollary 1} 
Suppose that $L$ is a connected CW complex with torsion-free
fundamental group and that the CW complex ${\wat L} = L \cup e^1 \cup
e^2$ is constructed by attaching a $1$--cell to $L$ and a 2--cell to
$L \cup e^1$.  If the inclusion map $j\co L \a \wat L$ induces a
surjection $j_*\co \pi_1L \a \pi_1{\wat L}$ then $j$ is a
simple-homotopy equivalence. \endproc

\prf 
This follows from elementary facts about the invariance of Whitehead
torsion under homotopy of attaching maps [\Co; Section 5] and the fact
that if ${\wat L} = L \cup e^1 \cup e^2$, where $e^1$ is a circle and
$e^2$ is a $2$-cell attached by a word $gt$, then $e^1$ is a free face
of $\wat L$ and $\wat L$ collapses to $L$ by an elementary
collapse. \endprf


\proclaim{Corollary 2} 
Suppose that $M$ is a connected $n$--manifold with torsion free
fundamental group and that $(W, M, M')$ is an h--cobordism with exactly
one handle of index one and one handle of index two and no other
handles (or dually with exactly one $n$--handle and one $ (n -
1)$--handle). Then $(W, M, M')$ is an s--cobordism.\endproc

\prf 
This is a consequence of the fact that a $k$-handle $D^k \times D^{n +
1 - k}$ collapses to its core union its attaching tube,
$D^k\times\{0\} \cup \partial D^k \times D^{n + 1 - k}$, see eg [\RS;
Chapter 6]. So the CW theory applies to the handlebody theory.\endprf



\sh{Background}

The surjectivity problem for group extensions and the question of
which Whitehead torsions can be realized were formulated by Cohen
[\Coh] and Metzler [\Me]; for more details on these problems and the
relevance of our results, see Section 5.

It will be useful to note from the outset that the conclusion
of the main theorem may be restated according to the following
lemma.

\proclaim{Lemma 1}If\ $G  \subset\, G*\gen{t}\ \
{\buildrel\pi\over \a}\ \ G*\gen{t}/\,\ngen{W}$\
where $W$ is a set of words in $G*\gen{t}$ then
$q = \pi\,\vert\,G$ is onto $\iff$ \ $gt$
lies in the kernel of $\pi$ for some $g \in G$.
\endproc
\prf $q$ is onto $\iff [\pi(t) \in \pi(G)] \iff [\pi(t) = \pi(g\inv)
\ {\rm for\ some}\ g \in G] \iff [\pi(gt) = 1] \iff
[gt \in {\rm kernel}(\pi)$\ for  some $g \in G]$. \endprf


Any element $w \in G*\gen{t}$ has a unique expression as a {\sl
reduced word}, $w = g_0t^{q_1}g_1t^{q_2}\ldots g_{n-1}t^{q_n}g_n$,
where $g_i\in G$ are non-trivial for $0<i<n$ and $q_i$ are non-zero
integers for each $i$.
The word $w$ is {\sl cyclically reduced} if further $g_n=1$ and if
$n>1$ then $g_0\neq1$.  Up to cyclic permutation there is a unique
cyclically reduced word in the conjugacy class of $w$, see eg [\CL,
Proposition 3.9].  
Since $\wat G$ depends only the conjugacy class of $w$, there is no
loss in assuming that $w$ is cyclically reduced and we shall do so
without comment from now on.  We call $\Sigma_{i=1}^nq_i$ the {\sl
exponent sum} of $t$ in $w$, denoted $\ex(w)$. The unreduced word
$t^{q_1}t^{q_2}\ldots t^{q_n}$ is called the {\sl $t$--shape} of $w$
and, thinking of $w=1$ as an equation over $G$, we call the elements
$g_i$ the {\sl coefficients} of $w$.

It is easy to see that if $q\co G \a \wat{G}$ is surjective then
$\ex(w) = \pm1$, since otherwise the abelianization of $\wat{G}/(q(G)
= 1)$ will be non-trivial.  So, replacing $w$ by $w^{-1}$ if
necessary, we may assume in our discussion that $\ex(w) = 1$.  Under
this hypothesis Klyachko [\Kl], in 1993, gave a brilliant argument to
prove the following theorem, which implies the Kervaire conjecture
[\Ke] in the case where $G$ is torsion-free.

\proclaim{Theorem}{\rm(Klyachko)}\qua 
If $G$ is a torsion-free group and $w \in G*\gen{t}$ with $\ex(w) = 1$
then the natural homomorphism $q\co G \a \wat{G} = {{G*\gen{t}}\over
\ngen{w}}$ is injective.
\endproc

An exposition (and extension) of Klyachko's theorem was given by Fenn
and Rourke [\FR] in 1996.  To prove our theorem we will use Klyachko's
result and his method, following closely the exposition in [\FR].  We
will quote some definitions and results from [\FR] and give those
proofs in detail for which the arguments differ and for which (proving
the contrapositive) the hypothesis is used that $w$ is not conjugate
to $gt$ for any $g \in G$.

\sh{Outline of the paper}

In Section 2 we consider a group $\Gamma$ in a slightly more general
situation than $G$ above. We assume (contrary to our Main Theorem)
that $w$ is not conjugate to $gt$ for any $g\in \Gamma$ but that some
$gt$ is in the kernel of $\Gamma*\gen{t} \a \wat \Gamma$. We show how
to construct a certain non-trivial CW subdivision of the $2$--sphere,
with edges labelled $t^{\pm1}$ and all but one corner labelled by an
element of $\Gamma$.

In Section 3 we prove our main theorem in a special case: We denote
$g^t = t\inv g t$. If $w$ has the form $w = b_0 a_0^t b_1 a_1^t \ldots
b_r a_r^t ct$, where the $a_i, b_i$ and $c$ are all elements of $G$
and $b_0 a_0^t b_1 a_1^t \ldots b_r a_r^t c \notin G$ then $q\co G \a
\wat{G}$ is not onto.

In Section 4, we complete the proof of the main theorem.  We use an
algebraic trick to parlay the result of Section 4 into a proof that,
in general, if $\ex(w) = 1$ and $w$ is not conjugate to $gt$ for any
$g \in G$ then $q\co G \a \wat{G}$ is not onto.


In Section 5 we briefly discuss the general surjectivity problem, in
which $n$ generators and $n$ relators are added to a group $G$.  We
give a bit of history and comment on the relevance of our result for
$n = 1$ to the general problem.

Finally, in Section 6 we extend our result to prove that if $w\in
G*\gen t$ is a word not of the form $gt$ whose $t$--shape is amenable
(see [\FR, \FRo]) then no word with $t$--shape $t^n$ can be in the
kernel of $\pi\co G\to\wat G = {G*\gen t\over\ngen w}$. 

\section{The cell subdivision lemma}

In this section we prove the cell subdivision lemma (below) which is
modelled on [\FR; Lemma 3.2].

The lemma uses the idea of a {\sl corner} of a 2--cell in a cell
subdivision $K$ of the 2--sphere.  This can be regarded as the
(oriented) angle formed by the two adjacent edges meeting at a vertex
(0--cell) in the boundary of the 2--cell.  If all the corners of a
2--cell are labelled by elements of a group, then a word can be read
around the 2--cell boundary by composing these elements either
unchanged or inverted according as the orientation of the corner
agrees or dissagrees with that of the 2--cell boundary.  Similarly if
all the corners at a vertex are labelled then a word can be read
around that vertex.  We shall always orient corners {\it clockwise},
thus if the above words are read {\it clockwise} for vertices and {\it
anticlockwise} for 2--cells, then no inversion is necessary.  See
figure 1 for an example: the word read around the boundary is $abc$;
after insertion of $t$ or $t\inv$ at the arrows (see part (e) of the
lemma below) it reads $tat\inv bt\inv c$.

\figure
\beginpicture
\setplotarea x from -2 to 2 , y from -0.5 to 4
\setcoordinatesystem units < .8500cm, .85000cm> point at 0 0
\linethickness=.5pt
\small
\putrule from -2 0 to 2 0
\setlinear
\plot -2 0  0 2  2 0 /
\circulararc 90 degrees from -.1 1.9 center at 0 2
\circulararc 45 degrees from -1.8 0 center at -2 0
\circulararc -45 degrees from 1.8 0 center at 2 0
\put {$\phantom a$} <0pt,0pt> [t] at 0 2  %%% to define top of picture
\put {$a$} <0pt,-2pt> [t] at 0 1.8
\put {$b$} <1.5pt,0pt> [lb] at -1.7 0.1
\put {$c$} <-1.5pt,.5pt> [rb] at 1.7 0.1
\arrow <7pt> [.1,.5] from  -1 1 to -.9 1.1
\arrow <7pt> [.1,.5] from   1.1 .9 to 1 1
\arrow <7pt> [.1,.5] from .1 0 to -.1 0
\endpicture
\caption{Reading the boundary of a 2--cell: $tat\inv bt\inv c$}
\endfigure



Let $H$ be a subgroup of a group $\Gamma$ and let $g\in \Gamma$.  We
say that $g$ is {\sl free relative to} $H$ if the subgroup $\gen{g,H}$
of $\Gamma$ generated by $g$ and $H$ is naturally the free product
$\gen{g} * H$ of an infinite cyclic group $\gen g$ with $H$. (Note in
particular that $g$ has infinite order.)

If $g, h$ are elements of a group let $g^h$ denote $h^{-1}gh$.

In this section and the next, we shall consider the following
working hypotheses: 

\proclaim{Working hypotheses}

Suppose that $H$ and $H'$ are two isomorphic subgroups of a group
$\Gamma$ under the isomorphism $h\to h^{\phi}$, $h\in H$.  Suppose that
for each $i$, $a_i, b_i$ are elements of $\Gamma$ such that $a_i$ is
free relative to $H$ and $b_i$ is free relative to $H'$.  Let $c$ be
an arbitary element of $\Gamma$.

Let $w_0$ be the word $$b_0a_0^tb_1a_1^tb_2a_2^t\cdots b_ra_r^tct$$ in
$\Gamma*\gen t$, where $r\ge 0$, and and let $W\subset \Gamma*\gen t$
be the set of words $\{w_0,h^t(h^{\phi})\inv\mid h\in H\}$.  Let $\ngen W$
be the normal closure of $W$ in $\Gamma*\gen t$ and let $\wat\Gamma$
denote $(\Gamma*\gen{t})/\ngen W$.\endproc

\proclaim{Cell subdivision lemma}

Assume the working hypotheses, above.  Suppose that, for some $g\in
\Gamma$, $gt$ is in the kernel of the natural map $\Gamma*\gen
t\to\wat\Gamma$.

Then there is a cell subdivision $K$ of the 2--sphere such that

\items
\item{\rm(a)} the edges (1--cells) of $K$ are oriented,

\item{\rm(b)} the corners (all oriented clockwise) are labelled 
by coefficients of elements $w$ or $w\inv$ for $w\in W$, with the
exception of one particular corner at one particular vertex $v_0$
which is unlabelled,

\item{\rm(c)} the clockwise product of the corner labelling around any
vertex is $1\in\Gamma$ except for $v_0$ where it is undefined,

\item{\rm(d)} there is one special 2--cell $e^2_\infty$ which contains 
the unlabelled corner and has boundary a single edge and the single
vertex $v_0$,

\item{\rm(e)} with the exception of $e^2_\infty$, the corner labels of 
any 2--cell (in anticlockwise order) are the coefficients of $w$ or
$w\inv$ for some $w\in W$ (up to cyclic rotation) with the property
that, if on passing from one corner to an adjacent corner the element
$t$ or $t^{-1}$ is inserted according to whether the intervening edge
is oriented in the same or opposite direction (see figure 1), then the
whole of $w$ or $w\inv$ is recovered,

\item{\rm(f)} the cell decomposition is irreducible in the following
senses:\nl 
{\rm type (1)}\qua there do not exist two 2--cells with an edge in common
(necessarily read as $t$ in one and $t\inv$ in the other) such that,
starting with one vertex of this edge, the words read in these
2--cells are inverses of each other,\nl 
{\rm type (2)}\qua there does not exist a chain of 2--gons with common
vertices $a,b$ such that the product of the corner labels in
the chain at $a$ (or, equivalently, at $b$) is $1\in\Gamma$,

\item{\rm(g)} the cell subdivision is non-degenerate in that there
exist at least two vertices and at least three 2-cells; in particular
there is a cell $e^2_1 \neq e^2_\infty$ whose boundary contains
$\partial e^2_\infty$ as a {\it proper} subset (see figure 5).
\enditems\endproc

\prf  The proof uses transversality as in the proofs of [\FR; Lemmas 3.1 
and 3.2].

Choose a 2--complex $L$ with $\pi_1(L)=\Gamma$ and form the 2--complex
$\wat L$ with $\pi_1(\wat L)=\wat\Gamma$ by attaching a 1--cell
$\gamma$ to the base point $*$ of $L$ (corresponding to $t$) and a
2--cell $\sigma_w$ with attaching map determined by $w$ for each $w\in
W$.

Since $gt$ is trivial in $\pi_1(\wat L)$ there is a map of a 2--disc
$f\co D^2\to\wat L$ whose boundary maps to $L\cup\gamma$ and which
represents $gt\in\pi_1(L\cup\gamma)=\Gamma*\gen t$.  Make $f$
transverse to the centres of the 2--cells $\sigma_w$. It follows that
the inverse images of small neighbourhoods of these centres is a
collection of disjoint discs $D_1,\ldots,D_m$ in the interior of
$D^2$.  By a radial expansion of $f$ on these discs we may assume that
their image is the whole of one of the $\sigma_w$. It follows that the
punctured disc $P={\rm closure}\bigl(D^2- (D_1\cup\cdots\cup
D_m)\bigr)$ is mapped by $f$ to $L\cup\gamma$. Let $p$ be the centre
of $\gamma$. Make $f|P$ transverse to $p$. Then $f^{-1}p$ is a
1--manifold $Z$ properly embedded in $P$. By a radial expansion along
$\gamma$ we can assume that $Z$ has a neighbourhood $N$ which is a
normal $I$--bundle, where each fibre is mapped by $f$ to $\gamma$
and closure$(P - N)$ is mapped by $f$ to $L$.

We now simplify the subset $\H_f= D_1\cup\cdots\cup D_m\cup N$ of
$D^2$ as follows. Suppose $N$ contains an annulus component $\cal A$
in the interior of $P$. Let $D'$ denote the interior disc of $D^2$
which bounds the interior boundary component of the annulus. Then
$D''=D'\cup\cal A$ is a sub disc of $D^2$ whose boundary is mapped to
$*$ by $f$.  We can then redefine $f$ so that $f(D'')=*$ leaving $f$
unchanged outside $D''$.

At this point $\H_f$ can be regarded as a collection of 0--handles
(the $D_i$) and 1--handles (the components of $N$) attached to the
0--handles and to $\d D^2$ (in fact there is precisely one 1-handle
attached to $\d D^2$ by one end) see figure 2. 
\figure\relabelbox\small
\epsfxsize 2truein\epsfbox{handles.eps}
\adjustrelabel <-4pt, 0pt> {a}{$\alpha$}
\adjustrelabel <-4pt, 0pt>  {dd2}{$\d D^2$}
\endrelabelbox
\caption {A view of $\H_f$}
\endfigure
We now prove that we
may assume that $\H_f$ is connected.  Suppose not.  Choose an
innermost component $C$.  Draw a simple loop $\alpha$ around $C$
separating it from the rest of $\H_f$.  Up to conjugacy $\alpha$
represents an element of $\pi_1(L)=\Gamma$ which is trivial in
$\pi_1(\wat L)=\wat\Gamma$.  But Klyachko proves that $\Gamma$ injects
in $\wat\Gamma$ (this is the precise content of [\FR; Theorem 4.1,
page 62]) and hence we may redefine $f$ so that the inside of $\alpha$
is mapped to $L$, which simplifies $\H_f$.

Note that the 0--handles can be labelled by elements $w$ or $w\inv$
for $w\in W$ according to the corresponding 2--cell of $\wat L$ and
orientation.   We say that
$\H_f$ is {\sl type (1) reducible} if there is a pair of 0--handles
labelled by $w$ and $w\inv$ (the same $w$) and joined by a 1-handle
which represents the same occurrence of $t$ (respectively $t\inv$) in
each word.  In this situation we can again simplify $\H_f$ without
changing $f|\d D^2$ by redefining $f$ near these 0--handles and
joining 1--handle (see figure 3).
\figure\relabelbox\small\let\ss\scriptstyle
\epsfxsize 4.5truein\epsfbox{cancel.eps}
\relabel {t}{$t$}
\relabel {a}{$\ss a$}
\relabel {b}{$\ss b$}
\relabel {c}{$\ss c$}
\relabel {d}{$\ss d$}
\relabel {w}{$\ss w$}
\relabel {a1}{$\ss a\inv$}
\relabel {b1}{$\ss b\inv$}
\relabel {c1}{$\ss c\inv$}
\relabel {d1}{$\ss d\inv$}
\relabel {w1}{$\ss w\inv$}
\endrelabelbox
\caption {Type (1) reduction of $\H_f$}
\endfigure
We say that $\H_f$ is {\sl type (2) reducible} if there is a chain of
0--handles (each having two 1--handles attached to it) labelled by
words $h_i^t(h_i^{\phi})\inv$, $i=1,2\ldots,q$ with $h_1h_2\ldots
h_q=1$ in $\Gamma$ (figure 4).
\figure\relabelbox\small\let\ss\scriptstyle
\epsfxsize 4.5truein\epsfbox{beads.eps}
\relabel {a}{$\alpha$}
\relabel {h1}{$\ss h_1$}
\relabel {h2}{$\ss h_2$}
\relabel {h3}{$\ss h_3$}
\relabel {h11}{$\ss (h_1^\phi)\inv$}
\relabel {h22}{$\ss (h_2^\phi)\inv$}
\relabel {h33}{$\ss (h_3^\phi)\inv$}
\relabel {hhh}{$h_1h_2h_3=1$}
\endrelabelbox
\caption {Type (2) reduction of $\H_f$}
\endfigure
If the chain forms a loop, the handlebody
is not connected and this chain and everything inside it may be
eliminated as indicated earlier.
Otherwise, the curve $\alpha$ indicated in figure 4 maps to
$tt\inv$ in $\Gamma*\gen t$ and there is another simplification given
by omitting this chain of 0--handles and redefining $f$ inside
$\alpha$ using the null-homotopy of $tt\inv$ in
$L\cup \gamma$. After these
simplifications there may now be more simplifications of the first two
types which can be performed.  Repeat all four until no more are
possible. Thus we can assume that $\H_f$ is connected and irreducible.

We now extend $\H_f$ to a handle decomposition $\H$ of $S^2\supset
D^2$ by letting the outside of $D^2$ be one 0--handle (denoted
$h^2_\infty$) and the regions of $D^2-\H_f$ be the 2--handles.

The required 2--complex $K$ is the dual complex to $\H$ obtained by
putting a vertex inside each 2--handle and joining by an edge across
each 1--handle.  The outside 2--cell is $e^2_\infty$ (containing
$h^2_\infty$) and has boundary containing a single vertex $v_0$.
Corners of 2--cells other than this corner are labelled by the
coefficient of the word $w$ or $w\inv$ labelling the 0--handle inside
the 2--cell opposite the corner.  See figure 5.

\figure\relabelbox\small\let\ss\scriptstyle
\epsfxsize 4truein\epsfbox{cells.eps}
\relabel {v0}{$v_0$}
\relabel {e21}{$e^2_1$}
\relabel {e2i}{$e^2_\infty$}
\relabel {t}{$t$}
\relabel {a}{$\ss a$}
\relabel {b}{$\ss b$}
\relabel {c}{$\ss c$}
\relabel {a1}{$\ss a$}
\relabel {b1}{$\ss b$}
\relabel {c1}{$\ss c$}
\endrelabelbox
\caption {The cell complex $K$ (shown dashed)}
\endfigure

The required properties of $K$ all follow from the construction:
1--cells are oriented by the orientations of the $I$--bundles
(1--handles) that they cross and properties (a) to (e) follow at once
(the word read around the boundary of a 2-cell is the label on the
contained 0--handle).  Property (f) follows from the irreducibility of
$\H_f$. 

Finally, property (g) uses the hypothesis that $r \geq 0$ (ie, that
$w_0$ is not conjugate to $gt$ for any $g \in \Gamma$). In order that
every $1$-handle of $\H$ have each end on some $\partial D^2_i$,
except that one of them has one end on $\partial D^2$, at least one of
the $0$-handles $D^2_i$ must be a $w_0$ or $w_0\inv$ handle. Because
$r\geq 0$, this handle must have at least three $1$-handles emanating
from it. Thus there have to be at least two $0$-handles inside $D^2$,
so that $K$ has at least three $2$-cells.  Since the handlebody closes
up, $D^2 - \H_f$ must have at least two components, resulting in at
least two vertices in $K$.
\endprf


\section{The key technical theorem}

In this section we prove the following result whose proof is modeled
on that of [\FR; Theorem 4.1].  We show that the hypotheses of the
cell subdivision lemma are self-contradictory.

\proclaim{Key Technical Theorem}

Assume the working hypotheses.  Then $gt$ is never in the kernel of
the natural map $\Gamma*\gen t\to\wat\Gamma$ for any $g\in \Gamma$.
\endproc

\rk{Remark}
Assuming this theorem, note that by Lemma 1 in the Introduction,
$\Gamma\to\wat\Gamma$ is not surjective. Therefore by taking $H$ and
$H'$ to be trivial, we can now deduce a special case of our main
theorem:
\smallskip
{\sl If the $t$--shape of $w = w_0$ is not $t$ (ie,
w is not conjugate to $gt$ for any $g \in \Gamma$) but is of the form
$t\inv tt\inv\ldots tt\inv tt$ then $q\co\Gamma\to\wat\Gamma$ 
is not surjective.}
\smallskip
In the next section we introduce an algebraic trick which will enable
us to deduce the general case, where $\ex(w)=1$ and $w$ is not
conjugate to $gt$ for any $g \in G$, from this special case.


\prf The proof relies heavily on the proof and terminology of
[\FR; Theorem 4.1, pages 62--64].
Assume that $gt$ is in the kernel of $\Gamma*\gen t\to\wat\Gamma$ where
$g\in\Gamma$.  By the cell subdivision lemma there is a cell subdivision
of $S^2$ with all 2-cells of the four types $I, I', II, II'$ illustrated
in Figure 6 with the exception of the special 2--cell $e^2_\infty$.  

\figure
\beginpicture
\setplotarea x from -2 to 5 , y from -4 to 4
\setcoordinatesystem units < 1.25000cm, 1.25000cm> point at 0 0
\linethickness=.5pt
\small
\putrule from -2 0 to 2 0
\put {$[2r+2,4r+1]$} [B] at 0 2.1  
\put {$0$} [t] at -2 -.1
\put {$1$} [t] at -1 -.1
\put {$2r$} [t] at 1 -.1
\put {$2r+1$} [t] at 2 -.1
\put {$c$} [t] at 0 1.8
\put {$b_0$} [lb] at -1.7 0.1
\put {$a_0$} [B] at -1 0.15
\put {$\ldots$} [B] at 0 0.15
\put {$a_r$} [rb] at 1.7 0.1
\put {$b_r$} [B] at 1 0.15 
\put {$I$} [] at 0 1
\setlinear
\plot -2 0  0 2  2 0 /
\circulararc 45 degrees from -1.8 0 center at -2 0
\circulararc -45 degrees from 1.8 0 center at 2 0
\circulararc 90 degrees from -.1 1.9 center at 0 2
\circulararc -180 degrees from -1.1 0 center at -1 0
\circulararc 180 degrees from 1.1 0 center at 1 0
\arrow <5pt> [.1,.5] from -1.5 0 to -1.6 0
\arrow <5pt> [.1,.5] from  -.5 0 to -.3 0
\arrow <5pt> [.1,.5] from  1.5 0 to 1.3 0
\arrow <5pt> [.1,.5] from  .4 0 to .6 0
\arrow <5pt> [.1,.5] from  -.9 1.1 to -1 1
\arrow <5pt> [.1,.5] from   1.1 .9 to 1 1
\setcoordinatesystem units < 1.3000cm, 1.3000cm> point at -5 -1
\circulararc -90 degrees from -1 0 center at 0 -1
\circulararc 90 degrees from -1 0 center at 0 1
\circulararc 90 degrees from .895 .095 center at 1 0
\circulararc 90 degrees from -.895 -.095 center at -1 0
\arrow <5pt> [.1,.5] from   0 .414 to -.141 0.407
\arrow <5pt> [.1,.5] from   0 -.414 to -.141 -0.407
\put {$II$} [] at 0 0
\put {$h^{-1}$} [] at .65 0 
\put {$h^{\phi}$} [] at -.7 0 
\put { } [] at 2 1
\setcoordinatesystem units < 1.25000cm, -1.25000cm> point at 0 -1
\putrule from -2 0 to 2 0
\put {$[1,2r]$} [t] at 0 2.1 
\put {$0$} [B] at -2 -.1
\put {$4r+1$} [B] at -1 -.1
\put {$2r+2$} [B] at 1 -.1
\put {$2r+1$} [B] at 2 -.1
\put {$c^{-1}$} [B] at 0 1.8
\put {$b_0^{-1}$} [lt] at -1.7 0.1
\put {$a_0^{-1}$} [t] at -1 0.15
\put {$\ldots$} [t] at 0 0.15
\put {$a_r^{-1}$} [rt] at 1.7 0.1
\put {$b_r^{-1}$} [t] at 1 0.15 
\put {$I'$} [] at 0 1
\setlinear
\plot -2 0  0 2  2 0 /
\circulararc -45 degrees from -1.8 0 center at -2 0
\circulararc 45 degrees from 1.8 0 center at 2 0
\circulararc -90 degrees from -.1 1.9 center at 0 2
\circulararc 180 degrees from -1.1 0 center at -1 0
\circulararc -180 degrees from 1.1 0 center at 1 0
\arrow <5pt> [.1,.5] from -1.5 0 to -1.6 0
\arrow <5pt> [.1,.5] from  -.5 0 to -.3 0
\arrow <5pt> [.1,.5] from  1.5 0 to 1.3 0
\arrow <5pt> [.1,.5] from  .4 0 to .6 0
\arrow <5pt> [.1,.5] from  -.9 1.1 to -1 1
\arrow <5pt> [.1,.5] from   1.1 .9 to 1 1
\setcoordinatesystem units < 1.3000cm, -1.3000cm> point at -5 -2
\circulararc 90 degrees from -1 0 center at 0 -1
\circulararc -90 degrees from -1 0 center at 0 1
\circulararc -90 degrees from .895 .095 center at 1 0
\circulararc -90 degrees from -.895 -.095 center at -1 0
\arrow <5pt> [.1,.5] from   0 .414 to -.141 0.407
\arrow <5pt> [.1,.5] from   0 -.414 to -.141 -0.407
\put {$II'$} <5pt,0pt> [] at 0 0
\put {$h^{}$} [] at .65 0 
\put {$({h^{\phi}})^{-1}$} <1pt,0pt> [] at -.5 0 
\put { } [] at 2 1
\endpicture
\caption{The 2-cells $I,I',II$ and $II'$}
\endfigure

As in [\FR] we give the two-sphere an orientation (``anticlockwise")
and give each $2$--cell of $K$ the induced orientation. A  traffic flow is
now defined, with a car running around the boundary of each $2$--cell in the
direction of the induced orientation  as follows:

At time 0 let a car on
the boundary of a country of type $I$ or $I'$ start at the corner labelled
$b_0$ or $b_0^{-1}$ and proceed in an anticlockwise manner with respect
to the orientation of the edge along which it is travelling, moving from
corner to corner in unit time except at the corner labelled $c$ or
$c^{-1}$ where it stops for $2r-1$ units. The times when the car is at
each corner are illustrated in figure 6. For countries of type $II$ or
$II'$ the car starts at the corner labelled $h^\phi$ or $({h^{\phi}})^{-1}$ 
and proceeds
in an anticlockwise manner moving from corner to corner in unit time.


For $e^2_\infty$ we need to consider also the 2--cell $e^2_1$ whose boundary
properly contains the boundary of $e^2_\infty$, see figure 5.  Let
$A$ be the car on the boundary of $e^2_\infty$ and $B$ the car on the
boundary of $e^2_1$.  Choose a point $\omega$ on $\d e^2_\infty$
different from $v_0$.  We shall engineer crashes between $A$ and $B$
to occur precisely at $\omega$.  Suppose that $B$ is approaching
$\omega$ then let $A$ approach $\omega$ from the opposite direction to
crash at $\omega$.  After the crash, let $A$ dawdle near $\omega$
until $B$ moves off $\d e^2_\infty$ (which it must, because $\d e^2_1$
{\it properly} contains $\d e^2_\infty$);  then let $A$ speed round to
just before $\omega$ where it again dawdles until $B$ again approaches
$\omega$ at which point the cycle repeats.

Recall from [\FR] that a {\it complete car crash} is said to occur when
two cars meet in the interior of an edge (necessarily going in opposite 
directions) or when a $N$ cars from $N$ neighbouring countries all meet
at a vertex of valency $N$.

Notice that, on $\d e^2_\infty$, the given flow has the property that
complete crashes occur at $\omega$ and nowhere else; in particular, no
complete crash occurs at $v_0$.  However there must be another
complete crash occurring at some other vertex of $K$. (This is for
exactly the same reasons as in the proof of [\FR; Theorem
4.1]. Properies 1 to 4 on pages 63--64 hold here also and the flow
satisfies the conditions of the Crash Theorem with Stops [\FR, Theorem
2.3, page 56].  So there must be another complete crash and, as in
[\FR], this must occur at a vertex.)

This leads to the identical contradiction as on [\FR; page 64]: The
flow has been chosen so that, at a vertex where all the cars come
together at the same time, the
labels around the corners are all $\{a,\, a\inv\}$ for some
coefficient $a=a_i$ of $w_0$ together with elements of $H$ or $\{b,\,
b\inv\}$ for some coefficient $b=b_i$ of $w_0$ together with elements
of $H'$.  For definiteness assume that we are in the former situation.
Then we can read an (unreduced) word of the form $a^\epsilon h_1 h_2
\dots h_{i_1} a^\epsilon h_1 h_2 \dots h_{i_2}a^\epsilon\dots$ which 
is 1 in $\Gamma$.  Now if this word contains a subword of the form
$a^\epsilon a^{-\epsilon}$ then $K$ is type (1) reducible and if it
contains a subword of the form $h_1 h_2 \dots h_i$ which is 1 in
$\Gamma$ then $K$ is type (2) reducible.  Since $K$ is irreducible
neither of these happen and the word either gives a non-trivial
relation in $\gen{a,H}$ contradicting the assumption that $a$ is free
relative to $H$ or reads $(a^\epsilon)^N = 1$ for $N\ge 1$ which also
contradicts the assumption that $a$ is free relative to $H$ (and in
particular has infinite order).
\endprf



\section{Proof of the main theorem}

In the light of the discussion in the Introduction and Lemma 1, we assume
that $\ex(w) = 1$ and, proving the contrapositive of the Main
Theorem, we assume
that $w$ is not conjugate to $gt$ for any $g \in G$. We must
prove that  
$$gt\hbox{\qua is not in the kernel of\qua }\pi\co G*\gen t\a \wat G
\ {\rm for \ any}\ g \in G. \eqno{(*)}$$

We now use Klyachko's algebraic trick described on pages 64--66 of
[\FR].  

Consider the homomorphism $\ex\co G*\gen t \to \Z$.  It is well known
that $K$, the kernel of $\ex$, is a free product of copies of $G$
generated by elements of the form $g^{t^\o}=t^{-\o}gt^\o$, $1 \neq g\in
G$.

Any element of $K$ has a canonical expression of the form
$k=g_1^{t^{\o_1}}\cdots g_r^{t^{\o_r}}$, where $\o_i\ne\o_{i+1}$ for
each $i$.   We shall call the $g_i^{t^{\o_i}}$ the {\sl canonical
elements} of $k$.  Let min$(k)$ be the minimum value of $\o_i$,
$i=1,\ldots,r$ and  max$(k)$ the maximum value. Fix a positive integer
$m$. Consider the following subgroups of $K$:
$$\eqalign{%
H&=\la k\in K\mid \,\hbox{\rm min}(k)\ge0, \hbox{\rm max}(k)\le m-2\ra  \cr
H'&=\la k\in K\mid \,\hbox{\rm min}(k)\ge1, \hbox{\rm max}(k)\le m-1\ra \cr
J&=\la k\in K\mid \,\hbox{\rm min}(k)\ge0, \hbox{\rm max}(k)\le m-1\ra  \cr}$$
and the following subsets:
$$\eqalign{%
X&=\{ k\in K\mid \,\hbox{\rm min}(k)=0, \hbox{\rm max}(k)\le m-1\}  \cr
Y&=\{ k\in K\mid \,\hbox{\rm min}(k)\ge0, \hbox{\rm max}(k)=m-1\}  \cr
Z&=\{ k\in K\mid \,\hbox{\rm min}(k)\ge1, \hbox{\rm max}(k)=m\}  . \cr}$$

\proclaim{Lemma 2}{\rm [\FR; Lemma 4.2, page 65]}\qua
Let $w\in G*\la t\ra $ satisfy $\ex(w)=1$. Then, after conjugation,
$w$ can be written as a product
$$b_0a_0^tb_1a_1^t\cdots b_ra_r^tct,$$
where $a_i\in Y, b_i\in X,i=0,\ldots, r$ and $c\in J$ for some $m>0$.

Furthermore, provided $w$ is not conjugate to $gt$ for some
$g\in G$, then $r\ge 0$ in the expression.\endproc 

\rk{Remark}The final sentence in the statement of lemma 2 is
not given in [\FR], but is immediate from the proof given there.



\proclaim{Lemma 3}{\rm [\FR; Lemma 4.3, page 66]}\qua
Suppose that $G$ is torsion-free, then any element $a$ of $Y$
is free relative to $H$.  Similary any element $b$ of $X$ is free
relative to $H'$.\qed\endproc

We can now complete the proof of the main theorem, which follows
closely the proof of [\FR; Theorem 4.4, pages 66--67].  By lemma 2 we
can assume that $w=b_0a_0^tb_1a_1^t\cdots b_ra_r^tct$, where $a_i\in
Y$, $b_i\in X$, $i=0,\ldots, r$ and $c\in J$ and $r\ge 0$.  We need to
think of each $a_i, b_i, c$ as functions of $t$ and for clarity we
shall introduce a new variable $s$.  To be precise let
$$w(s,t)\equiv b_0(t)a_0^s(t)\cdots b_r(t)a_r^s(t)c(t)s$$
where $s$ and $t$ are independent variables.

Write $\Gamma$ for $G*\la t\ra $ and let $H$, $H'$ be the subgroups
defined above.  There is an isomorphism $\phi:H\to H'$ given
by $h^\phi=h^t, h\in H$.

Lemma 3 gives the hypothesis of our key
technical theorem in Section 4, which implies that 
$$\hbox{$gs$\qua is never in the kernel of\qua $\Gamma*\gen s \to \wat 
\Gamma$.}\eqno{(**)}$$
The case $m=1$ is the special case (with $t$--shape $t\inv
tt\inv\ldots tt\inv tt$) covered by the proof in the last section, so
we may assume that $m>1$ and then $G\subset H\ne \emptyset$.


Each of the canonical elements of $a_i(t), b_i(t), c(t)$ is either in
$G$ or lies in $H^t$; moreover in $\wat{\Gamma}$ we have
$h^s=h^\phi=h^t$ for each $h\in H$.

Since $H$ is generated by elements of the form $t^{-i}gt^i$ for $i\le
m-2$ and since $h^s=h^t$ for each $h\in H$ it follows by induction on
$i$ that we can freely exchange $s$ and $t$ in products of elements of
the form $t^{-i}gt^i$ for $i\le m-1$. Thus we can exchange $s$ and $t$
in the coefficients of $w(s,t)$ and it follows that $w(s,s)=1$ in
$\wat{\Gamma}$.


Now consider the following commutative diagram 
%
$$\matrix{\Gamma=G*\gen t&\subset&\Gamma*\gen s&\a&\wat\Gamma\cr 
\cup&&\cup&&\uparrow\cr
G&\a&G*\gen s&\a&\widehat G\cr}$$ 
%
where $\widehat G={G*\gen s\over\ngen w}$.

By $(**)$, $gs\in \Gamma*\gen s$ does not map to $1\in \wat \Gamma$ 
for any $g \in G$. Therefore\ $gs \in G*\gen s$\ never maps to $1\in \wat G$.
This proves $(*)$ as required. \endprf



\section{The surjectivity problem and Whitehead torsion} 

If one adds $n$ generators $x_1, x_2, \ldots x_n$ and $n$ relators
$w_1, w_2, \ldots w_n$ to a group $G$ to form the group $\wat G$ then
one can ask whether the natural homomorphism $G \a \wat{G}$ is
injective.  If it is injective then one can ask whether it is
surjective.  Our main theorem answers this question completely for
torsion-free groups when $n = 1$.

The question of surjectivity, assuming injectivity, was raised by Cohen
[\Coh] in his study of Zeeman's conjecture.
Assuming the natural homomorphism is injective then one can associate
(after some normalization -- see [\Rot, pages 600-601]) to the set of
 words $w_1, \ldots w_n$ 
a Whitehead torsion element
$\tau \in {\rm Wh}(G)$ (the Whitehead group of $G$). Cohen conjectured
that if $\tau \neq 0$ then the injection cannot be onto.  In closely
related but independent work, in which he investigated 
inclusions of one 2-complex into another which are homotopy
equivalences, Metzler [\Me]  investigated the group theoretic
combinatorics and the set of Whitehead torsion elements which are associated to
such homotopy equivelences.  He named the set of Whitehead torsion elements
which can be realized by a relative 2--complex as ${\rm Wh}^\ast(G)$

Our main theorem appears to give evidence that  ${\rm Wh}^\ast(G) = 0$.
In fact, when $n = 1$ not only do we show that a necessary condition
for surjectivity is that the torsion of the $1 \times 1$ matrix is $0$,
but we show that, up to homotopy of the attaching map, the added one-
and two- cells can be collapsed away. However, one must be very cautious
concerning what this means for $n > 1$ in that

\items
\itemb not all Whitehead torsion
elements can be realized by $1\,\times \,1$ matrices, hence our result for
$n = 1$ in no way answers the question of whether ${\rm Wh}^\ast(G) = 0$, 

\itemb it is possible (an open conjecture) that ${\rm Wh}(G) = 0$ for all 
torsion-free groups.
\enditems

The only significant results on the surjectivity problem which we know
of for $n > 1$ are those of Rothaus [\Rot]. He develops an obstruction to
the surjectivity of the map $G \a \wat{G}$ in terms of representations
of $G$ into compact connected Lie groups. His theory had the following
application for dihedral groups, whose Whitehead groups are known to be
non-trivial.

\proclaim{Theorem}{\rm [\Rot; Theorem 11]}\qua If ${\rm p} \geq 5$ is an 
odd prime and $G = D_{2p}$ is the dihedral group of order 2$p$ and $n$
is any positive integer then there exist non-trivial Whitehead torsion
elements such that every injective homomorphism 
$G \a \wat{G} = {G*\gen{t_1, \ldots t_n}\over w_1 \ldots w_n}$
realizing this Whitehead torsion element is non-surjective.\endproc

Beyond Rothaus' work, the surjectivity problem for $n > 1$ is an
open and fascinating question.


\section{An extension and a question}

The main theorem (in the equivalent form given by lemma 1) can be
extended:

\proclaim{Extension of Main Theorem}

Let $G$ be a torsion free group and let $w\in G*\gen t$ be word which
is not of the form $gt$ and whose $t$--shape is amenable (see [\FR,
\FRo]) and consider the natural map $\pi\co G\to\wat G = {G*\gen
t\over\ngen w}$.  Let $x\in G*\gen t$ be any word with $t$--shape
$t^n$ for some $n>0$.

Then $x$ is not in the kernel of $\pi$.\endproc

The proof is very similar to the proof of the main theorem.  The cell
$e^2_\infty$ has $n$ edges all oriented the same way (``uphill'').
Notice that any other cell with an edge in common with $e^2_\infty$
has its car traverse that edge in the ``downhill" direction, since
adjacent cells induce opposite orientations on a common edge.  Choose
any point $\omega\in\d e^2_\infty$ not at a vertex.  The flow
constructed as in [\FR; page 68] for cells other than $e^2_\infty$ has
the property that there are times when all cars are going uphill and
hence are not on $\d e^2_\infty$.  This leaves time for car $A$ (on
$\d e^2_\infty$) to rush round from just after $\omega$ to just before
and hence there are no complete crashes on $\d e^2_\infty$ except at
$\omega$.  This leads to the identical contradiction as in the proof
of the main theorem.

The extension implies that all words of $t$--shape $t^n$ for some $n$ have
infinite order in $\wat G$.  This leads to the natural question:

\rk{Question}Suppose that $G$ is torsion-free and that $w$ is an
amenable word.  Is $\wat G$ torsion-free?
\ppar

If the answer is yes, then we can deduce that $G\to\wat G$ is never
surjective when $\wat G$ is obtained from $G$ by adding $n$ generators
and $n$ relators {\it one pair at a time}.

\references
\bye











gtmacros.tex
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%             gtmacros.tex            %%%%%%%%%%%%%%%
%%%%%%%%%%%             version 1.6             %%%%%%%%%%%%%%% 
%
%                       Colin Rourke   
%
%
%    These macros are recommended for use by authors submitting articles   
%    to Geometry and Topology or to Algebraic and Geometric Topology.  
%    They are intended to be used with plain TeX. Each macro is described 
%    briefly to make it clear how to use it (or to modify it to achieve
%    different results).  If you modify this file then please change its
%    name.  If you modify this file and use the modified file to 
%    format an article for submission to Geometry and Topology or
%    Algebraic and Geometric Topology, then please paste the modified
%    file into your main TeX file.  Do not submit it as a separate file.
%      
%    Instructions on using these macros are also given in  gtmacins.tex  
%    or  gtmacins.ps  or .pdf  available on the gt www pages or by 
%    anonymous ftp from the gt/info/macros directory.
%
%
\magnification=\magstephalf      % Sets default point size to 11pt.
%
%  Basic layout parameters :
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\vsize=7.5truein                 % Sets text height to 7.5 inches.
\hsize=5.2truein                 % Sets text width to 5.2 inches.
%
\newskip\stdskip                 % standard vertical space
\stdskip=6pt plus3pt minus3pt    % (slightly more stretchy
\medskipamount=\stdskip          % than the usual \medskip)
%
\parindent=0pt                   % Paragraphs are non-indented with
\parskip=\stdskip                % a little space between paragraphs. 
\abovedisplayskip=\stdskip       %  Reduces the space
\belowdisplayskip=\stdskip       %  around displays.
\mathsurround=0.75pt             % Gives a little extra space around maths.
\overfullrule=0pt                %  Prevents black boxes
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%   The following macro is for principal paragraph breaks ie
%   a paragraph break with a slightly larger space :
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\def\ppar{\par\goodbreak\vskip 8pt plus 4pt minus 4pt}     
%
%  The standard horizontal space for theorems, labels etc :
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\def\stdspace{\hskip 0.75em plus 0.15em\ignorespaces}
\let\qua\stdspace % useful abbreviation (3/4 of a quad)
%
%%%%%%%%%%%%%%            FONT MACROS            %%%%%%%%%%%%
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%           The following font macros define the AMS symbol 
%           and Euler-Fraktal fonts for use in text and
%           mathematics with appropriate size changes.
%           They also define two new control sequences  
%           \small  and  \large  (similar to those built
%           into LaTeX) which change the size of all fonts 
%           both in text and maths.  \small  is 10% smaller 
%           than normal and  \large  30% bigger.  The strange
%           size of the \large text fonts (10pt scaled 1315)
%           is because these macros are intended to be used
%           at \magstephalf.  The result is 10pt scaled 1440
%           (\magstep2) which is a standard font size.  If
%           you are borrowing these macros to use them at
%           another basic  \magnification, then you will
%           probably need to change 1315 to 1200 in the eleven
%           places marked ** below.  \large  will then be
%           20% bigger than normal.  Note that at \magstephalf
%           all the fonts come out roughly one point larger
%           than their size as defined in these macros.
%
%           The size-changing macros are based on Knuth's
%           \ninepoint and \eightpoint macros.
%
%
%    The macros are laid out in a way which makes it clear how to
%    add futher fonts (or delete unavailable ones) and how to add
%    further size changes.
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%    First comes a definition of  \hexnumber  which is needed to
%    refer to font families whose family number is not known :
%
\def\hexnumber#1{\ifcase#1 0\or 1\or 2\or 3\or 4\or 5\or 6\or 7\or 8\or
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%
%     Next we define the AMS symbol-a fonts at 13,10,9,7,6,5 points
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\font\thirtnmsa=msam10 scaled 1315    %%% **  see note above 
\font\tenmsa=msam10          \font\ninemsa=msam9
\font\sevenmsa=msam7         \font\sixmsa=msam6
\font\fivemsa=msam5
%%%%%%  (add further sizes here if you need them)
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%    and the standard size family for these fonts
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\newfam\msafam                  \textfont\msafam=\tenmsa
\scriptfont\msafam=\sevenmsa    \scriptscriptfont\msafam=\fivemsa
\edef\hexa{\hexnumber\msafam}        %  The msa family is  \fam\hexa
\def\msa{\fam\msafam\tenmsa}         %  \msa  switches to this family
%
%    Repeat these steps for the AMS symbol-b fonts
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\font\thirtnmsb=msbm10 scaled 1315   %%%  ** see note above
\font\tenmsb=msbm10      \font\ninemsb=msbm9
\font\sevenmsb=msbm7     \font\sixmsb=msbm6
\font\fivemsb=msbm5
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\newfam\msbfam                   \textfont\msbfam=\tenmsb       
\scriptfont\msbfam=\sevenmsb     \scriptscriptfont\msbfam=\fivemsb
\edef\hexb{\hexnumber\msbfam}    %  The msb family is \fam\hexb  
\def\msb{\fam\msbfam\tenmsb}     %  \msb switches to this family
%
%        Repeat for the Euler-Fraktal fonts 
%
\font\thirtneufm=eufm10 scaled 1315   %%% **  see note above 
\font\teneufm=eufm10                 \font\nineeufm=eufm9
\font\seveneufm=eufm7                \font\sixeufm=eufm6
\font\fiveeufm=eufm5
%%%%%%  (add further sizes here if you need them)
%
\newfam\eufmfam                    \textfont\eufmfam=\teneufm
\scriptfont\eufmfam=\seveneufm     \scriptscriptfont\eufmfam=\fiveeufm
\edef\hexf{\hexnumber\eufmfam}      % The Euler-Fraktal family is
\def\frak{\fam\eufmfam\teneufm}     % \fam\hexf and \frak switches to this
%
%%%  Add further fonts families here (using the same format) if you need
%    them.  The def of hexnumber is optional (it is only used for
%    \mathchardef 's).
%
%      Now we need to define the standard fonts (which are
%      already defined at 10,7 and 5 point) at 13,9 and 6 point:
%
%      Roman fonts:
\font\thirtnrm=cmr10 scaled 1315    %%%  ** see note above
\font\ninerm=cmr9                   \font\sixrm=cmr6   
%%%%%%  (add further sizes here if you need them)
%
%      Math italic fonts
\font\thirtni=cmmi10 scaled 1315    %%%  ** see note above 
\font\ninei=cmmi9                   \font\sixi=cmmi6  
%%%%%%  (add further sizes here if you need them)
%
%     Symbol fonts
\font\thirtnsy=cmsy10 scaled 1315   %%%  ** see note above
\font\ninesy=cmsy9                  \font\sixsy=cmsy6  
%%%%%%  (add further sizes here if you need them)
%
%     Bold face
\font\thirtnbf=cmbx10 scaled 1315   %%%  ** see note above 
\font\ninebf=cmbx9                  \font\sixbf=cmbx6  
%%%%%%  (add further sizes here if you need them)
%
%     The maths extension font (only defined at text size)
%
\font\thirtnex=cmex10 scaled 1315   %%%  ** see note above
\font\nineex=cmex9                  
%%%%%%  (add further sizes here if you need them)
%
%     Finally three fonts (text italic, slanted and typewriter type)
%     which are also only defined at text size
%
\font\thirtnit=cmti10 scaled 1315  %%%  ** see note above 
\font\nineit=cmti9                  
%%%%%%  (add further sizes here if you need them)
%
\font\thirtnsl=cmsl10 scaled 1315  %%%  ** see note above 
\font\ninesl=cmsl9                  
%%%%%%  (add further sizes here if you need them)
%
\font\thirtntt=cmtt10 scaled 1315  %%%  ** see note above 
\font\ninett=cmtt9                  
%%%%%%  (add further sizes here if you need them)
%
%
%     Now come the two main macros.  What  \small  does is to
%     change all the families of fonts from normal size which is
%     10,7,5  (ie 10pt text, 7pt subscript, 5pt subsubscript)
%     to 9,6,5.  \large  similarly changes to  13,9,7.  To make
%     other size changing macros, choose your three sizes, add
%     font size definitions if necessary and make the obvious changes
%     to one of these macros.  Change  \normalbaselineskip  and
%     \strutbox  dimensions to appropriate sizes as well.  To
%     add further fonts, insert them in each macro, using the
%     AMS fonts as a model.
%      
%
\def\small{%
%
%   redefine the sizes of the roman fonts :
%
\textfont0=\ninerm \scriptfont0=\sixrm \scriptscriptfont0=\fiverm
\def\rm{\fam0\ninerm}%       % ( \rm  sets \ninerm  in text mode
%                            %  and \fam0 in math mode)
%
%   and the math italic fonts :
%
\textfont1=\ninei \scriptfont1=\sixi \scriptscriptfont1=\fivei
%
%   and the symbol fonts :
%
\textfont2=\ninesy \scriptfont2=\sixsy \scriptscriptfont2=\fivesy
%
%   There is only one math extension font :
%
\textfont3=\nineex \scriptfont3=\nineex \scriptscriptfont3=\nineex
%
%   Next the bold font (named rather than numbered) :
%
\textfont\bffam=\ninebf \scriptfont\bffam=\sixbf
\scriptscriptfont\bffam=\fivebf \def\bf{\fam\bffam\ninebf}%
%
%   and the three text-only fonts : 
%
\textfont\itfam=\nineit \def\it{\fam\itfam\nineit}%
\textfont\slfam=\ninesl \def\sl{\fam\slfam\ninesl}%
\textfont\ttfam=\ninett \def\tt{\fam\ttfam\ninett}%
%
%   Now the three new families of AMS fonts :
%
%   AMS symbol-a
%
\textfont\msafam=\ninemsa \scriptfont\msafam=\sixmsa
\scriptscriptfont\msafam=\fivemsa \def\msa{\fam\msafam\ninemsa}%         
%
%   AMS symbol-b
%
\textfont\msbfam=\ninemsb \scriptfont\msbfam=\sixmsb
\scriptscriptfont\msbfam=\fivemsb \def\msb{\fam\msbfam\ninemsb}%         
%
%   Euler-Fraktal font
%
\textfont\eufmfam=\nineeufm  \scriptfont\eufmfam=\sixeufm
\scriptscriptfont\eufmfam=\fiveeufm \def\frak{\fam\eufmfam\nineeufm}%
%
%%%  Add further fonts families here if you need them.
%
%    Reset \normalbaselineskip and \strubox :
%
\normalbaselineskip=11pt%
\setbox\strutbox=\hbox{\vrule height8pt depth3pt width0pt}%
%
%    Set \normalbaselines and \rm (roman) as defaults :
%
\normalbaselines\rm
%
%    Reset some of the basic vertical skips:
%
\stdskip=4pt plus2pt minus2pt    
\medskipamount=\stdskip          
\parskip=\stdskip                
\abovedisplayskip=\stdskip       
\belowdisplayskip=\stdskip       
\def\ppar{\par\goodbreak\vskip 6pt plus 3pt minus 3pt}%     
%
%   And finally reset the size of section heads (see below):
%
\def\section##1{\global\advance\sectionnumber by 1
\vskip-\lastskip\penalty-800\vskip 20pt plus10pt minus5pt 
\egroup{\bf\number\sectionnumber\quad##1}\bgroup\small         
\vskip 6pt plus3pt minus3pt
\nobreak\resultnumber=1}%      % Reset resultnumber at start of section
%
}    %%%   End of  \small  macro      
%
%   Two useful abbreviations to keep track of \small material:
\def\beginsmall{\bgroup\small}
\let\endsmall\egroup
%
%
%    The \large  macro is similar (comments abbreviated):
%
%
\def\large{%
\textfont0=\thirtnrm \scriptfont0=\ninerm \scriptscriptfont0=\sevenrm
\def\rm{\fam0\thirtnrm}%
\textfont1=\thirtni \scriptfont1=\ninei \scriptscriptfont1=\seveni
\textfont2=\thirtnsy \scriptfont2=\ninesy \scriptscriptfont2=\sevensy
\textfont3=\thirtnex \scriptfont3=\thirtnex \scriptscriptfont3=\thirtnex
\textfont\bffam=\thirtnbf \scriptfont\bffam=\ninebf
\scriptscriptfont\bffam=\sevenbf \def\bf{\fam\bffam\thirtnbf}%
\textfont\itfam=\thirtnit \def\it{\fam\itfam\thirtnit}%
\textfont\slfam=\thirtnsl \def\sl{\fam\slfam\thirtnsl}%
\textfont\ttfam=\thirtntt \def\tt{\fam\ttfam\thirtntt}%
%   AMS symbol-a  :
\textfont\msafam=\thirtnmsa \scriptfont\msafam=\ninemsa
\scriptscriptfont\msafam=\sevenmsa \def\msa{\fam\msafam\thirtnmsa}%         
%   AMS symbol-b  :
\textfont\msbfam=\thirtnmsb \scriptfont\msbfam=\ninemsb
\scriptscriptfont\msbfam=\sevenmsb \def\msb{\fam\msbfam\thirtnmsb}%         
%   Euler-Fraktal font :
\textfont\eufmfam=\thirtneufm  \scriptfont\eufmfam=\nineeufm
\scriptscriptfont\eufmfam=\seveneufm \def\frak{\fam\eufmfam\teneufm}%
%%%% Add further fonts families here if you need them.
%   Reset \normalbaselineskip and \strubox and initialise :
\normalbaselineskip=16pt%
\setbox\strutbox=\hbox{\vrule height11.5pt depth4.5pt width0pt}%
\normalbaselines\rm}%
%     
\let\Large\large   %  for compatibility with latex
%
%   The next two lines define commonly used switches for
%   blackboard bold (\Bbb) and gothic type (\goth).  The   
%   \Bbb  switch is set to work in the same way as in amstex
%   and switches only the next character to blackboard bold.
\def\Bbb#1{{\msb#1}}
\def\goth{\frak}
%
%   To use the new AMS fonts you can either use the control
%   sequences \msa, \msb (alias \Bbb) and \frak (alias \goth) eg :
\def\N{{\frak N}}
\def\re{\Bbb R}
\def\doublecup{{\msa \char'144}}   % see the msam font table
%
%   or, more generally, make \mathchardef's (cf Knuth p155) eg :
\mathchardef\plussquare="0\hexa01
\mathchardef\nge="3\hexb0B
\mathchardef\maltesecross="0\hexa7A
\mathchardef\del="0\hexf01
%
%   or you can use the amstex names for all the new symbols by
%   inserting the line  \input amsnames  in your file directly
%   after \input gtmacros. 
%   This presupposes that you have collected a copy of the file
%   amsnames.tex  from the  gt/info/macros  ftp directory.
%
%
%   Finally we need a small capital font (for author(s)) :
%
\font\sc=cmcsc10
%
%%%%%%%%%%%%%%%%%       END OF FONT MACROS     %%%%%%%%%%%%%
%
%
%                 Knuth's \square macro :
%
\def\sqr#1#2{{\vcenter{\vbox{\hrule  height.#2truept
	\hbox{\vrule width.#2truept height#1truept 
	\kern#1truept \vrule width.#2truept}
	\hrule height.#2truept}}}}
%
\def\sq{\sqr55}    %   A small square for end-of-proofs. 
%                  %   (Define other size squares by varing the
%                  %   the two numbers.)
%
%
%      Style macros for section heads, theorem statements etc :
%   
%
\newcount\sectionnumber            %%%  Allocate registers to take
\newcount\resultnumber             %%%  section and result numbers.
\sectionnumber=0\resultnumber=1    %%%  Set these registers to 0 and 1
%
%   The \section macro produces a \large bold faced section heading
%   numbered to the left.  Pagebreaks are encouraged before the
%   start of the section and discouraged directly after the heading.
%   Typical use  \section{First steps}  with typical result :
%
%    1  First Steps     (set bold and \large)
%
\def\section#1{\global\advance\sectionnumber by 1
\xdef\nextkey{\number\sectionnumber}%      (used by the \key macro)
\vskip-\lastskip\penalty-800\vskip 20pt plus10pt minus5pt 
{\large\bf\number\sectionnumber\quad#1}         
\vskip 8pt plus4pt minus4pt
\nobreak\resultnumber=1}      % Reset resultnumber at start of section
%
%
%
%   Next a macro to set subheadings (like the  \section  macro
%   but without the number, with less space and set in standard size).
%
%   Typical use :  \sh{Example formats}
%
\def\sh#1{\vskip-\lastskip\ppar{\bf #1}\par\nobreak\medskip}         
%
%   The \proc ... \endproc macros ("proclaim") are for setting theorems, 
%   lemmas, conjectures etc with automatic numbering.  Typical use :    
%  
%    \proc{Theorem}Every lemon is yellow.\endproc
%
%   Typical result :
%     
%    Theorem 3.4  Every lemon is yellow.   

%   (with Theorem 3.4 set bold and a \stdspace of space before the 
%   statement set in slanted type).
%
\def\proc#1{\xdef\nextkey{\number\sectionnumber.\number\resultnumber}%
\vskip-\lastskip\ppar\bf%
\noindent#1\ \number\sectionnumber.\number\resultnumber
\stdspace\sl\global\advance\resultnumber by 1\ignorespaces}
\def\endproc{\rm\ppar} 
%
%  The \prf ... \endprf macros are for setting proofs.  The code
%  for \prf includes the code for \endproc, so there is no need to
%  type \endproc if the theorem is followed immediatedly by a proof.
%
\def\prf{\vskip-\lastskip\ppar\noindent{\bf Proof}%
\stdspace\rm}                            %  For start of proofs  
\def\qed{\hfill$\sq$\par\goodbreak\rm}   %  For end (or absence) of proofs
\def\endprf{\unskip\stdspace\hbox{}%     %  For end of proof (with
\hfill$\sq$\par\medskip}                 %  extra vertical space)  
\def\proof#1{\vskip-\lastskip\ppar\noindent{\bf#1}%
\stdspace\rm\ignorespaces}        %  For start of proof with alternative name
\let\endproof\endprf              %  \endproof is an alias for \endprf
%
%   Typical uses :    
%  
%    \proc{Theorem}Every lemon is yellow. \qed\endproc
%
%    \proc{Theorem}Every lemon is yellow.
%    \prf Use your eyes. \endprf
%
%    \proc{Theorem}Every lemon is yellow.
%    \proof{Proof of theorem} Use your eyes. \endprf
%
%   The next macro is a variant of the \proc macro.  It has
%   exactly the same result except that it omits the number.
%
%   Typical use :  
%    
%    \proclaim{Conjecture}Some oranges are yellow.\endproc
%
\def\proclaim#1{\vskip-\lastskip\ppar\bf%
\noindent#1\stdspace\sl\ignorespaces} 
\let\endproclaim\endproc
%
%   The next macro is a further variant for remarks, definitions etc.   
%   It omits the number and does not switch on slanted type.  
%  
%   Typical use :
%
%    \rk{Remark}Some lemons are thick-skinned.\endrk
%
\def\rk#1{\vskip-\lastskip\ppar{\bf #1}\stdspace\ignorespaces}                
\def\endrk{\par\medskip}
%
%   The next macro is for numbering equations etc, \label  produces the 
%   correct number  x.y  and advances the resultnumber register
%
%   Typical use :
%
%     $$fx=7\eqno{\bf\label}$$
%
%   result :
%
%                           fx = 7                           3.5
%
\def\label{\xdef\nextkey{\number\sectionnumber.\number\resultnumber}%
\number\sectionnumber.\number\resultnumber
\global\advance\resultnumber by 1}
%
%
%
%   The next macros are to automate external references.  To use them 
%   type \reflist ..... \endreflist near the beginning of your paper, 
%   where  .... is the list of references in alphabetical order 
%   and in  the form  \key{KEY}  reference    where "KEY" is a 
%   string of characters which reminds you of the reference.   
%   Separate  references with a blank line or a \par.   Eg 
%
%     \reflist
%
%     ..... more references ....
%
%     \key{Kn-84} {\bf D Knuth}, {\it The TeXbook}, Addison--Wesley (1984)
%
%     ..... more references ....
%
%     \endreflist
%
%   Then type  \references  where you wish the references to be printed
%   (normally near the end of the paper).  To refer to Knuth type
%   for example    see Knuth [\ref{Kn-84}, page 320]   and the correct
%   numerical reference will be printed.  Edit the \references macro
%   to change the formatting (if desired).
%   There is an alternative \refkey for \key, provided your KEY contains
%   only letters.  The syntax is:
%
%     \reflist
%
%     ..... more references ....
%
%     \refkey\Knuth  {\bf D Knuth}, {\it The TeXbook}, Addison--Wesley (1984)
%
%     ..... more references ....
%
%     \endreflist
%
%   \key{Knuth}  has exactly the same maening as \refkey\Knuth and you
%   can mix the two syntaxes if you want.  But \refkey\Kn-84
%   would not work.  It would set Kn as the KEY and -84 would get printed!
%
\newcount\refnumber              %  Register for reference numbers
\refnumber=1                     %  set initially to 1.
\long\def\reflist#1\endreflist{%
\long\def\thereflist{#1}{\def\refkey##1##2\par{\xdef##1{\number\refnumber}%
\global\advance\refnumber by 1}%
\def\key##1##2\par{\expandafter\xdef%
\csname##1\endcsname{\number\refnumber}%
\global\advance\refnumber by 1}#1\par}}
\long\def\references{%
\penalty-800\vskip-\lastskip\vskip 15pt plus10pt minus5pt 
{\large\bf References}\ppar %`References' is set \large bold with space around.
{\leftskip=25pt\frenchspacing    % The list of references is set 
\small\parskip=3pt plus2pt       % \small  with small spaces between,
\def\refkey##1##2\par{\noindent  % numbers in [,]'s and set just to the
\llap{[##1]\stdspace}\ignorespaces##2\par}         % left of a 25pt margin.
\def\key##1##2\par{\noindent  
\llap{[\ref{##1}]\stdspace}\ignorespaces##2\par}  
\def\,{\thinspace}\thereflist\par}}
%
%   Next a footnote macro (with automatic numbering) which sets the
%   footnote  \small.
%
%   Typical use :
%         ..... are yellow.\fnote{By yellow here we mean Britsh
%    Standard colour BS3320.} 
%
\newcount\footnotenumber         % Register for footnote number
\footnotenumber=1                % set initially to 1
\def\fnote#1{\xdef\nextkey{\number\footnotenumber}%
{\small\ifnum\footnotenumber>9\parindent=14pt%
\else\parindent=10pt\fi\footnote{$^{\number\footnotenumber}$}%
{\hglue-5pt#1}\global\advance\footnotenumber by 1}}
%
%
%   Next macros for handling figures with automatic numbering (using 
%   TeX's \midinsert to float the figure to a suitable place).
%   
%   The \figure ... \endfigure macro centres the figure and adds
%   an automatically numbered label  Figure XX  after it.
%
%   If you have a caption, then type \caption{caption text} 
%   somewhere between \figure and \endfigure.  The macro
%   will then add  Figure XX: caption text  after the figure.
%
%   If you want an unnumbered or uncentred figure, then use TeX's raw 
%       \midinsert Figure instructions \endinsert  
%   and if you want a numbered figure label in the same style then
%   use \caption{caption text} outside of  \figure ... \endfigure.
%
%   If you need just the label Figure XX  outside of  \figure ... \endfigure
%   then type  \figurelabel .
%
\newcount\figurenumber          % register for figure number
\figurenumber=1                 % set initially to 1
\def\caption#1{\xdef\nextkey{\number\figurenumber}%
\cl{\small Figure \number\figurenumber: #1}%
\global\advance\figurenumber by 1}
\def\figurelabel{\xdef\nextkey{\number\figurenumber}%
\cl{\small Figure \number\figurenumber}%
\global\advance\figurenumber by 1}
%
\long\def\figure#1\endfigure{{\xdef\nextkey{\number\figurenumber}%
\let\captiontext\relax\def\caption##1{\xdef\captiontext{##1}}%
\midinsert\cl{\ignorespaces#1\unskip\unskip\unskip\unskip}\vglue6pt\cl{\small 
Figure \number\figurenumber\ifx\captiontext\relax\else: \captiontext
\fi}\endinsert\global\advance\figurenumber by 1}}
%
%   Macros for self-correcting internal references.
%
%   There are two macros  \key{KEY}  and  \ref{KEY} .
%
%   The \key macro sets up KEY as a key for whatever number is 
%   being referenced and the \ref macro converts the KEY into 
%   that number.  Type \key after a  \section or \proc or 
%   \label or \fnote or \figure or \caption or \figurelabel .
%
%   Example:
%
%       \section{Introduction}\key{intro}
%       \proc{Theorem}\key{MainTh}Lemons are yelloy\endproc
%       Here we follow\fnote{Follow in the sense of Dickens}
%       \key{Dickens-note}the crowd ....  
%
%       In section \ref{intro}
%       we stated theorem \ref{mainTh} and noted (see footnote 
%       \ref{Dickens-note}) ...
%
\def\nextkey{??}   %  initialise \nextkey (which is reset by all the
%                     numbering macros)
%
\def\key#1{\expandafter\xdef\csname #1\endcsname{\nextkey}}
%
\def\ref#1{\expandafter\ifx\csname #1\endcsname\relax
\immediate\write16{Reference {#1} undefined}??\else
\csname #1\endcsname\fi}
%
%   Note:  If the KEY contains only letters then \KEY has exactly the
%   same meaning as \ref{KEY} so in the example you could have:
%
%       In section \intro\ we ....
%
%   The \key will work at any time after the macro which sets the
%   number, provided no other macro which sets a number has been used. 
%
%   Macros for forward references:
%              =======
%   The \key \ref macros ONLY work for backwards references.  If you  
%   want to use forwards references, then type \useforwardrefs  near
%   the beginning of your file.  The KEY's are then stored in an
%   auxiliary  .ref  file and you then suffer the same disadvantage as
%   when using LaTeX that you must TeX the file twice to get
%   the references correct.
%
%   To use a forward ref type \ref{KEY}.  (You can type the
%   alternative  \KEY  but you'll get an error on first TeX'ing 
%   if the \KEY is not yet defined.) 
%
%   The macro also allows external references to be listed at the end 
%   of the file (if you wish to).  (Indeed they can be typed anywhere
%   before the \references command.)  You can combine the reference list
%   and the \references command by typing the references (using the
%   same syntax as before) between the commands \biblio and \endbiblio 
%   (don't type \references or they'll be printed twice).
%
\newread\gtinfile
\newwrite\gtreffile
%
\def\useforwardrefs{
\openin\gtinfile\jobname.ref
%
\ifeof\gtinfile
\closein\gtinfile
\immediate\write16{No file \jobname.ref}
\else
\closein\gtinfile
\input \jobname.ref
\fi
%
\immediate\openout\gtreffile \jobname.ref
%
%   Adapt \key :
%
\def\key##1{{\def\\{\noexpand}%
\expandafter\xdef\csname ##1\endcsname{\nextkey}%
\immediate\write\gtreffile{\\\expandafter\\\def\\\csname ##1\\\endcsname%
{\nextkey}}}}
%
%  Adapt macros for external references:  
%
\long\def\reflist##1\endreflist{%
\long\def\thereflist{##1}{\def\refkey####1####2\par{\xdef####1{%
\number\refnumber}{\def\\{\noexpand}\immediate\write\gtreffile
{\\\def\\####1{\number\refnumber}}}\global\advance\refnumber by 1}%
\def\key####1####2\par{\expandafter\xdef%
\csname####1\endcsname{\number\refnumber}%
{\def\\{\noexpand}\immediate\write\gtreffile
{\\\expandafter\\\def\\\csname ####1\\\endcsname{\number\refnumber}}}
\global\advance\refnumber by 1}##1\par}}
%
\long\def\biblio##1\endbiblio{\reflist##1\endreflist\references}%
%
%  Adapt obselete key macros (\numkey, \seckey and \figkey):
%
\def\numkey##1{{\def\\{\noexpand}%
\xdef##1{\number\sectionnumber.\number\resultnumber}
\immediate\write\gtreffile{\\\def\\##1%
{\number\sectionnumber.\number\resultnumber}}}}
%
\def\seckey##1{{\def\\{\noexpand}\xdef##1{\number\sectionnumber}
\immediate\write\gtreffile{\\\def\\##1{\number\sectionnumber}}}}
%
\def\figkey##1{\xdef##1{\number\figurenumber}%
{\def\\{\noexpand}\immediate\write\gtreffile%
{\\\def\\##1{\number\figurenumber}}}
\number\figurenumber\global\advance\figurenumber by 1}
%
}   %  end of \useforwardrefs
%
%
%   The next five macros are obselete and have been superseeded by
%   the general \key macro above.  They are included merely to 
%   maintain backward compatibility for the package:
%
%
\def\figkey#1{\xdef#1{\number\figurenumber}%
\number\figurenumber\global\advance\figurenumber by 1}
%
\def\fig#1#2\endfig{%
\midinsert\cl{#2}\vglue6pt\cl{\small Figure #1}\endinsert}
%
\def\newfig{\number\figurenumber\global\advance\figurenumber by 1}
%
\def\numkey#1{\xdef#1{\number\sectionnumber.\number\resultnumber}}
%
\def\seckey#1{\xdef#1{\number\sectionnumber}}
%
%   End of obselete macros.
%
%
%   The next macro is a version of the verbatim macro given by Knuth.
%
%   This macro produces a "verbatim" printout of
%   any ASCII string which does not contain the symbol "
%   (TeX files do not usually contain " 's).
%   More precisely, everything between consecutive pairs
%   of " 's is printed verbatim in the typewriter font cmtt.
%   For an explanation of how the macro works, see Knuth pp 420-1.
%
%   There are two switches: \verb (which switches the macro on)
%   and \brev which switches the macro off (the default).  When
%   the macro is switched off the symbol " has its usual 
%   meaning for TeX.  To use the macro, type \verb before use
%   and the use " to switch verbatim on and off.  Be careful
%   not to use " for any other purpose.  There is no need to
%   switch the macro off again unless you need to use " for
%   some other purpose (eg making  \mathchardef 's).  Note 
%   that the macro MUST BE OFF before inputting  amsnames.tex .
%
%   Whether the macro is on or off you can always use the
%   control sequence \dq (double quote) for " e.g.
%   \mathchardef\sum=\dq1350  is perfectly valid.
%   The control sequence \ttq is an abbreviation for
%   {\tt\dq}.  Thus "\ttq" will produce " (in cmtt)
%   inside a verbatim quote.
%
%
\def\dq{"}   %  define a code for " so it can be used when \verb is on
\def\ttq{{\tt\dq}}  %  code for " in cmtt
%
\def\verb{\catcode`\"=\active}       %  The main
\def\brev{\catcode`\"=12}            %  switches.
%
\brev                                %  Prime switches and
\verb                                %  switch on.
%
{\obeyspaces\gdef {\ }}              
{\catcode`\`=\active\gdef`{\relax\lq}}
\def"{%
\begingroup\baselineskip=12pt\def\par{\leavevmode\endgraf}%
\tt\obeylines\obeyspaces\parskip=0pt\parindent=0pt%
\catcode`\$=12\catcode`\&=12\catcode`\^=12\catcode`\#=12%
\catcode`\_=12\catcode`\~=12%
\catcode`\{=12\catcode`\}=12\catcode`\%=12\catcode`\\=12%
\catcode`\`=\active\let"\endgroup}
%
\brev      %   Finally switch the macro off (for safety)
%
%   Macros for itemised lists.   Typical use :
%    
%    \items
%    \item{(i)}Colours must be defined.
%    \item{(ii)}Colour cards may not be cited.
%    \enditems
%
%   Result :
%
%    (i)  Colours must be defined. 
%   (ii)  Colour cards may not be cited.
%
%
\def\items{\par\leftskip = 25pt}           % Start of itemised list         
\def\enditems{\par\leftskip = 0pt}         % end of itemised list   
\def\item#1{\par\leavevmode\llap{#1\stdspace}%
\ignorespaces}                             % labelled item
\def\itemb{\item{$\bullet$}}               % bulleted item.
%
%   The \quote ... \endquote macros are for typesetting quotations :
%
\def\quote{\par\leftskip=25pt\rightskip=25pt\it}
\def\endquote{\par\leftskip=0pt\rightskip=0pt\rm}
%
%   A few useful abbreviations :
%
\def\co{\colon\thinspace}    %  Colon with correct spacing for maps.
\def\np{\vfil\eject}         %  Forced page break (new page).
\def\nl{\hfil\break}         %  New line.
\def\cl{\centerline}         %  Centerline
\def\gt{{\mathsurround=0pt\it $\cal G\mskip-2mu$eometry \&\ 
$\cal T\!\!$opology}}        %  The journal title in recommended style
\def\gtm{{\mathsurround=0pt\it $\cal G\mskip-2mu$eometry \&\ 
$\cal T\!\!$opology $\cal M\mskip-1mu$onographs}}    %  for monographs
\def\agt{{\mathsurround=0pt\it$\cal A\mskip-.7mu$lgebraic \&\ 
$\cal G\mskip-2mu$eometric $\cal T\!\!$opology}}  % AGT
%
%    Finally some macros for automatic title page or header generation.
%    To use them type your header information using the following  
%    example as a guide :
%
%    Note that \\ is used as standard separator (for lines in \title and
%    \address, between authors and between email addresses or URL's)
%    and that \email, \url and \secondaddress are optional.
%

% Example:  \title{A short spoof paper\\with a two-line title}
% =======   \authors{Albert Einstein\\Leonardo da Vinci}
%           \address{IAS\\Princeton}\secondaddress{Renaissance\\Venice}
%           \email{ae@ias.princeton.edu\\ldv@ren.ven.hist}
%           \abstract 
%           A short spoof paper with a very short abstract.
%           \endabstract 
%           \primaryclass{00-01, 00-02}\secondaryclass{68-00, 68-01}
%           \keywords{Short, spoof, paper}
%           \maketitlepage
%
%
%    The title page or header will then be generated automatically.
%
%
%    Define the various ingredients of the title page:
%
\def\title#1{\def\thetitle{#1}}
\def\shorttitle#1{\def\theshorttitle{#1}}
%
\def\author#1{\edef\previousauthors{\theauthors}
 \ifx\theauthors\relax\def\theauthors{#1}\else
 \def\theauthors{\previousauthors\par#1}\fi}
\def\shortauthors#1{\def\theshortauthors{#1}}
\let\authors\author\let\secondauthor\author        % aliases
%
\def\address#1{\edef\previousaddresses{\theaddress}
 \ifx\theaddress\relax\def\theaddress{#1}\else
 \def\theaddress{\previousaddresses\par\vskip 2pt\par#1}\fi}
\let\addresses\address                             % alias
\def\secondaddress#1{\edef\previousaddresses{\theaddress}
 \ifx\theaddress\relax\def\theaddress{#1}\else
 \def\theaddress{\previousaddresses\par{\rm and}\par#1}\fi}   
\def\shortaddress#1{\def\theshortaddress{#1}}
%
\def\email#1{\edef\previousemails{\theemail}
 \ifx\theemail\relax\def\theemail{#1}\else
 \def\theemail{\previousemails\hskip 0.75em\relax#1}\fi}
\let\emails\email\let\emailaddress\email\let\emailaddr\email  % aliases
\def\secondemail#1{\edef\previousemails{\theemail}
 \ifx\theemail\relax\def\theemail{#1}\else
 \def\theemail{\previousemails\hskip 0.75em{\rm and}\hskip 0.75em
 \relax#1}\fi}
%
\def\url#1{\edef\previousurls{\theurl}
 \ifx\theurl\relax\def\theurl{#1}\else
 \def\theurl{\previousurls\hskip 0.75em\relax#1}\fi}
\let\urls\url\let\urladdress\url\let\urladdr\url      % aliases
\def\secondurl#1{\edef\previousurls{\theurl}
 \ifx\theurl\relax\def\theurl{#1}\else
 \def\theurl{\previousurls\hskip 0.75em{\rm and}\hskip 0.75em
 \relax#1}\fi}
%
\long\def\abstract#1\endabstract{\long\def\theabstract{#1}}
%
\def\primaryclass#1{\def\theprimaryclass{#1}}
\let\subjclass\primaryclass                        % alias
\def\secondaryclass#1{\def\thesecondaryclass{#1}}
\def\keywords#1{\def\thekeywords{#1}}
%
%  Set \\ to \par and title page items to \relax to initialise macros :
%
\let\\\par\let\thetitle\relax\let\theshorttitle\relax
\let\theauthors\relax\let\theshortauthors\relax
\let\theaddress\relax\let\theshortaddress\relax
\let\theemail\relax\let\theurl\relax
\let\theabstract\relax\let\theprimaryclass\relax
\let\thesecondaryclass\relax\let\thekeywords\relax
%
%
%
%   Basic title page layout (edit this macro if you
%   wish to adjust the title page layout) :
%
\long\def\maketitlepage{    % start of definition of \maketitlepage

\vglue 0.2truein   % top margin

% title :
%
{\parskip=0pt\leftskip 0pt plus 1fil\def\\{\par\smallskip}{\large
\bf\thetitle}\par\medskip}   

\vglue 0.15truein 

% authors :
%
{\parskip=0pt\leftskip 0pt plus 1fil\def\\{\par}{\sc\theauthors}
\par\medskip}%
 
\vglue 0.1truein 

% address(es) email's and URL's (with switches to detect whether the
% optional items have been used) :
%
{\small\parskip=0pt
{\leftskip 0pt plus 1fil\def\\{\par}{\sl\theaddress}\par}
\ifx\theemail\relax\else  % email address?
\vglue 5pt \def\\{\stdspace{\rm and}\stdspace} 
\cl{Email:\stdspace\tt\theemail}\fi
\ifx\theurl\relax\else    % URL given?
\vglue 5pt \def\\{\stdspace{\rm and}\stdspace} 
\cl{URL:\stdspace\tt\theurl}\fi\par}

\vglue 7pt 

{\bf Abstract}

\vglue 5pt

\theabstract

\vglue 7pt 

{\bf AMS Classification numbers}\quad Primary:\quad \theprimaryclass\par

Secondary:\quad \thesecondaryclass

\vglue 5pt 

{\bf Keywords:}\quad \thekeywords

\np  % page break at the end of the title page

}    % end of definition of \maketitlepage
%
%    % \makeshorttitle (for general preprints) doesn't take a new page
%
\long\def\makeshorttitle{    % start of definition of \makeshorttitle

%\vglue 0.2truein   % top margin

% 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 

% address(es) email's and URL's (with switches to detect whether the
% optional items have been used) :
%
{\small\parskip=0pt
{\leftskip 0pt plus 1fil\def\\{\par}{\sl\ifx\theshortaddress\relax
\theaddress\else\theshortaddress\fi}\par}
\ifx\theemail\relax\else  % email address?
\vglue 5pt \def\\{\stdspace{\rm and}\stdspace} 
\cl{Email:\stdspace\tt\theemail}\fi
\ifx\theurl\relax\else    % URL given?
\vglue 5pt \def\\{\stdspace{\rm and}\stdspace} 
\cl{URL:\stdspace\tt\theurl}\fi\par}

\vglue 10pt 

% abstract and classification numbers (with switches):

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

{\bf AMS Classification}\stdspace\theprimaryclass
\ifx\thesecondaryclass\relax\else; \thesecondaryclass\fi\par
{\bf Keywords}\stdspace \thekeywords\par}
\vglue 7pt
}    % end of definition of \makeshorttitle
%
\let\maketitle\makeshorttitle        %% alias
%
%    %%%% \makeagttitle (for AGT) similar to \makeshorttitle but
%         with addresses omitted (they go at the end)
%
%%%% publication info and test defaults:

\def\volumenumber#1{\def\thevolumenumber{#1}}
\def\volumeyear#1{\def\thevolumeyear{#1}}
\def\pagenumbers#1#2{\def\startpage{#1}\def\finishpage{#2}}
\def\published#1{\def\publishdate{#1}}
\def\received#1{\def\receiveddate{#1}}
\def\revised#1{\def\reviseddate{#1}}
\let\reviseddate\relax
%% Defaults for authors to use to check layout
\volumenumber{X}
\volumeyear{20XX}
\pagenumbers{1}{XXX}
\published{XX Xxxember 20XX}

\long\def\makeagttitle{   %%% start of definition of \makeagttitle
\agt\hfill      %   Journal title (top left) 
%   logo placeholder (top right)
\hbox to 60truept{\vbox to 0pt{\vglue -14truept{\bf [Logo here]}\vss}\hss}
%
\break
{\small Volume \thevolumenumber\ (\thevolumeyear)
\startpage--\finishpage\nl
Published: \publishdate}

\vglue .2truein

% 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


%%%%% Macro to typeset addresses (typically at the end of the paper)

\def\Addresses{\bigskip
{\small \parskip 0pt \leftskip 0pt \rightskip 0pt plus 1fil \def\\{\par}
\sl\theaddress\par\medskip \rm Email:\stdspace\tt\theemail\par
\ifx\theurl\relax\else\smallskip \rm URL:\stdspace\tt\theurl\par\fi}}

\def\agtart{%   Full mock-up of AGT article style (for authors to test with)
%  get print centerpage:
\hoffset 14truemm
\voffset 31truemm
\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 numbers are printed here
\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 declaration is printed here\hfill\else
\agt, Volume \thevolumenumber\ (\thevolumeyear)\hfill\fi}\vss}}
%  force \agttitle
\let\maketitle\makeagttitle\let\makeshorttitle\makeagttitle}


\endinput
%
%    Version 1.1   9.6.96
%    Version 1.2   spaces tightened, References \large  27.10.97 
%    Version 1.3   automatic title page macros  11.11.97
%    Version 1.4   vertical spacing improved  24.12.97
%    Version 1.5   \qua, and \makeshorttitle added,  internal refs overhauled 
%                  (based on \key and \ref), \useforwardrefs added 9.12.98
%    Version 1.6   update for AGT: \agtart and \makeagttitle added 15.10.00
gtoutput.tex
%%%
%%%  This version of  gtoutput.tex  is intended to finish formatting
%%%  papers published in Geometry & 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 G&T papers.
%%%
%%%
%%%                                             Colin Rourke  14.9.2000
%%%
%%%  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}

%  get print centerpage:

\ifplaintex 
\hoffset 14truemm
\voffset 31truemm
\else
\headsep 23pt
\footskip 35pt
\hoffset -4truemm
\voffset 12.5truemm
\fi


%  load pictex if not already loaded :
\expandafter\ifx\csname beginpicture\endcsname\relax
\expandafter\ifx\csname documentclass\endcsname\relax
\input pictex \else\font\fiverm=cmr5
\input prepictex \input pictex \input postpictex \fi\fi

\def\gt{{\mathsurround=0pt\it $\cal G\mskip-2mu$eometry \&\ 
$\cal T\!\!$opology}}        %  journal title in recommended style

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

%  define the various new ingredients of the title page 

\def\lognumber#1{\def\thelognumber{#1}}
\def\volumenumber#1{\def\thevolumenumber{#1}}
\def\papernumber#1{\def\thepapernumber{#1}}
\def\volumeyear#1{\def\thevolumeyear{#1}}
\def\samplenumber#1{\def\thesamplenumber{#1}\def\thevolumenumber{#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\coverauthors#1{\def\thecoverauthors{#1}}
\def\asciiauthors#1{\def\theasciiauthors{#1}}
\def\asciiaddress#1{\def\theasciiaddress{#1}}
\def\asciiemail#1{\def\theasciiemail{#1}}
\long\def\asciiabstract#1{\long\def\theasciiabstract{#1}}
\def\asciikeywords#1{\def\theasciikeywords{#1}}
\def\shortauthors#1{\def\theshortauthors{#1}}
\def\shorttitle#1{\def\theshorttitle{#1}}

%  initialise

\let\\\par\let\thelognumber\relax
\let\thevolumenumber\relax\let\thepapernumber\relax
\let\thevolumeyear\relax\let\thesamplenumber\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\theasciiemail\relax\let\theshortauthors\relax\let\theshorttitle\relax
\let\thecoverauthors\relax

\long\def\maketitlep{   % start of definition of \maketitlep

\count0=\startpage

\gt\hfill      %   Journal title (top left) 
%    Logo (top right) :
\beginpicture
\setcoordinatesystem units <0.33truein, 0.33truein> point at 2.2 0.9
\setplotsymbol ({$\cal G$})
\plotsymbolspacing=9truept
\circulararc 315 degrees from 0 1 center at 0 0
\setplotsymbol ({$\cal T$})
\circulararc 315 degrees from 1 -1 center at 1 0
\endpicture
%   end of logo
%
\break
{\small\ifx\thesamplenumber\relax % sample?  
Volume \else Sample
\fi\thevolumenumber\ (\thevolumeyear)
\startpage--\finishpage\nl
Published: \publishdate}
\vglue 0.5truein plus 0.4fil minus 0.1truein

% title
{\parskip=0pt\leftskip 0pt plus 1fil\def\\{\par\smallskip}{\ifplaintex\large
\else\Large\fi\bf\thetitle}\par\medskip}   

\vglue 0pt plus 0.1fil 

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

 
\vglue 0pt plus 0.1fil 

%address(es)
{\small\parskip=0pt\let\newline\\
{\leftskip 0pt plus 1fil\def\\{\par}{\sl\theaddress}\par}
\expandafter\ifx\theemail\relax    % email address?
\relax\else\vglue 5pt plus 0.02fil minus 2pt\def\\{\stdspace{\rm 
and}\stdspace} 
\cl{Email:\stdspace\tt\theemail}\fi
\ifx\theurl\relax                  % URL given?
\relax\else\vglue 5pt plus 0.02fil minus 2pt\def\\{\stdspace{\rm 
and}\stdspace}
\cl{URL:\stdspace\tt\theurl}\fi\par}

\vglue 7pt plus 0.3fil minus 3pt

{\bf Abstract}
\vglue 5pt plus 0.1fil minus 2pt

\theabstract

\vglue 7pt plus 0.3fil minus 3pt

{\bf AMS Classification numbers}\quad Primary:\quad \theprimaryclass

Secondary:\quad \thesecondaryclass

\vglue 5pt plus 0.3fil minus 2pt

{\bf Keywords}\quad \thekeywords

\vglue 10pt plus 0.5fil minus 5pt

{\small  Proposed: \theproposer\hfill Received: \receiveddate\nl
Seconded: \theseconders\hfill 
\ifx\reviseddate\relax                         % paper revised?
Accepted: \accepteddate                        % no
\else
Revised: \reviseddate                          % yes
\fi}
\eject
}       %  end of definition of \maketitlep

\let\maketitlepage\maketitlep
\let\maketitle\maketitlepage

%%% Headers and footers

\font\phead=cmsl9 scaled 950
\font\lhead=cmsl9 scaled 1050
\font\pnum=cmbx10 scaled 913
\font\lnum=cmbx10 
\font\pfoot=cmsl9 scaled 950
\font\lfoot=cmsl9 scaled 1050
\ifplaintex
\headline{\vbox to 0pt{\vskip -4.5mm\line{\small\phead\ifnum
\count0=\startpage ISSN 1364-0380 (on line)
1465-3060 (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
\gt, Volume \thevolumenumber\ (\thevolumeyear)\hfill\fi}\vss
}}
\else
\makeatletter
\def\@oddhead{{\small\lhead\ifnum\count0=\startpage ISSN 1364-0380 (on line)
1465-3060 (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
\gt, Volume \thevolumenumber\ (\thevolumeyear)\hfill\fi}
\def\@evenfoot{\@oddfoot}
\makeatother
\fi

\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:pppp}
\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}
\immediate\write\gtoutfile{Title: \ifx\theasciititle\relax
\thetitle\else\theasciititle\fi}
\immediate\write\gtoutfile{Subj-class: GT or SG or MG etc}
\immediate\write\gtoutfile{MSC-class: \theprimaryclass\ifx\thesecondaryclass\relax\else, \thesecondaryclass\fi}
\immediate\write\gtoutfile{Journal-ref: Geom. Topol. \thevolumenumber
(\thevolumeyear) \startpage-\finishpage}
\immediate\write\gtoutfile{Comments: Published by Geometry and Topology at}
\immediate\write\gtoutfile{\s\s http://www.maths.warwick.ac.uk/gt/GTVol\thevolumenumber/paper\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{\maketitlep\makeheadfile}
\let\maketitle\maketitlepage

\endinput
%%%
%%%  This version of  gtoutput.tex  is intended to finish formatting
%%%  papers published in Geometry & 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 G&T papers.
%%%
%%%
%%%                                             Colin Rourke  14.9.2000
%%%
%%%  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}

%  get print centerpage:

\ifplaintex 
\hoffset 14truemm
\voffset 31truemm
\else
\headsep 23pt
\footskip 35pt
\hoffset -4truemm
\voffset 12.5truemm
\fi


%  load pictex if not already loaded :
\expandafter\ifx\csname beginpicture\endcsname\relax
\expandafter\ifx\csname documentclass\endcsname\relax
\input pictex \else\font\fiverm=cmr5
\input prepictex \input pictex \input postpictex \fi\fi

\def\gt{{\mathsurround=0pt\it $\cal G\mskip-2mu$eometry \&\ 
$\cal T\!\!$opology}}        %  journal title in recommended style

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

%  define the various new ingredients of the title page 

\def\lognumber#1{\def\thelognumber{#1}}
\def\volumenumber#1{\def\thevolumenumber{#1}}
\def\papernumber#1{\def\thepapernumber{#1}}
\def\volumeyear#1{\def\thevolumeyear{#1}}
\def\samplenumber#1{\def\thesamplenumber{#1}\def\thevolumenumber{#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\coverauthors#1{\def\thecoverauthors{#1}}
\def\asciiauthors#1{\def\theasciiauthors{#1}}
\def\asciiaddress#1{\def\theasciiaddress{#1}}
\def\asciiemail#1{\def\theasciiemail{#1}}
\long\def\asciiabstract#1{\long\def\theasciiabstract{#1}}
\def\asciikeywords#1{\def\theasciikeywords{#1}}
\def\shortauthors#1{\def\theshortauthors{#1}}
\def\shorttitle#1{\def\theshorttitle{#1}}

%  initialise

\let\\\par\let\thelognumber\relax
\let\thevolumenumber\relax\let\thepapernumber\relax
\let\thevolumeyear\relax\let\thesamplenumber\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\theasciiemail\relax\let\theshortauthors\relax\let\theshorttitle\relax
\let\thecoverauthors\relax

\long\def\maketitlep{   % start of definition of \maketitlep

\count0=\startpage

\gt\hfill      %   Journal title (top left) 
%    Logo (top right) :
\beginpicture
\setcoordinatesystem units <0.33truein, 0.33truein> point at 2.2 0.9
\setplotsymbol ({$\cal G$})
\plotsymbolspacing=9truept
\circulararc 315 degrees from 0 1 center at 0 0
\setplotsymbol ({$\cal T$})
\circulararc 315 degrees from 1 -1 center at 1 0
\endpicture
%   end of logo
%
\break
{\small\ifx\thesamplenumber\relax % sample?  
Volume \else Sample
\fi\thevolumenumber\ (\thevolumeyear)
\startpage--\finishpage\nl
Published: \publishdate}
\vglue 0.5truein plus 0.4fil minus 0.1truein

% title
{\parskip=0pt\leftskip 0pt plus 1fil\def\\{\par\smallskip}{\ifplaintex\large
\else\Large\fi\bf\thetitle}\par\medskip}   

\vglue 0pt plus 0.1fil 

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

 
\vglue 0pt plus 0.1fil 

%address(es)
{\small\parskip=0pt\let\newline\\
{\leftskip 0pt plus 1fil\def\\{\par}{\sl\theaddress}\par}
\expandafter\ifx\theemail\relax    % email address?
\relax\else\vglue 5pt plus 0.02fil minus 2pt\def\\{\stdspace{\rm 
and}\stdspace} 
\cl{Email:\stdspace\tt\theemail}\fi
\ifx\theurl\relax                  % URL given?
\relax\else\vglue 5pt plus 0.02fil minus 2pt\def\\{\stdspace{\rm 
and}\stdspace}
\cl{URL:\stdspace\tt\theurl}\fi\par}

\vglue 7pt plus 0.3fil minus 3pt

{\bf Abstract}
\vglue 5pt plus 0.1fil minus 2pt

\theabstract

\vglue 7pt plus 0.3fil minus 3pt

{\bf AMS Classification numbers}\quad Primary:\quad \theprimaryclass

Secondary:\quad \thesecondaryclass

\vglue 5pt plus 0.3fil minus 2pt

{\bf Keywords}\quad \thekeywords

\vglue 10pt plus 0.5fil minus 5pt

{\small  Proposed: \theproposer\hfill Received: \receiveddate\nl
Seconded: \theseconders\hfill 
\ifx\reviseddate\relax                         % paper revised?
Accepted: \accepteddate                        % no
\else
Revised: \reviseddate                          % yes
\fi}
\eject
}       %  end of definition of \maketitlep

\let\maketitlepage\maketitlep
\let\maketitle\maketitlepage

%%% Headers and footers

\font\phead=cmsl9 scaled 950
\font\lhead=cmsl9 scaled 1050
\font\pnum=cmbx10 scaled 913
\font\lnum=cmbx10 
\font\pfoot=cmsl9 scaled 950
\font\lfoot=cmsl9 scaled 1050
\ifplaintex
\headline{\vbox to 0pt{\vskip -4.5mm\line{\small\phead\ifnum
\count0=\startpage ISSN 1364-0380 (on line)
1465-3060 (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
\gt, Volume \thevolumenumber\ (\thevolumeyear)\hfill\fi}\vss
}}
\else
\makeatletter
\def\@oddhead{{\small\lhead\ifnum\count0=\startpage ISSN 1364-0380 (on line)
1465-3060 (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
\gt, Volume \thevolumenumber\ (\thevolumeyear)\hfill\fi}
\def\@evenfoot{\@oddfoot}
\makeatother
\fi

\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:pppp}
\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}
\immediate\write\gtoutfile{Title: \ifx\theasciititle\relax
\thetitle\else\theasciititle\fi}
\immediate\write\gtoutfile{Subj-class: GT or SG or MG etc}
\immediate\write\gtoutfile{MSC-class: \theprimaryclass\ifx\thesecondaryclass\relax\else, \thesecondaryclass\fi}
\immediate\write\gtoutfile{Journal-ref: Geom. Topol. \thevolumenumber
(\thevolumeyear) \startpage-\finishpage}
\immediate\write\gtoutfile{Comments: Published by Geometry and Topology at}
\immediate\write\gtoutfile{\s\s http://www.maths.warwick.ac.uk/gt/GTVol\thevolumenumber/paper\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{\maketitlep\makeheadfile}
\let\maketitle\maketitlepage

\endinput

rlepsf.tex
%%%%%%%%%%%%%%%%%%%%%%   rlepsf.tex   %%%%%%%%%%%%%%%%
%                     Version 0.0 (alpha test)
%
%    Comments, problems etc to :   cpr@maths.warwick.ac.uk
%
%    Macros to relabel an eps file with TeX labels,  
%    written by Michael Greene, with TeX front end 
%    written by Colin Rourke. 
%
%    Designed for use with the macro package epsf.tex 
%    and dvips converter (written by Rokicki and Knuth) 
%    and with eps output from a variety of drawing 
%    packages (tested with output from xfig and Adobe 
%    Illustrator, and with hand-drawn figures). 
%    
%    To use this package type  \input rlepsf  (in place of
%    the usual  \input epsf ).  Then place each eps file
%    in a  \relabelbox ... \endrelabelbox  with relabelling
%    instructions after the  \epsfbox{..} .
%
%    The eps file is called exactly as usual with epsf.tex
%    typical instructions being  \epsfbox{filename.eps}
%    and   \epsfxsize dimension \epsfbox{filename.eps}
%    (where "dimension" is the required finished width of 
%    the diagram).
%
%    There are three (re)labelling macros:
%
%       \relabel {pslabel}{TeX label}
%       \adjustrelabel <dx,dy> {pslabel}{TeX label}  
%       \extralabel <x,y> {extra label}
%
%    \relabel  uses the ps file position for the new TeX
%    label (which can be any TeX output).  The new label
%    is positioned with same reference point (left-hand 
%    end of baseline) as the original ps label.  
%    \adjustrelabel  adjusts the position by  dx,dy  and 
%    is useful for fine tuning the position of the
%    new label.  \extralabel  allows extra TeX labels to be
%    inserted.  The positions x,y are measured from the
%    bottom right-hand corner of the box (so x is typically
%    negative) !  The  re/extra-labelling instructions can
%    be given in any order and any TeX dimensions can
%    be used for x,y,dx,dy.   
%
%    The result is a box of dimensions the original epsbox
%    but with new TeX labels.  This can then be inserted in
%    $$ ... $$  or \centerline{ ... }  for display, or
%    combined with other boxes to form a larger display.
%
%    The  \relabelbox ... \endrelabelbox  forms a TeX group;
%    thus instructions for resizing text for the new labels
%    (eg  \small ) will remain local to this group.
%
%    There are two main uses for the package :   
%
%    (1)  Creating graphics with TeX labels :   
%    Draw the graphic with dummy ps labels (all different and in
%    some common font eg Times-Roman).  Then use \relabel 'ling 
%    instructions to replace the dummy labels by the genuine TeX
%    labels.  Use roughly the same size dummy labels as the final
%    TeX labels and then the position will probably be perfect
%    first time.  Adjust the positions if necessary by editing
%    \relabel  to  \adjustrelabel  with appropriate  <dx,dy>.
%
%    (2)  Editing the ps labels in an existing eps file to TeX
%    labels :   
%    Use  \relabel  and  \adjustrelabel  to replace the 
%    existing labels.  You may have to use  \extralabel  for
%    some of the new labels (or to edit the eps file to change
%    repeated ps labels to be different), see note (2) below.
%
%    Notes:  
%    (1)  All ps labels are removed whether or not
%    instructions for relabelling are given. 
%    (2)  If the eps file has two or more labels the
%    same, then only one will be replaced.  Therefore
%    if you are designing a diagram from scratch then
%    use different temporary ps labels for your labels
%    even if you wish them to finish the same.  If
%    you are relabelling an existing eps diagram with TeX
%    labels then use \extralabel's to replace the other 
%    label(s) or edit the eps file to change the repeated
%    labels to different ones.
%    (3)  The macros are sensitive to extra spaces and 
%    the above syntax should be used.  
%    (4)  The output will not view correctly on a dvi
%    viewer which will typically show the new labels
%    superimposed at the bottom right-hand corner.  
%    However those produced by  \extralabel  will be shown
%    correctly positioned.
%
%    A typical example of use is appended.   CPR 19-5-96
%   
%    Debugged to work with latest version of dvips by CPR
%    26-5-96.  Debugged to work with LaTeX, CPR 31-5-96
%
%
\input epsf
%
% Initialise for MTG control
%
\special{! (MTG set-up) pop userdict /MTGdict 99 dict put MTGdict begin
  /Mshow {[ MTGdict /rllist get aload length 2 add -1 roll
    currentpoint transform ] MTGdict /rllist 3 2 roll put} def
  /findlabel {/sought exch def
    /MTGx 3 index def /MTGy 2 index def aload length 3 idiv dup 0 eq
      {pop}
      {{2 index sought eq
          {/MTGy exch def /MTGx exch def pop}
          {pop pop pop}
        ifelse}
      repeat}
    ifelse exch MTGx exch sub exch MTGy exch sub idtransform translate
    } def
  end}%
%
\def\relabelbox{%
  \hbox\bgroup%
  \special{ps:(MTG takes control) pop 
    /MTGsavestate save def /p /show load def MTGdict begin gsave
    /rllist [] def /save {false} def /restore {pop} def /show {Mshow} def
    /ashow {Mshow pop pop} def /widthshow {Mshow pop pop pop} def /awidthshow
    {Mshow 5 {pop} repeat} def /xshow {pop Mshow} def /yshow {pop Mshow} def
    /xyshow {pop Mshow} def /cshow {Mshow pop} def /kshow {Mshow pop} def}%
   \special{ps:/a {moveto} bind def}%
}%
\def\endrelabelbox{%
  \special{ps:(MTG signs off) pop grestore end /restore {restore} bind def
    MTGsavestate restore}\egroup%
}%
\def\relabel #1#2 {%
  \special{ps:/a {} def}%
  \special{ps:gsave /a {moveto} bind def transform MTGdict /rllist get (#1) findlabel}%
  \smash{\rlap{#2}}%
  \special{ps:grestore}%
}%
\def\adjustrelabel <#1,#2> #3#4 {%
  \special{ps:/a {} def}%
  \special{ps:gsave /a {moveto} bind def transform MTGdict /rllist get (#3) findlabel}%
  \smash{\rlap{\kern #1 \raise #2\hbox{#4}}}%
  \special{ps:grestore}%
}%
\def\extralabel <#1,#2> #3 {\smash{\rlap{\kern #1 \raise #2\hbox{#3}}}}%
\endinput
%
%%%%   Typical example of use :
%
\input rlepsf.tex
The line before the figure.\hfil\break
\centerline{%
\relabelbox            
\epsfxsize 3truein \epsfbox {figure1.eps}  
%%  Give label size instruction (eg \small) here. 
\adjustrelabel <0pt,1pt> {Ab}{$A^b$}
\adjustrelabel <-4pt,0pt> {P}{$P$}
\relabel {Bc}{$B_c$}
\extralabel <-.3truein,0.3truein> {First extra label}
\adjustrelabel <1pt,0pt> {IP"}{$\int P''$}
\adjustrelabel <1pt,0pt> {P'}{$P'$}
\extralabel <-1.5cm,1.5cm> {Second extra label}
\endrelabelbox
}
The line after the figure.\bye

