\section*{2) Enhanced Surgery Presentations}\label{pg-2}
\

Assume we are given a compact  $n$-manifold $X$ and a Morse Function
$f:X\to [0,1]\,$. It is an elementary fact (see [Mi])  that if $f$ has a 
singularity of index $j$ with non-degenerate critical value $a$, then the 
sublevel manifold $X^{\leq a+\epsilon}=f^{-1}([0,a+\epsilon])\,$ is
homeomorphic to $X^{\leq a-\epsilon}$ with a $j$- handle $\hh^j$ attached 
at its boundary. This means, we have an embedding $S^{j-1}\times
D^{n-j}\hookrightarrow f^{-1}(a-\epsilon)\subset \partial X^{\leq
a-\epsilon}\,$, and  we attach $\hh^j=D^j\times D^{n-j}\,$, along the 
respective piece in the boundary $\partial \hh_j\,=\,\partial D^j\times
D^{n-j}\,\cup\,D^j\times \partial D^{n-j}\,$.   

The classical link presentations of [Wc] and [Li] of three folds result
from attaching two handles to a four ball such that
$M=\partial\bigl(D^4\cup\hh^2\cup\ldots\cup\hh^2\bigr)\,$. The theorem of
Kirby  (see [Ki])
states that two such  presentations (with the same signature)
 can be related by sliding the handles across each other.
Both results rely on the fact that the manifold $S^3=\partial D^4$ we
surger on is connected and simply connected. 
Neither is  true for the manifolds $\H\bigl(\bar g^+,l,\bar g^-\bigr)\,$.

For these reasons we shall admit presentations that involve  also surgeries
with one handles. Most of this section is devoted to describe in detail the 
calculus of ``bridged links'', which is the corresponding generalization of the
link calculus for closed manifolds. Some elements of this description were also
used in [FR]. We shall also discuss the calculus of ``bridged ribbon graphs''
analogous to [RT0] or [RT] for generic, planar projections of the link
diagrams. 
\s


 
\subparagraph{2.1) Some Cerf Theory}\label{pg-21}
\
\s


A presentation for a manifold using a Morse function $f$ as above is only
defined if the critical points and critical values of $f$ are non-degenerate.
Since the  space of these functions is disconnected it is not always
possible to deform the presentation of one function into the presentation
given by another function. Nevertheless, if we admit also functions of
codimension one we obtain again a connected space. 
\s

The content of Cerf's theory [Ce] is to determine the connectivity of function
spaces depending on their codimensions,  and the behavior of the singularities  
once a path of function passes through a lower stratum. In this section we
shall review the elements that are relevant for our purposes. We start by
introducing the  function spaces, that will describe the bridged link
presentations and its moves.
\s


{ \em 2.1.1) Excellent and Codimension One Functions}\label{pg-211}
\s


For given $W$ and boundary function $\h$ introduce
 a Riemannian structure such that the gradient flow of $\h$ is parallel to
the boundary piece $\psi\Bigl([0,1]\times \Sigma\Bigr)\,$. As usual we
introduce the space ${\cal F}$ of smooth functions on $W$ which coincide
with  $\h$ near $\partial W\,$. Also, we denote by 
${\cal F}^0$ the ``excellent functions'', which are Morse functions with
distinct critical values, and by ${\cal F}^1\,$ the codimension one functions.
The latter set is the union  of  the set ${\cal F}^1_{\alpha}$ of functions
with distinct 
critical values and only one degenerate critical point, which is a birth
(or death) point, and the set ${\cal F}^1_{\beta}\,$  of Morse functions 
for which  exactly two critical values coincide.

We denote the
stratification of codimension one:

$
\hphantom{xxxxxxxxxxxxxxxxxxxxx}\f_T=\fo\cup\fla\cup\flb
$

\noindent
We also denote by 

$
\hphantom{xxxxxxxxxxxxxxxxxxxxx}p\f\subset\f\qquad\qquad p\fo\subset\fo\qquad {\rm etc.}
$ 

the subspaces of functions for which all  singularities have 
index one or two, all index two singularities have higher values than
the index one singularities, and all singularities are in general position.  

Correspondingly, the codimension one strata in there are denoted

$
\hphantom{xxxxxxxxxxxxxx}\;p\fla,\quad\;p\flb\,,\quad\;p\fo_i\qquad {i=1,2}\qquad\;p\fo_{12}\;.
$

They are the functions with one 1-2 birth point, functions with exactly two critical
values coinciding, excellent functions for which the ascending and
descending manifolds of two index $i$ singularities
intersect in a single trajectory, and functions where the descending
manifold of a 2-singularity and the ascending manifold of a 1-singularity
intersect with an intermediate level three fold in a curve and a sphere that
are tangential to each other in exactly one point ( and transversal in all
other). 

Their union with $p\fo$ is denoted
$p\f_T\,$. 

It is occasionally useful to label the singularities of index
one (three) as to  which components of the surgered manifold are connected 
by the respective one handles.

\begin{defi}
\ben


\item 
The {\em  label}  of a critical point $c$ of index one (three) is the pair
$(\lambda,\mu)$ of connected 
components of $\h^{-1}(0)\,$ which are intersected by the descending
(ascending) manifold of $c$. 
\item For a function $f\in p\fo\,$ we denote by $\breve M$ the level
manifold $f^{-1}(y)\,$, where $\,y\,$ lies above all critical values of  index
one and below all critical values of index two.

\een  
\end{defi}

It is clear that $\breve M$ can be obtained from $\H$ by either connecting
different component to each other or to copies of $S^1\times S^2\,$.
In fact the connection prescription may be entirely determined from the set
of labels of the index one singularities. 


\s


{\em 2.1.2) Paths and Deformations of Paths of Functions}\label{pg-212}
\s

A basic result in [Ce] asserts that the space $\f_T$ is 0-connected and any
path in $\f_T$ can be deformed to a path which is transversal to the lower
dimensional strata $\fl\,$.  By the general  theory of stable manifolds, a
 presentation changes along a path in $\fo$
only by an isotopy.
 A complete
set of  ``moves'' for presentations on the same four fold are  thus given
by the (transversal) passages  through $\fl\,$.

 A generic path of functions in $\f_T$ is conveniently  illustrated by a graphic.
This is a collection of paths $\bigr(f_t(c^i_t),t\bigl)\,$ where
$c^i_t$ is a continuous family of critical points of $f_t$. 
The point where $f_t$ passes through
$\fla$ is given by a beak joining an index $i$ and an index $i+1$ critical
point. Going through $\flb$ corresponds to crossing of two components of the
graphic. Also, we draw a dashed line at points of $t$ where the ascending
and the descending manifolds of two critical points of the same index
intersect each other in a trajectory.

In particular, a path through the codimension one strata of $p\f_T$ is
presented by the pieces of a graphic given in~\ref{fig-codone}.


\begin{figure}[ht]
\begin{center}\ 

\epsfbox{pics/2-2.eps}
\end{center}

\caption{Codimesion-One Strata in Graphics}
\label{fig-codone}

\end{figure}

In the graphic for the intersection of the manifolds of index one critical
points we also indicated the  possible labels.
\medskip

 
The next more general result in [Ce] concerns  deformations of generic
paths. They may always be chosen such  that the paths pass transversally
through singularities of 
codimensions two. For more general  
results on functions with framings and a useful and humorous summary of
singularities of higher codimensions see [Ig]. The list of elementary
deformations can be  illustrated by graphics as in [Ce], 
[Ki], and [HW, pg26]:

\ben
\item {\em Independent Trajectories}: If the descending manifolds of a  trajectory
$c_t$ of singularities do not intersect the ascending manifolds of another
trajectory $c'_t$ for any $t\,$ then the two trajectories can be moved
independently from each other. (e.g., [HW,pg.65]). For the relevant 
examples see [Ki].
\medskip
  
\item {\em Triangle Lemma} : For $i_1+i_3\leq 3\,$, $inf(i_1,i_3)\leq i_2-1\,$ or 
$i_1=i_2=i_3\leq 2\,$ we have the move:

\begin{center}\ 

\epsfbox{pics/2-tria.eps}
\end{center}

\bigskip

\bigskip\ 

\item {\em  Beak Lemma}: If $i>0$ we have the move
\begin{center}\ 

\epsfbox{pics/2-beak.eps}
\end{center}


\item {\em  Dovetail Lemma}:


\begin{center}\ 

\epsfbox{pics/2-dove.eps}
\end{center}

\een
 

The remaining moves are obtained from the above by reflections $t\to 1-t$
and $f\to 1- f\,$.  


\subparagraph{2.2) Bridged Link Presentations}\label{pg-22}
\
\s

We introduce an enhanced presentation of a three fold by adding both
one and two handles to another compact, oriented three fold, which does 
not have to be connected or simply connected. 
\s 


{\em 2.2.1) Surgery Presentation from Bridged Links}\label{pg-221}\nopagebreak
\s

From the structure of the singularities of a  function $f\in \fo$ we obtain
 a
presentation of the cobordism $\h^{-1}(1)$ by considering the intersections 
($\cong S^0,{\rm or }\,\cong S^1$) of the descending (or unstable) manifolds of
the critical points with 
$\h^{-1}(0)\,$. These yield together with framings of their normal bundles
a unique surgery prescription. To be more precise, let us introduce the
following standard handles:
\s  

{\it 1)A two handle $\hh^2\,=\,D^2_a\times D^2_b$ and two intervals $I,J\cong [-1,1]$
such that $D^2_a=J\times I\,$.
\s

2)A 2-sphere $S^2\subset {\bf R}^3$ with meridian $S^1=\{(x,y,0)\in S^2\}$
and involution $\rho:S^2\to S^2\,:\,(x,y,z)\to(x,y,-z)\,$.
Also denote by $B^3$ the oriented ball bounded by $S^2$ and by $D^2_{\pm}$
the upper and the lower hemisphere.} 
 \s

 On $\H=\H(\bar m^+,l,\bar m^-)$ for arbirary $l$
and a function $f\in p\fo $  a {\em bridged link presentation}
consists of the following data:

\s

{\it 1) An orientation preserving embedding $\phi=\bigl(\phi_1,\phi_1',\ldots,\phi_m'\bigr)$ 
of $2m$ copies of $B^3$.}

\s

The pairs of balls are neighborhoods of the points where the descending
trajectories of the index one singularities intersect $\H$. 
The three fold $\breve M$ just above
all index one singularities is then (for a suitable given Riemann structure)
 naturally
homeomorphic to the manifold obtained by removing the interiors of the
images of the three balls and identifying the pairs of boundaries using the
isomorphisms
$$
\bar\phi_j :=\phi_j'\circ \rho \circ \phi_j^{-1}\;.
$$

Occasionally, we shall prefer the picture of an actual handle attachement,
where we glue in an additional $S^2\times [0,1]$ in between the boundary components. 
\s

The surgery is described further by considering the intersections of the
descending manifolds of the index two singularities with $\breve M\,$. Close
to the singularity, the normal (Morse) form of $f$ determines - up to
homotopy -
a trivialization of the normal bundle. This, in turn, is determined by a
non vanishing section of the normal bundle in any level manifold, i.e., an
embedding of the ribbon: 
$
R:=J\times \partial D^2_b \,\hookrightarrow \breve M\;, 
$
or more precisely of a link of ribbons. Undoing the index one surgery we
obtain the next ingredient of the surgery data:
\s
     
{\it 2) A map $\tau=(\tau_1,\ldots,\tau_s)\,:\,R\amalg\ldots\amalg R\,
\hookrightarrow \,\H$ of $s$ copies of the ribbon $R$ which is an embedding
into the complement of the balls $im(\phi)$ except for a finite number of
intervals 
$J\times \{p\}\subset R$. The right and
left sided limits $\tau_j^{\pm}\,:\,J\times \{p\}\to \H$ are embeddings
into  spheres $\phi_{k(p)}(S^2)$ and $\phi_{k(p)}'(S^2)$ such that}

$$
\bar \phi_k\tau_j^{\pm}=\tau^{\mp}_j\;.
$$
\s

 Since we will be dealing with non-simply connected manifolds (with
no canonical framings of tangent bundles over the one-skeletons) we 
preferred here the language of ribbons over that of framing numbers. 

A typical picture of the ``bridged link'' is given in Figure~\ref{fig-typp}. The
action of the flip $\rho$ is indicated by the invariant
meridians, drawn here as dashed lines. 


\begin{figure}[ht]
\begin{center}\ 
\epsfxsize=5.5in
\epsfbox{pics/2-3.eps}
\end{center}


\caption{Bridged Link}\label{fig-typp}
\end{figure}


It should be kept in mind  that the two spheres may lie in different
components of $\H$ depending on the label of the critical point, and 
are thus not connected by any cancelling ribbon (as in move 4) of the next
section).

The surgery on a component of the resulting ribbon link in $\breve M$ is given as usual by
extending the embedding to the tubes $D^2_a\times \partial D^2_b\supset
J\times 0\times \partial D^2_b\,$. Then we remove the images and glue in 
$\partial D^2_a\times D^2_b$ with reversed orientation. 
\s


{\em 2.2.2) Moves for Bridged Links}\label{pg-222}
\
\s

In this section we compile a list of ``moves'' of bridged link
presentations that do not change the homeomorphy class of the 
presented three fold. Some of them come from deforming the Morse
function through a codimension one stratum, some are obtained by
modification of the bounding four fold. 
\s


1)\ub{\em Isotopies} An isotopy is a superposition of isotopies
of the three balls in $\H$ and of the ribbon link in $\breve M$. 
More precisely, after identifying a pair of embedded
spheres and their vicinity via the maps $\phi$ and $\phi'$ 
with the respective standard regions in $\bf R^3\,$, we may consider a pair of regions
$F\subset S^2$ and $F'=\rho(F)\subset {S^2}'\,$. A collar
$(U,F)\cong(F\times I,F\times 0)$ (opposite to the bounded balls) of one region,
can be moved
continuously to the other ending with the isomorphism:
$$
F\times I\,\stackrel{\rho\times (1-t)}{\hbox to 1.5cm{\rightarrowfill}}
\,{F'\times I}\;.
$$
Extending this isotopy (or its inverse) to the identity outside of $F$ and $F'\,$, we
can move pieces of the link diagram through a
gluing sphere as indicated in Figure~\ref{fig-isot}.

We confined generic presentations to have  to 1- and 2-singularities
in general positions; yet,  an isotopy over a surgery sphere also involves situations
where the surgery spheres are not always transversal to the ribbons.
At these points we can deform our path such that we pass only through the
stratum $p\fo_{12}\,$. This means only one ribbon is tangential to the
sphere at a time, and this configuration results from pushing a small 
ribbon loop through the sphere.  


\begin{figure}[ht]

\begin{center}\ 
\epsfxsize=5.5in
\epsfbox{pics/2-4.eps}
\end{center}

\caption{Isotopy over Surgery Sphere}\label{fig-isot}
\end{figure}
\s

2) \ub{\em One- (Handle) Slides} They occur when  the Morse function passes
through $\fo_1$ and the  ascending manifold of an
index one singularity $Lo$ is moved transversally through one of the 
descending trajectories
of another index  one singularity, $Hi\,$, with higher value. Considering a
level manifold in between the two singularities it is clear that this 
corresponds
to moving a ball with the attached ribbons through a gluing sphere, as 
was described above for links.   
A  typical situation is suggested in Figure~\ref{fig-1slid}.

\begin{figure}[ht]
\begin{center}\ 
\epsfxsize=5.3in
\epsfbox{pics/2-5.eps}
\end{center}


\caption{One Slide}\label{fig-1slid}
\end{figure}

\s

3) \ub{\em Two- (Handle) Slides} This is the Kirby move, denoted ${\cal O}_2$ in
[Ki]. It occurs when the path of  
Morse functions passes through $\fo_2\,$, and the 
intermediate manifolds of
the two index two singularities are moved through each other transversally.
At the point where the descending two fold of the singularity $Hi\,$ 
with higher
value goes through the other critical point $Lo$ we first have the 
corresponding
ribbons tangential along small pieces of their  boundaries. To describe the result
of the slide, the ribbon of $Lo$
is cut along its middle and the closer part is connected to the ribbon 
$Hi$ as
depicted  in Figure~\ref{fig-2slid}. On an intermediate level manifold this
is just an isotopy of the ribbon $Hi$ over the surgered piece along $Lo\,$.

\begin{figure}[ht]
\begin{center}\ 
\epsfxsize=5.3in
\epsfbox{pics/2-6.eps}
\end{center}


\caption{Two Slide}
\label{fig-2slid}
\end{figure}

\s

4) \ub{\em 1-2- (Handle) Cancellation} 
 The Cancellation Lemma of Smale (see, e.g. [HW] pg. 172) states that 
when for consecutive  singularities $c_1$ and $c_2$ 
of index $i$ and $i+1$ the intersection
of the descending  manifold of $c_2$ intersects the ascending manifold of
$c_1$ in exactly one trajectory, then $c_1$ and $c_2$ can be canceled 
against each other.
 This is achieved  along a path of codimension one functions passing
through $\fla$ ,i.e., an $i\,-\,i+1\,-$ birth- or death-point.
\s

Here we are only interested in the situation where $f$ passes through 
$p\fla\,$, and where the singularities are presented by a pair of balls $c_1$
 and a ribbon $c_2$ intersecting the
gluing sphere of $c_1$ in exactly one interval but no other spheres. In
this
situation we remove in addition to the balls a solid tube $D^2\times I$
around $c_2$ and
collapse the opposite hemispheres of the total removed region using $\rho$
and the framing of $c_2$, see Figure~\ref{fig-canc}.

\begin{figure}[ht]

\begin{center}\ 

\epsfbox{pics/2-7.eps}
\end{center}


\caption{1-2-Cancellation}
\label{fig-canc}
\end{figure}


In order to explicitly show that this move preserves  the surgered manifold
we perform the
one surgery,
where we already removed the solid torus of the two surgery. This
corresponds to identifying the remaining hemispheres, which leaves us with 
Figure~\ref{fig-canc-pr}, where the torus is empty. Completing the two
surgery we obtain the right hand side of 
Figure~\ref{fig-canc}.

\begin{figure}[ht]

\begin{center}\ 

\epsfbox{pics/2-8.eps}
\end{center}


\caption{Cancellation Surgery}
\label{fig-canc-pr}

\end{figure}


  
\s
 
5) \ub{\em  Modification (Handle Trading)} This  move  also changes the
bounding four manifold, exchanging an
index one singularity (or handle) for an index two singularity. Suppose we 
have a ribbon-component $R=S^1\times [1,2]$ which can be extended to an
embedding of a disc $D=S^1\times [2,0]/S^1\times 0$, (which may intersect
other ribbons-components nicely). Then we duplicate $D$ and push the two 
copies away from each other. We complete each of the discs to a sphere, so
that possible ribbons are attached to the outside, and define the maps
$\phi$ and $\phi'$ such that $\bar\phi$ is the correct identification of
the hemispheres $D$, see Figure~\ref{fig-modi}.

\begin{figure}[ht]

\begin{center}\ 

\epsfbox{pics/2-9.eps}
\end{center}


\caption{Modification}
\label{fig-modi}
\end{figure}


The procedure of modification is described in [Wa]. To see  that
the identification of the hemispheres is the correct one we shall make the
argument explicit: We parametrize a tubular neighborhood of $R$ by
$J\times I \times \partial D_b^2$ and the thickened, bounded disc by
$J\times D_c^2$, with identification $\partial D^2_b\,=\,\partial D^2_c$.
Instead of removing only the solid torus we also take out the thickened disc and
glue it back later. The opposite torus $\partial D_a\times D_b\,$ consists of
two parts $J\times 0\times D_b^2$ and $Z\times D_b^2$ with
$Z=\bigl(\partial D_a^2-J\times 0\bigr)$, which are identified 
along two copies of $D_b\,$. If we glue  the second part into the empty
region along $Z\times \partial D_b^2\,$, we obtain th region in
Figure~\ref{fig-modi-pr} where the insides of the two spheres $D^2_b\cup
D^2_c$ are missing.

\begin{figure}[ht]

\begin{center}\ 
\epsfxsize=5.3in
\epsfbox{pics/2-10.eps}
\end{center}


\caption{Modification Surgery}
\label{fig-modi-pr}
\end{figure}

The other part, $J\times 0\times D_b^2$  is first glued to the extra piece  $J\times
D^2_c$ along the cylinder $J\times \partial D^2$ giving $J\times S^2$ and
the included ribbons are $J$ times an interval on $S^2$. 
Shrinking $J$ to a point, it is clear from the picture 
that we obtain exactly the desired one surgery.


\medskip

To describe equivalences of cobordisms we need to include two more moves,
which change the signature of $W$ and the 
number of components  of $\H$:

\medskip

6)\ub{\em Signature} Add an unknotted, ribbon separated from the rest by a
sphere  with framing $\pm 1$. This is the Kirby move ${\cal O}_1$ and
results from 
connecting a projective space ${\bf CP}^2$ ($\overline{{\bf CP}^2}\,$) to $W\,$. We sometimes denote
such a link component by $\FLO\,$ ($\frL\,$).

\medskip

7)\ub{\em 0-1-Cancellation} If for a bridged link presentation an $S^3$
component of $\H$ 
contains nothing but a single surgery sphere we may omit the $S^3$-component,
and the respective pair of surgery spheres. (The content of the presentation in
this  situation is to connect the $S^3$ component to another component which is
of course trivial.)  

\medskip

Finally, let us introduce a move which is a combination of the previous
ones:
\medskip

8)\ub{\em $\eta$-move, $^0\bigcirc\mkern-13mu\bigcirc^0$-move} 
  An $\eta$-move is the addition  or subtraction of  a pair
of ribbons $\,^0\bigcirc\mkern-15mu\subset^{\cdots}_{\cdots}\,$ where one
ribbon extends to a disc which is intersected exactly 
once by the second ribbon $\cal C$ but no other ribbon of the presentation.
As described in the proof of Proposition~\ref{prop-conn} this corresponds
to the two-surgery $\,W\,\to\,(W-S^1\times D^3)\cup D^2\times S^2\,$.

When the ribbon $\cal C$ can also be contracted to an annulus, and the pair
of ribbons can be isolated from the rest,  we call the associated $\eta$-move an
$^0\bigcirc\mkern-13mu\bigcirc^0$-move. It corresponds to the surgery $W\to
W\# S^2\times S^2\,$. Using two slides shows that
$^0\bigcirc\mkern-13mu\bigcirc^0= ^0\bigcirc\mkern-13mu\bigcirc^{2n}\,$.
Thus, we shall tacitly assume that the
$^0\bigcirc\mkern-13mu\bigcirc^0\,$-move also implies addition and removal
of an isolated  $^0\bigcirc\mkern-13mu\bigcirc^1\,=\,\FLO\cup\frL\,$-piece. This corresponds
to connected summing with the non-orientable $S^2$-bundle $S^2\tilde\times S^2\,$.

 
 


\subparagraph{2.3) Equivalences of Bridged Links}\label{pg-23}
\
\s


{\em 2.3.1) Completeness of Moves for Bridged Links}\label{pg-231}
\s


A surgery presentation of a three manifold ( e.g., a morphism of $\wct\,$),
may be changed by either changing the bounding four fold $W$ within its
 4+1-cobordism class or by changing the handle decomposition of $W\,$.
 As indicated in the beginning of this chapter it is equivalent to speak
of excellent functions on $W$ instead of handles so that we may use the
results of Cerf's Theory from Section 2.1) and the correspondence between
codimension one strata of $p\f_T\,$ and the handle slides cancellations
given in Section 2.2.2). We know that any two excellent functions in
$p\f_T\,$ can be connected by a path in $\f_T\,$. Yet, this space is too large as it
contains singularities of all indices so that we have to deal with all
types of handles. For moves between bridged links we are only interested in
decompositions of the form (\ref{eq-PPP}), which is equivalent to
considering only functions in $p\f_0\subset p\f_T\,$. We need the following
refinement of the connectivity result from [Ce].
\medskip

\blm\label{lm-12-cerf}
The space $p\f_T\,$ is 0-connected.
\elm

{\em Proof :} The elimination of components of a graphic with index zero and
index four  can be literarily taken from [Ki]. The trajectories of index three
singularities start and end in birth - and death - points, and, similarly as for
index one trajectories in [Ki], can be rearranged as disjoint trajectories
as indicated in first
graphic of Figure~\ref{fig-cerf-pf}, using the Beak Lemma. 
Introducing a double Dovetail we find the second graphic, and another
application of the Beak Lemma and the Principle of Independent Trajectories
yields the third picture. Removing the remaining Dovetails we find a graphic 
which contains only index one and two singularities. By the Principle of
Independent Trajectories we can move all index one trajectories below the
index two trajectories. We thus end up with a path in $\f_T\,$.
\hfill$\Box\,$.
\s
 
\begin{figure}[ht]

\begin{center}\ 

\epsfbox{pics/2-11.eps}
\end{center}


\caption{Elimination of Three Handles}
\label{fig-cerf-pf}
\end{figure}

Put in a different language, Lemma~\ref{lm-12-cerf} states that all
handle decomposition of a four fold of the form~(\ref{eq-PPP}) can be
related to each other by isotopies, 1- or 2-slides, and 1-2-cancellations.
If we combine this result with Proposition~\ref{prop-conn} we arrive at the
following completeness of moves:


\begin{cor}\label{co-compl} If two bridged links on $\H$ present the same manifold with
boundary, then they 
are related by a sequence of the moves {\em 1)- 8)} from Section 2.2.2).
\end{cor}
\medskip


Note, that the modification has been added to the list, even though it
was not used in the preceding arguments. 

In fact it can be expressed as 
an $\eta$-move, where the long ribbon $\cal C$ is chosen as
a connecting strip between the two spheres, followed by a cancellation
along $\cal C\,$.
\s


{\em 2.3.2) Reduction of Moves}\label{pg-232}
\s

It turns out that the complete set of relations between bridged links, as listed in
Corollary~\ref{co-compl}, is highly redundant. It is not hard to imagine
that there are many possibilities to select a minimal subset of moves from
it, which generates all other moves and equivalences.
 Let us assume that the ribbons passing through a general surgery sphere
penetrate at given intervals $I_1\ldots I_k\ldots\,$. 
Then the set of moves that has been proposed in Theorem~\ref{TWO} of the
introduction follows from the following reduction:
\medskip


\bpp\label{pp-quint-mv}

Two bridged link presentations yield the same three manifold if they are related 
by the following moves:
\ben
\item  {\rm Isotopies} Aside from general isotopies of surgery spheres and 
ribbons, which are constant along at the intervals $I_j$, we have isotopies
where ribbon pieces are pushed through surgery spheres of different
components. These can be written as combinations of

a) Opposite actions of the braid group of the two spheres.

b) An untangled, unbraided loop being pushed through. 

\item {\rm One Slides} We may assume that the $Hi$ -sphere is pushed
through $Lo$ first, followed by the attached ribbons. More precisely, before the
move we have ribbons $r_1\ldots r_m$ going through $I_1\ldots I_m$ of $Lo$
and ribbons $r_{m+1}\ldots r_{n+m}$ going through $I_1\ldots I_n$ of $Hi\,$.
After the move we have $r_1\ldots r_{n+m}$ going through $I_1\ldots
I_{n+m}$ of {\rm Lo}. Also, we only consider one slides where the two $Lo\,$
spheres are in different components. 

\item void

\item {\rm Cancellation} We only consider cancellation with no other
ribbons but the cancelling one attached. 

\item {\rm Modification} As stated.

\item {\rm Signature} As stated.

\item void

\item void

\een

\epp

{\em Proof:} An isotopy over a surgery sphere can be deformed into an isotopy
where the strands in a vicinity of the sphere are always radial, except for
passages through the stratum $p\fo_{12}\,$. The latter give rise to the
case {\em 1.b)}. The ambiguity of pushing the strands into the prescribed
positions $I_j\,$ are given by elements of the braid group as described in
{\em 1.a)} and $2\pi$-twists of the strands, which can be reexpressed as the
combination of two loops and one braid.  


In order to verify 2)
we deform a one slide  such that first the $Hi$-sphere is pushed through
$Lo$, (the attached ribbons going through $Lo$ will then be loops at $Lo$,)
and then the ribbons at $Hi$. Conjugating everything with isotopies at $Lo$
we can confine ourselves to the situation where the ribbons are ordered as
described.  
 
If we have the general situation of a cancellation as in
Figure~\ref{fig-canc} we may push the cancelling ribbon $c_2$ to the side
an use modifications as in Figure~\ref{fig-canc-red} to reduce the
cancellation to an isolated one. The reduction can also be found by using
two slides. Specifically,  we may separate one penetrating ribbon after the other
from the surgery spheres by sliding it over the extra cancelling ribbon.
   
\begin{figure}[ht]

\begin{center}\ 

\epsfxsize=5.7in
\epsfbox{pics/2-12.eps}
\end{center}


\caption{Specialization of Cancellation}
\label{fig-canc-red}
\end{figure}

The two-slides can be expressed in terms of cancellations. This is shown in
Figure~\ref{fig-2slide-repl} where we  move the surgery sphere $A$  along 
the $Lo$ ribbon of  the two-slide.
 
\begin{figure}[ht]

\begin{center}\ 
\epsfxsize=5.7in
\epsfbox{pics/2-13.eps}
\end{center}


\caption{Elimination of Two-Slides}
\label{fig-2slide-repl}
\end{figure}

Move 6) is omitted if we are only interested in cobordisms of $\wct\,$.
It is easy to see that the removal of $\,^0\bigcirc\mkern-15mu\subset^{\cdots}_{\cdots}\,$
as in move 8) can also be given by a modification at the annulus followed by a
1-2-cancellation along the ribbon $\cal C\,$. 
Finally, if we have a pair of surgery  spheres in the same component we may 
move them close to each other and replace the moves {\em 1a), 1b)} and {\em
2)} by ordinary isotopies through an annulus, which are  conjugated by a modification.  
\hfill$\Box$
\s

The following, alternative  set of generating moves among ordinary links constitues the
proper generalization of Kirby's original calculus to non-simply connected
manifolds:


 
\bpp\label{pp-red1}
Suppose for two ordinary link presentations on a connected not necessarily
simply connected manifold $M_0\,$ the surgered manifolds are homeomorphic.


\noindent
Then we can relate the
presentations by a sequence of the following moves
\begin{enumerate}
\item Isotopies
\item ${\cal O}_2$-moves 
\item $\eta$-move
\item ${\cal O}_1$-moves\quad.
\end{enumerate}
If we consider only cobordisms in $\wct$ the ${\cal O}_1$-move is again
omitted.
\medskip

\noindent
If $M_0\,$ is also simply connected we may replace the $\eta$-move by the 
$^0\bigcirc\mkern-13mu\bigcirc^0$-move.

\epp

Proposition~\ref{pp-red1} can be directly derived from
Proposition~\ref{pp-quint-mv}, thus yielding a more structured proof of
Kirby's original theorem. The strategy is to consider the formal spaces of
bridged links $\bigl({\cal BL}\bigr)$ and ordinary links $\bigl({\cal
L}\bigr)$ in $M_0\,$ and define the quotient space $\overline{\bigl({\cal BL}\bigr)}$     
 and $\overline{\bigl({\cal L}\bigr)}$, where the relations from the two
previous Propositions have been imposed. We certainly have an inclusion
$\i\,:\,\bigl({\cal L}\bigr)\hookrightarrow \bigl({\cal BL}\bigr)\,$
and it is implied in the proof of Proposition~\ref{pp-quint-mv} that this
factors into a map $\bar{\i}\,:\,\overline{\bigl({\cal
L}\bigr)}\to\overline{\bigl({\cal BL}\bigr)}\,$. If we choose a path
between associated surgery spheres for each bridged link we can define a
map $p\,:\,\bigl({\cal BL}\bigr)\to \bigl({\cal L}\bigr)\,$ by moving
the spheres close to each other along the paths and elimnating them with 
a modification. What needs to be established is that $p\,$ factors into a
map $\bar p\,:\,\overline{\bigl({\cal BL}\bigr)}\to\overline{\bigl({\cal
L}\bigr)}\,$, which is independent of the choice of the recombination 
paths. 
It is then obvious that $\bar p$ and $\bar{\i}\,$ are inverses of each
other so that the two link-calculi are equivalent on connected manifolds.
A detailled discussion will be given in a separate paper, [Ke3].$\smile$
\medskip

In this context the interpretation of the $\eta$-move is to compensate
for the recombination ambiguity of the isolated cancellation diagram.
For the same reason the $\eta$-move was introduced in [Ki], when
trajectories of one-handles that started and ended in different birth and
death points had to be eliminated. The graphic of the function is 
changed as indicated in Figure~\ref{fig-graca}, where the vertical lines
indicate modifications of the fourfold.   
If the recombination is, e.g., done
along the trajectories ending at the birth point, the first modification is
an addition of a $^0\bigcirc\mkern-13mu\bigcirc^0$-element, and the second
a subtraction of an $\eta$-configuration, where $\cal C$ runs along the
trajectories of the surgery spheres.     

\begin{figure}[ht]

\begin{center}\ 

\epsfbox{pics/2-14.eps}
\end{center}


\caption{Elimination of One-Handles}
\label{fig-graca}
\end{figure}

\medskip

The fact that for simply connected manifolds $M_0\,$ a general $\eta$-move is the
composite of ${\cal O}_2$-moves and a
$^0\bigcirc\mkern-13mu\bigcirc^0$-move has been proven in [Ki]. A slightly
more direct argument starts from a general PL-homotopy of the curve $\cal
C\,$. This is made transversal, such that only at a finite number of times
there may be single crossings of $\cal C$ with another segment $\cal X$, which
can be a piece of $\cal C$ it self or another ribbon. If we move the
annulus aroud $\cal C$ to this crossing it can be substituted by a
two-slide of $\cal X$ over the annulus. For the change of framing of $\cal C$,
but also the reexpression of the $^0\bigcirc\mkern-13mu\bigcirc^0$-move,
using auxiliary $\FLO$'s and ${\cal O}_1$-moves see [Ki].
 
Finally, note that the reduction of the $\eta$-move can also be taken from
its five dimensional interpretation. In particular, it is clear that the
attaching curve $\cal C\,$ for five dimensional 2-handle can be contracted inside
of $M_0\times I\cup\hh^2\cup ... \cup\hh^2\,$.   


\subparagraph{2.4) Bridged Ribbon Graphs}\label{pg-24}
\
\s

The construction of the invariants of [RT] from a link presentation is
preceded by a reduction of the presentation to spaces of so called
ribbon graphs, which are generic projections of the links into ${\bf R}^2\,$.
 
In this paragraph we shall describe the analogous space of {\em bridged} ribbon
graphs, which will contain not only ribbons but also special pairs of
coupons. The relations we will impose on this space are the ones that
appear  in Proposition~\ref{pp-quint-mv} and the usual  relations that take
care of isotopies and the ambiguity of choosing a projection.  


We confine ourselves to  presentation in a union of $K$ handlebodies $H_g$
or spheres $S^3\,$. They are given by the union  
of $K$ diagrams of bridged  ribbon graphs in ${\bf R}^2\,$. 
Clearly, the interior of $H_g$ is homeomorphic to
${\bf R}^3$ with $g$ strands, extending to $\infty$, removed. 
Thus in a projection of a ribbon diagram into ${\bf R}^2$ we present the
strands as infinite, parallel strips in vertical direction. The pieces 
of an ordinary  directed ribbon graph as in [RT0] are then given by the
elements depicted  in Figure~\ref{fig-cdr-elem}. 


\begin{figure}[ht]

\begin{center}\ 

\epsfbox{pics/2-15.eps}
\end{center}


\caption{Ribbon Segments}
\label{fig-cdr-elem}


\begin{center}\ 

\epsfbox{pics/2-16.eps}
\end{center}


\caption{Pairs of Coupons}
\label{fig-br-elem}
\end{figure}

Here the black strands are the ones representing the holes in $H_g$.
In a bridged ribbon graph we have in addition pairs of coupons, one with strands 
going up (labeled $(j,+)$) and one with the same number of strands going down.
The two coupons, as in Figure~\ref{fig-br-elem}, live  in different
${\bf R}^2$'s of the K-component  ribbon-graph, if they present surgery
spheres in different handlebodies. 


We have the usual relations for
projections of ribbon graphs, e.g., $Rel_1-Rel_{10}$ of [RT0].
In these relations we also include auxiliary black strands, whenever they
make sense. 
Isotopies of the gluing sphere lead to the moves $Rel_{11}-Rel_{13}\,$.

The last relation also takes care of the ambiguity of identifying a surgery
sphere with a $+$ or a $-$ coupon.
It is  understood that the upside down versions of these pictures are
also included in the set of moves. 
\medskip

Next, we describe the $1a)$ and $1b)$ Isotopies, which are also used to
resolve the ambiguity of moving general strands into the standard positions
$I_1\ldots I_n\,$. The moves are given by the pictures for
$Rel_{14}-Rel_{15}$ and their mirror images.  


The reduced one slide is given by  $Rel_{16}$ and reflections. The
modification and the  
reduced cancellation are given by $Rel_{17}-Rel_{18}$. We remind ourselves
that $Rel_{17}$ does not imply the relations $Rel_{14}$ and $Rel_{15}\,$, since
the coupons may be in different components. 


Finally, we have the signature move $Rel_{19}$, which is the inclusion of an
isolated, unknotted ribbon $\FLO$ with one twist.  


\nopagebreak
\begin{figure}[ht]

\begin{center}\ 
\epsfysize=1.2in
\epsfbox{pics/2-17.eps}
\end{center}


\caption{$Rel_{11}$}
\label{fig-rel-11}

\end{figure}

\begin{figure}[ht]

\begin{center}\ 

\epsfysize=.9in
\epsfbox{pics/2-18.eps}
\end{center}


\caption{$Rel_{12}$}
\label{fig-rel-12}
\end{figure}
\begin{figure}[hb]

\begin{center}\ 
\epsfxsize=5.2in
\epsfbox{pics/2-19.eps}
\end{center}


\caption{$Rel_{13}$}
\label{fig-rel-13}
\end{figure}
\begin{figure}[ht]

\begin{center}\ 

\epsfbox{pics/2-20.eps}
\end{center}


\caption{$Rel_{14}$}
\label{fig-rel-14}
\end{figure}


\begin{figure}[hb]

\begin{center}\ 

\epsfbox{pics/2-21.eps}
\end{center}


\caption{$Rel_{15}$}
\label{fig-rel-15}
\end{figure}
\begin{figure}[ht]

\begin{center}\ 

\epsfbox{pics/2-22.eps}
\end{center}


\caption{$Rel_{16}$}
\label{fig-rel-16}
\end{figure}

 
\begin{figure}[ht]

\begin{center}\ 

\epsfbox{pics/2-23.eps}
\end{center}


\caption{$Rel_{17}\qquad
Rel_{18}$}
\label{fig-rel-17} 
\end{figure}


\parbox{6in}{
\bpp \label{pp-BL-moves}
\
\nopagebreak
\ben
\item Two directed, bridged, ribbon graphs are projections of 
bridged link presentations of the same manifold if and only if they are
related by relations $Rel_1-Rel_{19}$.

\item If the coupons $(j,\pm)$ lie in the same component of the presentation the
moves $Rel_{13}, \,Rel_{14}, \,Rel_{15}\,$, and $ Rel_{16}$ follow from the
other moves. 
In particular, if the presentation has only one component
$Rel_1-Rel_{12}$ and $Rel_{17}-Rel_{19}\,$ form a complete set of moves
for presentations of  cobordisms in $\ct$.

\item For  presentations of {\em connected} morphisms in $\wct$ we can omit
$Rel_{19}\,$.   
\een  
\epp}
{"config": "arxiv", "file": "math9806114/BL2.tex"}