The intersection of spheres in a sphere and \\ a new geometric meaning of the Arf invariant
Eiji Ogasa
Department of Mathematical Sciences
Komaba
Tokyo 153
JAPAN \thanks{ {1991 Mathematics Subject Classification.} Primary 57M25, 57Q45 \nl Keywords: the Arf invariant, spin cobordism group, pass-moves, realizable pair of links, surgery of submanifolds \nl This research was partially supported by Research Fellowships of the Promotion of Science for Young Scientists.}
math0003089.tex
%The intersection of spheres in a sphere and 
%a new geometric meaning of the Arf invariant

%%2000 March ni xxx.lanl.gov ni okuru 
%University of Tokyo Preprint sereis,   UTMS 95-7
%57M25, 57Q45 
 
%Let S^3_i be a 3-sphere embedded in the 5-sphere S^5 (i=1,2). 
%Let S^3_1 and S^3_2 intersect transversely. 
%Then the intersection C of S^3_1 and S^3_2 
%is a disjoint collection of circles.  
%Thus we obtain a pair of 1-links, C in S^3_i (i=1,2), 
%and a pair of  3-knots, S^3_i in S^5 (i=1,2). 
%Conversely let (L_1,L_2) be a pair of 1-links and 
%(X_1,X_2) be a pair of 3-knots. 
%It is natural to ask whether the pair of 1-links (L_1,L_2) 
%is obtained as the intersection of the 3-knots X_1 and X_2 as above. 
%We give a complete answer to this question. 
%Our answer gives a new geometric meaning of the Arf invariant of 1-links. 


%Let f be a smooth transverse immersion S^3 into S^5.  
%Then the self-intersection  C consists of double points.  
%Suppose that C is a single circle in S^5. 
%Then f^{-1}(C)  in S^3 is a 1-knot or a 2-component 1-link.   
%There is a similar realization problem. 
%We give a complete answer to this question. 





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%%%%%%    TEXT START    %%%%%%
\begin{document}


\title{
The intersection of spheres in a sphere and \\ 
a new geometric meaning of the Arf invariant
}
\author{
 Eiji Ogasa\\
 ogasa@ms.u-tokyo.ac.jp\\
Department of Mathematical Sciences, 
 University of Tokyo\\ 
 Komaba, Tokyo 153, JAPAN  
 \thanks{ 
 {\it 1991 Mathematics Subject Classification.} Primary 57M25, 57Q45 \nl
 Keywords: the Arf invariant, spin cobordism group, pass-moves, 
realizable pair of links, surgery of submanifolds \nl
This research was partially supported by Research Fellowships
of the Promotion of Science for Young Scientists.}
}
\date{}
\maketitle


\noindent{\bf Abstract}

Let $S^3_i$ be a 3-sphere embedded in the 5-sphere $S^5$ ($i=1,2$). 
Let $S^3_1$ and $S^3_2$ intersect transversely. 
Then the intersection $C=S^3_1\cap S^3_2$  
is a disjoint collection of circles.  
Thus we obtain a pair of 1-links,   $C$ 
in $S^3_i$ ($i=1,2$), 
and a pair of  3-knots, $S^3_i$ in $S^5$ ($i=1,2$). 
Conversely let $(L_1,L_2)$ be a pair of 1-links and $(X_1,X_2)$ be a pair 
of 3-knots. 
It is natural to ask whether 
the pair of 1-links $(L_1,L_2)$    %ireta 
is obtained as the intersection of 
the 3-knots $X_1$ and $X_2$  % ireta
as above. 
We give a complete answer to this question. 
Our answer gives a new geometric meaning of the Arf invariant 
of 1-links. 

Let $f:S^3 \longrightarrow S^5$ be a smooth transverse immersion.  
Then the self-intersection  $C$ consists of double points.  
Suppose that $C$ is a single circle in $S^5$. 
Then $f^{-1}(C)$  in $S^3$ is a 1-knot or a 2-component 1-link.   
There is a similar realization problem. 
We give a complete answer to this question. 



\np

\section{  Introduction and Main results } 

Let $S^3_i$ be a 3-sphere embedded in the 5-sphere $S^5$ ($i=1,2$). 
Let $S^3_1$ and $S^3_2$ intersect transversely. 
Then the intersection $C=S^3_1\cap S^3_2$  
is a disjoint collection of circles. 
Then $C$ in $S^3_i$ is a 1-link ($i=1,2$).
Note that the orientation of $C$ is induced by 
that of $S^3_1$ , 
that of $S^3_2$ and that of $S^5$.  
Thus we obtain a pair of 1-links,   $C$ 
in $S^3_i$ ($i=1,2$), 
and a pair of  3-knots, $S^3_i$ in $S^5$ ($i=1,2$). 

Conversely let $(L_1,L_2)$ be a pair of 1-links and $(X_1,X_2)$ be a pair 
of 3-knots. It is natural to ask whether 
 the pair of 1-links $(L_1,L_2)$ % ireta
is obtained as the 
intersection of 
the 3-knots $X_1$ and $X_2$ % ireta 
as above. 
We give a complete answer to this question. (Theorem 1.1.)

To state our results we need some definitions.    
 
An  {\it (oriented) (ordered) $m$-component n-(dimensional) link}
 is a smooth, oriented submanifold $L=\{K_1,...,K_m\}$ of $S^{n+2}$, 
which is the ordered disjoint union of $m$ manifolds, each PL homeomorphic 
to the $n$-sphere.  
If $m=1$, then $L$ is called a {\it knot}.  
(See [1], [2], [8], [9]. )
%(See [CS], [CO], [Ke], [KW]. )
 
We say that n-links $L_1$ and $L_2$ are {\it equivalent} 
if there exists an orientation preserving diffeomorphism 
$f:$ $S^{n+2}$ $\rightarrow$ $S^{n+2}$ 
such that $f(L_1)$=$L_2$  and 
$f\vert_{L_1}:$ $L_1$ $\rightarrow$ $L_2$ is 
an orientation preserving diffeomorphism.    


{\bf Definition  }  
$(L_1, L_2, X_1, X_2)$ is called a 
{\it 4-tuple of links } if the following conditions (1), (2) and (3) hold.


(1)  
  $L_i=(K_{i1},...,K_{im_i})$  is an oriented ordered $m_i$-component 
1-dimensional link  $(i=1,2).$  

(2)  
   $m_1=m_2.$

(3)  
    $X_i$ is a 3-knot. 


{\bf Definition   } 
A 4-tuple of links $(L_1, L_2, X_1, X_2)$ is said to be  {\it realizable } if 
there exists a smooth transverse 
immersion  $f:S^3_1\coprod S^3_2 \longrightarrow S^5$   
with the  following properties.  
We assume that the orientations of $S^3_1, S^3_2$ 
and $S^5$ are given. 


(1)  
 $f\vert S^3_i$ is a smooth embedding. 
  $f(S^3_i)$ in $S^5$ is equivalent to the  3-knot $X_i (i=1,2).$  

(2)
 For  $C=f(S^3_1)\cap f(S^3_2)$, 
the inverse image  $f^{-1}(C)$ in $S^3_i$ is equivalent to the 1-link 
$L_i (i=1,2).$   
Here, the orientation of $C$ is induced 
naturally from the preferred orientations of $S^3_1, S^3_2,$ and $S^5$,   
 and an arbitrary order is given to the components of $C$.  



The following theorem characterizes the realizable 4-tuples of links. 



{\bf Theorem 1.1}    {\it 
 A 4-tuple of links $(L_1, L_2, X_1, X_2)$ is realizable  if and only if 
 $(L_1, L_2, X_1, X_2)$  satisfies one of the following conditions (1) and (2). 
 

(1)
Both $L_1$ and $L_2$ are proper links, and 

\vskip3mm
\hskip2cm $\mathrm{Arf}(L_1) = \mathrm{Arf}(L_2).$
\vskip3mm

(2)
 Neither $L_1$ nor $L_2$ is a proper link, and 

\vskip3mm
\hskip2cm
$\mathrm{lk}(K_{1j}, L_1-K_{1j})\equiv
  \mathrm{lk}(K_{2j}, L_2-K_{2j})  \mathrm{mod 2}$ 
  for  all $j$. 
}
\vskip3mm


In the case where $L_i$ is a 1-component link, 
that is, $L_i$ is a knot $K_i$, we have the following corollary. 

{\bf Corollary 1.2}   {\it 
 For 1-knots  $K_1$ and $K_2$, a 4-tuple of links  
 $(K_1, K_2, X_1, X_2)$ is realizable    
if and only if 


\vskip3mm
\hskip2cm $\mathrm{Arf}(K_1) = \mathrm{Arf}(K_2).$
\vskip3mm
}  



{\bf Note.}   
Theorem 1.1 and Corollary 1.2 give 
a new geometric meaning of the Arf invariant. 

{\bf Note 1.2.1.}     
The  problem (26) in [3]
%[F] 
says: Investigate ordinary sense slice 1-links,  
where {\it ordinary sense slice 1-links} 
are 1-links which are obtained as follows: 
Let $S^2$ be in 
$ R^4= R^3\times R.$ 
%$\Bbb R^4=\Bbb R^3\times\Bbb R.$ 
Then 
$S^2\cap [ R^3\times\{0\}]$ in $ R^3\times\{0\}$ is a 1-link. 
%$S^2\cap [\Bbb R^3\times\{0\}]$ in $\Bbb R^3\times\{0\}$ is a 1-link. 
By using Theorem 1.1, the author 
gives  an  answer to this  problem.    
The answer is: 
for every ordinary sense slice 1-link     
we can define the Arf invariant and it is zero (see [18]).
%([O3]).   


Let $f:S^3 \longrightarrow S^5$ be a smooth transverse immersion.  
Then the self-intersection  $C$ consists of double points.  
Suppose that $C$ is a single circle  in $S^5$. 
Then the $f^{-1}(C)$  in $S^3$ is a 1-knot or a 2-component 1-link.   
There is a similar realization problem. 
We consider which 1-knots 
(resp. 2-component 1-links ) we obtain as above. 
We give complete answers. 


{\bf Theorem 1.3}    {\it 
 Let $f:S^3 \longrightarrow S^5$ be a smooth transverse immersion.  
Then the self-intersection  $C$ consists of double points.  
Suppose that $C$ is a single circle  in $S^5$. 

(1)    Any 2-component 1-link is realizable 
as $f^{-1}(C)$  in $S^3$ for an immersion $f$.   

(2)    Any 1-knot is realizable 
as $f^{-1}(C)$  in $S^3$ for an immersion $f$.   
} 


{\bf Remark. }  
Suppose that $K_1$ is the trivial 1-knot, 
$K_2$ is the trefoil 1-knot, and $X_1$ and $X_2$ are 3-knots.  
Suppose that $L=(L_1, L_2)$ is a split 1-link such that 
$L_1$ is the trivial 1-knot and $L_2$ is the trefoil knot.   
 Then, by Corollary 1.2,  a 4-tuple of links $(K_1, K_2, X_1, X_2)$   
is not realizable.  
But, by Theorem 1.3 (1),  the two component split link $L$ 
 is realizable as in Theorem 1.3 (1). 


%In [O1],[O2],[O4] 
In [16],[17],[18],[19] 
the author discussed some topics which are related to this paper. 
%In [O4] 
In [19] he discussed the intersection of three 4-spheres in a 6-sphere. 
%In [O2] 
In [17] he discussed the intersection of two $(n+2)$-spheres in an $(n+4)$-sphere. 
%In [O1] 
In [16] he discussed the following: 
Let $L=(K_1, K_2)$ be a 2-link in $S^4=\partial B^5$. 
Take a slice disc $D^3_i$ in $B^5$  for each component $K_i$. 
He discussed the intersection of two slice discs $D^3_1$ and $D^3_2$ in the 5-ball $B^5$.  
In [18] he applied Theorem 1.1 to Fox's problem as we state in Note 1.2.1.  



 
{\bf Problem 1.4. } 
Suppose each of $S^3_1, S^3_2$, and  $S^3_3$ is a 3-sphere embedded in $S^5$.  
Suppose $S^3_i$ and $S^3_j$ intersect transversely. 
Suppose each of $S^3_1\cap S^3_2$, $S^3_2\cap S^3_3$, and $S^3_3\cap S^3_1$ 
is a single circle. 
Then we have a triple of 1-links, 
$L_i$=($S^3_i\cap S^3_j$, $S^3_i\cap S^3_k$) in $S^3_i$,  
where $(i,j,k)=(1,2,3), (2,3,1), (3,1,2)$. 

Which triple of 1-links do we obtain like this?

Do we characterize such triple by the Arf invariants, the linking numbers, and 
the Saito-Sato-Levine invariants? 
(See [20] for the definition of the Saito-Sato-Levine invariant.) 


In [19] the author discussed a higher dimensional version of Problem 1.4. 



This paper is organized as follows. 
In \S 2 
we review  
spin cobordism and the Arf invariant. 
In \S 3
 we discuss 
a necessary condition 
for the realization of 4-tuple of links. 
We find the obstruction for the realization in the spin cobordism group
$\Omega^{\mathrm{spin}}_{\star}.$ 
 In \S 4 
 we discuss a sufficient condition for the realization of 4-tuple of links. 
We carry out surgeries of submanifolds to carry out an (un)knotting  operation. 
Theorem 1.1 is deduced from \S 3 and \S 4. 
In \S 5 
we prove Theorem 1.3. 
In \S 6 
we give a problem.





%\end{document}



\section{ 
Spin cobordism and the Arf invariant}

In this section we review some results on 
the Arf invariant and spin cobordism. 
%See [Kf1] and [Kf2] 
See [5] and [6] 
for the Arf invariant. 
%See [Ki] 
See [10] 
for spin structures and spin cobordism.  
 

We suppose that, when we say $M$ is  a spin manifold,  
$M$ is oriented.

Recall that a {\it proper link} is an $m$-component 1-link    $L=\{K_1,...,K_m\}$ 
such that 
$ \mathrm{lk}(K_j, L-K_j)$ $=\sum_{1\leqq i \leqq m, i\not= j}$ 
$ \mathrm{lk}(K_j, K_i)$ 
is an even number for each $K_j$. 



Let $L=(K_1,...,K_m)$ be an $m$-component 1-link. 
Let $F$ be a Seifert surface for $L$.  
We induce a spin structure $\sigma$ on $F$ from the unique one on $S^3$.  
We induce a spin structure $\sigma_i$ on $K_i$ from $\sigma$ on $F$.  
Then we have: 

{\bf Proposition 2.1}   {\it 
Under the above condition, 
for each $i$, 

mod 2  $\mathrm{lk}$ $(K_i, L-K_i)\quad
=[(K_i, \sigma_i)]\in\Omega_1^{\mathrm{spin}}$.   

In particular, $L$ is a proper link if and only if each $[(K_i, \sigma_i)]=0$. 
} 


Suppose that $L$ is a proper link. Take $(F, \sigma)$ as above. 
Let $\hat{F}$ be the closed  surface obtained 
from $F$ by attaching disks to the boundaries. 
Let $\hat{\sigma}$ be the unique extension of 
$\sigma$ over $\hat{F}$. 
Then we have:

{\bf Proposition 2.2}   {\it 
  Under the above condition, 
 $\mathrm{Arf}$ $(L)$= $[(\hat{F}, \hat{\sigma})]$ $\in\Omega_2^{\mathrm{spin}}$.  }
 
 
Although they may be folklore, 
the author gives a proof of Proposition 2.1 and 
that of Proposition 2.2 in the appendix. 


\section{ 
A necessary condition for the realization of 4-tuple of links  
}

In this section we discuss a necessary condition 
for the realization of a 4-tuple of links. That is, 
we prove the  following two propositions. 
  
{\bf Proposition 3.1}   {\it 
If $(L_1, L_2, X_1, X_2)$ is realizable then 

$\hskip36pt
\ \mathrm{lk}(K_{1j}, L_1-K_{1j})
\equiv
\ \mathrm{lk}(K_{2j}, L_2-K_{2j}) \quad \mathrm{mod}\hskip2pt2
$\quad
for all $j$. 
\newline
In particular,  
$L_1$ is a proper link 
if and only if 
$L_2$ is a proper link.  
}


{\bf Proposition 3.2}   {\it 
Let $L_1$ and $L_2$ be proper links. 
If $(L_1, L_2, X_1, X_2)$
is realizable then 
$$\mathrm{Arf}(L_1) = \mathrm{Arf}(L_2).$$ 
}


In order to prove them, we prepare a lemma. 


Let $M_1$ and  $M_2$ be codimension one submanifolds of 
an $n$-dimensional compact spin manifold $N$. 
Suppose that $M_1$ and $M_2$ are compact oriented manifolds. 
Suppose that $M_1$, $M_2$, and $N$ may have the boundary and the corner.  
Let $M_1$ and  $M_2$  intersect transversely.  
Suppose $M_i$ may be embedded in the boundary (resp. the corner ) of $N$. 

We induce a spin structure $\sigma_i$ on $M_i$ from $N$. 
We induce a spin structure $\xi_i$ on $M_1\cap M_2$ from  $\sigma_i$ on $M_i$.  
($i=1,2$). 
Then it is easy to prove:

{\bf Lemma }   {\it 
$\xi_1$ and $\xi_2$ are same.  
} 

The spin structure $\xi_1$=$\xi_2$ on   $M_1\cap M_2$ is called 
 {\it the unique spin structure induced by $M_1$, $M_2$ and $N$}. 
 
%$$\mathrm{Table 3.1}$$

\vskip1cm
\begin{tabular} {llllc}
&&$M_1$&&\\
&
\rotatebox[origin=c]{45}{$\subset$}
&&
\rotatebox[origin=c]{-45}{$\subset$}
&\\
$M_1\cap M_2$&&&&$N$\\
&
\rotatebox[origin=c]{-45}{$\subset$}
&&
\rotatebox[origin=c]{45}{$\subset$}
&\\
&&$M_2$&&\\

\end{tabular}
\vskip1cm

%\vskip1cm
%\hskip3cm   Table 3.1
%\vskip1cm




{\bf Proof of Proposition 3.1.}
Let $f:S^3_1\amalg S^3_2\i S^5$ be an immersion to realize 
the 4-tuple of links $(L_1, L_2, X_1, X_2)$.  
Let $C=\amalg C_i$ denote $f(S^3_1)\cap f(S^3_2)$.  


We abbreviate $f(S^3_i)$ to  $ S^3_i.$   
Let $V_i$ be a Seifert hypersurface for $X_i$, i.e.,  

$$S^3_i=\partial V_i\subset V_i\subset S^5.$$

We  make $V_1$ and $V_2$ intersect transversely.  
We induce a spin structure $v_i$ on $V_i$ from the unique one on $S^5$.

Put $W=V_1\cap V_2$. 
We have: 

%$$\mathrm{Table 3.2}$$

\vskip1cm
\begin{tabular} {llllc}
&&$V_1$&&\\
&
\rotatebox[origin=c]{45}{$\subset$}
&&
\rotatebox[origin=c]{-45}{$\subset$}
&\\
$W$&&&&$S^5$\\
&
\rotatebox[origin=c]{-45}{$\subset$}
&&
\rotatebox[origin=c]{45}{$\subset$}
&\\
&&$V_2$&&\\
\end{tabular}
\vskip1cm

%\vskip1cm
%\hskip3cm   Table 3.2
%\vskip1cm




We give $W$ the unique spin structure $w$ induced by $V_1$, $V_2$ and $W$. 



Here, we see that 
$\partial W=(\partial V_1 \cap V_2)\cup (V_1 \cap \partial V_2).$  
Put  $F_1$=$\partial V_1 \cap V_2$.  
Put  $F_2$=$V_1 \cap \partial V_2$. 
Then $F_1$ ( resp. $F_2$ ) is a Seifert surface for $L_1$ ( resp. $L_2$ ). 

We induce a spin structure on $\partial V_i=S^3_i$ from $v_i$ on $V_i$. 
Note that it is the unique one on $S^3_i$.  

We have:

%$$\mathrm{Table 3.3}$$

\vskip1cm
\begin{tabular} {llllc}
&&$S^3_1$&&\\
&
\rotatebox[origin=c]{45}{$\subset$}
&&
\rotatebox[origin=c]{-45}{$\subset$}
&\\
$F_1$&&&&$V_1$\\
&
\rotatebox[origin=c]{-45}{$\subset$}
&&
\rotatebox[origin=c]{45}{$\subset$}
&\\
&&$W$&&\\
\end{tabular}
\vskip1cm

%\vskip1cm
%\hskip3cm   Table 3.3
%\vskip1cm



We give  $F_1$ the unique spin structure $\sigma_1$ induced by $S^3_1$, $W$ and $V_1$.  


We have:

%$$\mathrm{Table 3.4}$$

\vskip1cm
\begin{tabular} {llllc}
&&$W$&&\\
&
\rotatebox[origin=c]{45}{$\subset$}
&&
\rotatebox[origin=c]{-45}{$\subset$}
&\\
$F_2$&&&&$V_2$\\
&
\rotatebox[origin=c]{-45}{$\subset$}
&&
\rotatebox[origin=c]{45}{$\subset$}
&\\
&&$S^3_2$&&\\

\end{tabular}
\vskip1cm

%\vskip1cm
%\hskip3cm   Table 3.4
%\vskip1cm



We give  $F_2$ the unique spin structure $\sigma_2$ induced by $S^3_2$, $W$ and $V_2$.  


We have:

%$$\mathrm{Table 3.5}$$

\vskip1cm
\begin{tabular} {llllc}
&&$F_1$&&\\
&
\rotatebox[origin=c]{45}{$\subset$}
&&
\rotatebox[origin=c]{-45}{$\subset$}
&\\
$C$&&&&$W$\\
&
\rotatebox[origin=c]{-45}{$\subset$}
&&
\rotatebox[origin=c]{45}{$\subset$}
&\\
&&$F_2$&&\\
\end{tabular}
\vskip1cm

%\vskip1cm
%\hskip3cm   Table 3.5
%\vskip1cm



We give $C_j$ the unique spin structure  $\tau_j$ induced 
by $F_1$, $F_2$ and $W$. 



Then 
 mod2 lk$(K_{1j}, L_1-K_{1j})$=mod2 lk$(K_{2j}, L_2-K_{2j})$  
=$[(C_j, \tau_j)]$ $\in\Omega_1^{\mathrm{spin}}$  
for all $j$. 
This completes the proof of Proposition 3.1. 

We confirm that we have:

%$$\mathrm{Table 3.6}$$  

\vskip1cm
\begin{tabular} {llllllllc}
&&&&$S^3_1$&&&&\\
&&&\rotatebox[origin=c]{45}{$\subset$}
&&
\rotatebox[origin=c]{-45}{$\subset$}
&&&\\
&&$F_1$&&&&$V_1$&&\\
&
\rotatebox[origin=c]{45}{$\subset$}
&&
\rotatebox[origin=c]{-45}{$\subset$}
&&
\rotatebox[origin=c]{45}{$\subset$}
&&
\rotatebox[origin=c]{-45}{$\subset$}
&\\
$C$&&&&$W$&&&&$S^5$\\
&
\rotatebox[origin=c]{-45}{$\subset$}
&&
\rotatebox[origin=c]{45}{$\subset$}
&&
\rotatebox[origin=c]{-45}{$\subset$}
&&
\rotatebox[origin=c]{45}{$\subset$}
&\\
&&$F_2$&&&&$V_2$&&\\
&&&\rotatebox[origin=c]{-45}{$\subset$}
&&
\rotatebox[origin=c]{45}{$\subset$}
&&&\\
&&&&$S^3_2$&&&&\\
\end{tabular}
\vskip1cm


%\vskip1cm
%\hskip3cm   Table 3.6
%\vskip1cm





{\bf Proof of Proposition 3.2.}
We can make the corner of $W$ smooth. 
Note 
$\partial (W, w)$ =
$({F_1},  {\sigma_1})\cup ({F_2},  {\sigma_2}$. 
Take  $\hat{F_i}$ and $\hat{\sigma_i}$ as above. 
Then we have: 

$\mathrm{Arf}(L_1) + \mathrm{Arf}(-L_2)=
\mathrm{Arf}(L_1) + \mathrm{Arf}(L_2)=
[(\hat{F_1},  \hat{\sigma_1})]
+[(\hat{F_2},  \hat{\sigma_2})]
=[\partial (W, w)] 
=0\in \Omega^{\mathrm{spin}}_2\cong Z_2. $
%=0\in \Omega^{\mathrm{spin}}_2\cong\Bbb Z_2. $


Therefore  
$\mathrm{Arf}(L_1) = \mathrm{Arf}(L_2).$


%\input 01part1.tex  
%\input appendix.tex


\section{ 
 A sufficient condition for the realization of 4-tuple of links  
}

In this section we discuss 
a sufficient condition 
for the realization of a 4-tuple of links.  
That is, 
we prove the following proposition. 

{\bf Proposition 4.1}  {\it 
A  4-tuple of links $(L_1, L_2, X_1, X_2)$ is realizable  
if $(L_1, L_2, X_1, X_2)$  satisfies one of 
the  conditions (1) and (2) of Theorem 1.1.  
}

It is easy to prove that Proposition 4.1 is equivalent to the following Proposition 4.2. 


{\bf Proposition 4.2 }  {\it 
Let $X_1$ and $X_2$ be the trivial 3-knots. 
Let $L_1$ and $L_2$ be 1-links.  
Suppose that $L_1$ and $L_2$  satisfies 
one of the  conditions (1) and (2) of Theorem 1.1.  
Then the 4-tuple of links $(L_1, L_2, X_1, X_2)$ 
is realizable.   
}


We prove 

{\bf  Lemma 4.3 } {\it      
Let $X_1$ and $X_2$ be the trivial 3-knots. 
Let $L$ be a 1-link.   
Then the 4-tuple of links $(L, L, X_1, X_2)$ is realizable.   
}




{\bf Proof of Lemma 4.3. }
 Let $f:S^3_1\coprod S^3_2 \e S^5$ 
be an embedding  such that  
$f (S^3_1\coprod S^3_2)$ in $S^5$ 
is equivalent to the  trivial 3-link. 
We  take a chart $(U, \phi)$  of $S^5$ 
with the following properties (1) and (2). 

(1)
 $\phi:U\cong  R^5 = \{(x,y,z,u,v)\vert x,y,z,u,v\in   R\}$
=$ R^3\x  R_u\x R_v$.

(2)
$U\cap f(S^3_1)=\{(x,y,z,u,v)\vert u=0, v=0\}= R^3_1$ 

$U\cap f(S^3_2)=\{(x,y,z,u,v)\vert u=1, v=0\}= R^3_2$  


%(1)
% $\phi:U\cong \Bbb R^5 = \{(x,y,z,u,v)\vert x,y,z,u,v\in  \Bbb R\}$
%=$\Bbb R^3\x \Bbb R_u\x\Bbb R_v$.

%(2)
%$U\cap f(S^3_1)=\{(x,y,z,u,v)\vert u=0, v=0\}=\Bbb R^3_1$ 

%$U\cap f(S^3_2)=\{(x,y,z,u,v)\vert u=1, v=0\}=\Bbb R^3_2$  


%$$\mathrm{Figure 4.1}$$
$$\mathrm{Figure 1}$$

Obviously Lemma 4.3 follows from Lemma 4.4.

{\bf  Lemma 4.4 }     {\it 
There exists an immersion  $g: S^3_1\coprod S^3_2 \longrightarrow S^5$ 
with the following conditions. 


(1)
$g\vert_{ S^3_2} =f\vert_{ S^3_2}$.  

(2)
$g\vert_{ S^3_1}$ is isotopic to $f\vert_{ S^3_1}$. 

(3)
$g\vert_{ S^3_1-g^{-1}(U)}$ = $f\vert_{ S^3_1-g^{-1}(U)}$.   
Hence $g(S^3_1)\cap g(S^3_2)$ $\subset U$.

(4)
$g(S^3_1)\cap g(S^3_2)$ in 
$g(S^3_1)$ and that in $g(S^3_2)$ are both equivalent to the 1-link $L$.  

(5)
%$g(S^3_1)\cap\{(x,y,z,u,v)\vert u=0, v\in\Bbb R\}$= 
$g(S^3_1)\cap\{(x,y,z,u,v)\vert u=0, v\in R\}$= 
 $g(S^3_1)\cap\{(x,y,z,u,v)\vert u=0, v=0\}$.  
} 


{\bf Proof of Lemma 4.4. }
We modify the embedding $f$ to construct an immersion  $g$.  


In 
$ R^3_1$ %$\Bbb R^3_1$ 
we take the 1-link $L$ and a Seifert surface $F$ for $L$.     
Let $N(F)$

=$F\times\{t\vert -1\leqq t\leqq1\}$ 
be a tubular neighborhood of $F$ in 
$ R^3_1$.     
%$\Bbb R^3_1$.     


We define a subset $E$ of 
%$N(F)\times \Bbb R_u \times \Bbb R_v $
$N(F)\times  R_u \times  R_v $
$=\{(p,t,u,v)\vert p\in F, -1\leqq t\leqq 1, u\in R, v\in R \}$
%$=\{(p,t,u,v)\vert p\in F, -1\leqq t\leqq 1, u\in \Bbb R, v\in \Bbb R \}$
so that  

$E=\{(p,t,u,v)\vert p\in F, \quad$ 
$ 0\leqq u\leqq \frac{\pi}{2},\quad 
t=k\cdot \mathrm{cos} u, \quad
v=k\cdot \mathrm{sin} u, \quad$ 
$-1\leqq k\leqq 1\}$.  


Put  $P$=$\overline{\partial E-(\partial E\cap f(S^3_1))}$.
Put  $Q$=$\overline{f(S^3_1)-(\partial E\cap f(S^3_1))}$.
Note that $\partial P$=$\partial Q$=$\partial N(F)$.  
Put $\Sigma=$$P\cup Q$.
Then, by the construction, $\Sigma$ is a 3-sphere embedded in $S^5$ and 
is the trivial 3-knot.  

% ireru 
\vskip3mm
{\bf Note.} 
In $U$ the following hold. 

(1)   $g(S^3_1)\cap\{(x,y,z,u,v)\vert u=0, v\in R\}$ 
        $=g(S^3_1)\cap\{(x,y,z,u,v)\vert u=0, v=0\}$ 
        $=\overline{g(S^3_1)-N(F)}$

(2)   $g(S^3_1)\cap\{(x,y,z,u,v)\vert 0<u\leqq\frac{\pi}{2}, v\in R\}$ 
      is Int $N(F)$. 



(3) Let $0<u'<\frac{\pi}{2}$.  
 
     $g(S^3_1)\cap\{(x,y,z,u,v)\vert u=u', v>0\}$ 
         is diffeomorphic to Int $F$. 
     
(4)  Let $0<u'<\frac{\pi}{2}$.  

     $g(S^3_1)\cap\{(x,y,z,u,v)\vert u=u', v<0\}$ 
     is diffeomorphic to Int $F$. 


(5) Let $0<u'<\frac{\pi}{2}$. 

  $g(S^3_1)\cap\{(x,y,z,u,v)\vert u=u', v=0\}$ 
    is diffeomorphic to $L$. 


(6)  Let $0<u'<\frac{\pi}{2}$.  

$g(S^3_1)\cap\{(x,y,z,u,v)\vert u=u', v\in R\}$ 
is diffeomorphic to $\partial(N(F))$. 

Let $F_i$ be diffeomorphic to $F$ ($i=1,2$). 
Recall $F$ is a compact oriented surface with boundary. 
We identify $\partial F_1$ with $\partial F_2$ 
to obtain $F_0=F_1\cup F_2$. 
Note $F_0$ is diffeomorphic to $\partial(N(F))$. 




(7)    $g(S^3_1)\cap\{(x,y,z,u,v)\vert u=\frac{\pi}{2}, v\in R\}$ 
      is diffeomorphic to $N(F)$. 


(8)    $g(S^3_1)\cap\{(x,y,z,u,v)\vert u=\frac{\pi}{2}, v=0\}$ 
      is diffeomorphic to $F$. 

(9)    $g(S^3_1)\cap\{(x,y,z,u,v)\vert 0<u\leqq\frac{\pi}{2}, v=0\}$ 
      is diffeomorphic to Int $F$. 



\vskip3mm
% ireta 


%In Figure 4.2,  
In Figure 2, 3, 4,   
$\Sigma\cap U$ and $g(S^3_2)\cap U$  are drawn. 
There, we replace 
$ R^3\x R_u\x R_v$ with 
$ R^2\x R_u\x R_v$.  
%$\Bbb R^3\x \Bbb R_u\x\Bbb R_v$ with 
%$\Bbb R^2\x \Bbb R_u\x\Bbb R_v$.  


%$$\mathrm{Figure 4.2}$$

$$\mathrm{Figure 2}$$

$$\mathrm{Figure 3}$$

$$\mathrm{Figure 4}$$



By the construction,  $\Sigma\cap$ $g(S^3_2)$  in  $\Sigma$ and 
that in $g(S^3_2)$  are both equivalent to $L$. 
Define $g\vert_{ S^3_1}$ so that $g(S^3_1)$=$\Sigma$. 


This completes the proof of Lemma 4.4 and therefore Lemma 4.3.  


{\bf Note.} 
 Lemma 4.3 gives an alternative proof of 
the results of [4]
%[GM] 
and Lemma 1 of [15].
%[N]. 


In order to prove Proposition 4.2, we review the pass-moves. 
%See [Kf1] and [Kf2] for detail. 
See [5] and [6] for detail. 



{\bf Definition } (See [5][6].)
%(See  [Kf1] and [Kf2].)
Two 1-links are {\it pass-move equivalent}
 if one is obtained from the other 
by a sequence of pass-moves. 
See 
%Figure 4.3 
Figure 5 
for an illustrations of the pass-move. 

%$$\mathrm{Figure 4.3}$$

$$\mathrm{Figure 5}$$



In $B^3_a$ there are four arcs. 
Each of four arcs may belong to different components of the 1-link. 
We do not assume the four arcs belong to one component of the 1-link. 


The following propositions are essentially proved in [5]. %[Kf1]. 
A proof is written in the appendix of [14]. %[MN].  


{\bf Proposition 4.5 } (See [5][14].)  
   {\it 
%(See [Kf1][MN].) 
Let $L_1$ and $L_2$ be 1-links. 
Then  $L_1$ and $L_2$ are pass-move equivalent if and only if 
$L_1$ and $L_2$
 satisfy 
one of the  conditions (1) and (2) of Theorem 1.1. 
}



{\bf Proposition 4.6 }   (See [5][14].)
 {\it 
%(See [Kf1][MN].) 
Let $L_1$ and $L_2$ be 1-links. 
Suppose that $L_1$ is pass-move equivalent to $L_2$. 
Let $F$ be an oriented Seifert surface for $L_2$ 
such that the genus of $F$ is not less than  the genus of $L_1$. 
Then there exists a disjoint union of 3-balls $B^3$ 
such that  $B^3\cap F$ is  as  in 
%Figure 4.4 
Figure 6 
and the pass-moves in all $B^3$ change $L_2$ to $L_1$ 
}


%$$\mathrm{Figure 4.4}$$

$$\mathrm{Figure 6}$$



Let $L_1$ and $L_2$ be 1-links. 
Suppose $L_1$ and $L_2$ satisfy one of 
the conditions (1) and (2) of Theorem 1.1.  
Then by Proposition 4.5  
$L_1$ is obtained from $L_2$ by a sequence of pass-moves. 
We choose a Seifert surface $F$ for $L_2$ and 
the disjoint union of 3-balls $B^3$ in $S^3$ 
as in Proposition 4.6.    

Take a 1-link ($Y_{1}$, $Y_{2}$) in each $B^3$ as in 
%Figure 4.5. 
Figure 7. 
By considering the Kirby moves of framed links (in [10]), 
it is easy to prove:

%$$\mathrm{Figure 4.5}$$
 
 $$\mathrm{Figure 7}$$



{\bf Lemma 4.7}  
 {\it 
We carry out 0-framed surgeries along all $Y_{1}$ and all $Y_{2}$.    
After these surgeries,  we have the following.  
(1)
$S^3$ becomes  a  3-sphere  again. 
(2)
each $B^3$ becomes a 3-ball again,  
(3) $L_2$ in the old sphere $S^3$ changes to $L_1$ in the new 3-sphere. 
} 
% along ni naosita, (1)to (2) wo koukan 


We go back to the proof of Proposition 4.2. 

As in Lemma 4.4, 
take an immersion $g: S^3_1\coprod S^3_2 \longrightarrow S^5$ 
to realize $(L_2, L_2, X_1, X_2)$, where $X_i$ is the trivial 3-knot.  

Obviously Proposition 4.2 follows from Lemma 4.8. 


 {\bf Lemma 4.8 } 
 {\it 
There exists an immersion  $h: S^3_1\coprod S^3_2 \longrightarrow S^5$ 
with  the following conditions. 



(1)
$h\vert_{ S^3_2} =g\vert_{ S^3_2}$.  

(2)
$h( S^3_1 ) $ in $S^5$ is equivalent to the trivial 3-knot.  

(3)
$h\vert_{ S^3_1-g^{-1}(U)}$ = $g\vert_{ S^3_1-g^{-1}(U)}$. 
Hence $h(S^3_1)\cap h(S^3_2)$ $\subset U$.

(4)
$h(S^3_1)\cap h(S^3_2)$ in $h(S^3_1)$ is equivalent to $L_1$. 
$h(S^3_1)\cap h(S^3_2)$ in $h(S^3_2)$ is equivalent to $L_2$. 

(5)
%$g(S^3_1)\cap\{u\neq0, v\in\Bbb R\}$=$h(S^3_1)\cap\{u\neq0, v\in\Bbb R\}$  
$g(S^3_1)\cap\{u\neq0, v\in R\}$=$h(S^3_1)\cap\{u\neq0, v\in R\}$  
}


{\bf Proof of Lemma 4.8. } 
We  modify the immersion $g$ to define an immersion $h$ as follows. 

As in the proof of Lemma 4.4, 
we take $L_2$ and a Seifert surface $F$ for $L_2$ in $g(S^3_1)$. 
Suppose 
$L_2\subset R^3_1$ and $F\subset U$. 
%$L_2\subset \Bbb R^3_1$ and $F\subset U$. 
Suppose the genus of $F$ satisfies the condition in Proposition 4.6. 
Take 3-balls $B^3$  in $g(S^3_1)$  as in Proposition 4.6.   
Take ($Y_1$, $Y_2$) in $B^3$ as in Lemma 4.7.  
See Int $N(F)$ in $g(S^3_1)$. 
Int $N(F)$ is not in 
%$\Bbb R^3_1$ 
$ R^3_1$ 
and is in $\{u>0\}$. 
But by Proposition 4.6, 
we can suppose $Y_1$ and $Y_2$ are in 
$ R^3_1$.  
%$\Bbb R^3_1$.  
 


Take  an embedded 2-disc $h^{'2}_{1}$  in 
$\{(x,y,z,u,v)\vert u=0, v\geqq 0 \}$ 
so that 
$h^{'2}_{1}$  meets  $\{(x,y,z,u,v)\vert u=0, v=0 \}$ 
at $Y_1$ transversely. 
Suppose that $h^{'2}_{1}$ is embedded trivially. 


Take  an embedded 2-disc $h^{'2}_{2}$  in 
$\{(x,y,z,u,v)\vert u=0, v\leqq 0 \}$ 
so that 
$h^{'2}_{2}$  meets  $\{(x,y,z,u,v)\vert u=0, v=0 \}$ 
at $Y_2$ transversely. 
Suppose that $h^{'2}_{2}$ is embedded trivially. 

%See Figure 4.6.
See Figure 8.

%$$\mathrm{Figure 4.6}$$

$$\mathrm{Figure 8}$$

Let $h^2_{1}$ be a tubular neighborhood of $h^{'2}_{1}$ 
in $\{(x,y,z,u,v)\vert u=0, v\geqq 0 \}$.  


Let $h^2_{2}$ be a tubular neighborhood of $h^{'2}_{2}$ 
in $\{(x,y,z,u,v)\vert u=0, v\leqq 0 \}$.  

Then $h^2_{1}$ and $h^2_{2}$ are diffeomorphic to the 4-ball.    
Furthermore we can regard $h^2_i$  
as a 4-dimensional 2-handle  attached to  
$g(S^3_1)$ along $Y_i$  with 0-framing. 


Put $R=g(S^3_1)$ $-\{g(S^3_1)\cap\partial h^2_1\}$ 
$-\{g(S^3_1)\cap\partial h^2_2\}$.
Put $S_1=$ $\partial h^2_1$ $-\{g(S^3_1)\cap\partial h^2_1\}$.  
Put $S_2=$ $\partial h^2_2$ $-\{g(S^3_1)\cap\partial h^2_2\}$.  
Here, we consider all $h^2_i$.  
Put $\Lambda$=$R\cup S_1\cup S_2$.

Then we can regard  $\Lambda$ as the result of 
0-framed surgeries on  $g(S^3_1)$ along all $Y_1$ and $Y_2$. 
Then the pass-moves are carried out in all $B^3$. 

Therefore $\Lambda$ is an embedded 3-sphere in $S^5$. 
Furthermore  $\Lambda\cap g(S^3_2)$ in  $\Lambda$ is equal to the 1-link $L_1$   
and $\Lambda\cap g(S^3_2)$ in  $g(S^3_2)$ is equal to the 1-link $L_2$.  

Put $h\vert_{ S^3_1}$ so that  $h(S^3_1)$=$\Lambda$. 
By the above construction, $h$ satisfies the conditions (1)(3)(4)(5) in Lemma 4.8.  

  

We prove: 

{\bf Lemma 4.9} 
  {\it 
$h\vert(S^3_1)$ is equal to the trivial 3-knot.  
}



{\bf Proof.}  
By the construction, $h(S^3_1)$ bounds a 4-manifold 
represented by the disjoint union of some copies of 
the  framed link in 
%Figure 4.7.  
Figure 9. 

%$$\mathrm{Figure 4.7}$$
$$\mathrm{Figure 9}$$


%See [Ki] 
See [10] 
for dot-circles. 
Hence this 4-manifold is diffeomorphic to the 4-ball. 
Therefore $h(S^3_1)$ bounds a 4-ball. 
This completes the proof of Lemma 4.9 and therefore Lemma 4.8. 
This completes the proof of Proposition 4.2 and therefore Proposition 4.1. 



\section{ The  proof of Theorem 1.3  }


{\bf Lemma 5.1.1 } 
 {\it 
Let $L$ be the trivial 2-component 1-link. 
There exists a self-transverse immersion  
$f:S^3\i B^5$ with the following properties. 

(1)
The singular point set (in $B^5$) is a single circle $C$.  

(2)
$f^{-1}(C)$ in $S^3$ is equivalent to $L$. 

(3)
$f(\overline{S^3-N(L)})$ $\subset \partial B^5$
and 
 $f(\mathrm{Int}N(L))$ $\subset$ $\mathrm{Int}B^5$, where 
 $N({L})$ is a tubular neighborhood of $L$ in $S^3$.  
}


{\bf Lemma 5.1.2 } 
 {\it 
Let $L$ be the Hopf link. 
There exists a self-transverse immersion  
$f:S^3\i B^5$ satisfying with the properties (1), (2) and (3) in Lemma 5.1.1. 
}


{\bf Lemma 5.1.3 }   
  {\it 
Let $L$ be the trivial 1-knot. 
There exists a self-transverse immersion  
$f:S^3\i B^5$ satisfying  with the properties (1), (2) and (3) in Lemma 5.1.1.  }



{\bf Proof of Lemma 5.1.1 }
Take a chart of $(U, \phi)$ of $B^5$ such that  
$\phi (U)$=
%$\{(x,y,z,v,t)\vert$ $x,y,z,v \in \Bbb R,  t\leqq0 \}.$  
$\{(x,y,z,v,t)\vert$ $x,y,z,v \in R,  t\leqq0 \}.$  


Put $F$=
%$\{(x,y,z,v,t)\vert$ $x,y\in \Bbb R,  z\geqq0, v=0, t\leqq0 \}.$  
$\{(x,y,z,v,t)\vert$ $x,y\in R,  z\geqq0, v=0, t\leqq0 \}.$  

Put $A$=
%$\{(x,y,z,v,t)\vert$ $x,y\in \Bbb R,  z=0, v=0, t\leqq0 \}.$  
$\{(x,y,z,v,t)\vert$ $x,y\in R,  z=0, v=0, t\leqq0 \}.$  

We can regard $U$ as the result of rotating $F$ around the axis $A$. 

Take an immersed 2-disc $D$ in $F$ as in 
%Figure 5.1.  
Figure 10, 11. 

%$$\mathrm{Figure 5.1}$$


$$\mathrm{Figure 10}$$

$$\mathrm{Figure 11}$$


% ireta

\vskip3mm
{\bf Note. }    
In $F$ the following hold. 

(1) $\{(x,y,z,v,t)\vert$ $x,y\in R,  z\geqq0, v=0, t=0 \}\cap D$ 
  is diffeomorphic to $\overline{D-(two 2-discs)}$. 
  
  
(2) $\{(x,y,z,v,t)\vert$ $x,y\in R,  z\geqq0, v=0, t<0 \}\cap D$ 
  is a union of the interior of two 2-discs, 
  where the intersection of the two 2-discs is one point. 
  The point is $p=(1,0,0,0,-1)$. 
  
(3) Let $-1<t'<0$.

 $\{(x,y,z,v,t)\vert$ $x,y\in R,  z\geqq0, v=0, t=t' \}\cap D$ 

is the Hopf link in 

 $\{(x,y,z,v,t)\vert$ $x,y\in R,  z\geqq0, v=0, t=t' \}$. 

(4) $\{(x,y,z,v,t)\vert$ $x,y\in R,  z\geqq0, v=0, t=-1 \}\cap D$ 
  is a union of two circles, 
  where the intersection of the two circles is one point.  
The point is  $p=(1,0,0,0,-1)$. 
 
(5) Let $-2<t'<-1$. 

 $\{(x,y,z,v,t)\vert$ $x,y\in R,  z\geqq0, v=0, t=t' \}\cap D$ 

is the trivial link in 

 $\{(x,y,z,v,t)\vert$ $x,y\in R,  z\geqq0, v=0, t=t' \}$. 


(6) $\{(x,y,z,v,t)\vert$ $x,y\in R,  z\geqq0, v=0, t=-2 \}\cap D$ 

is a disjoint union of two 2-discs. 

\vskip3mm

%ireta


As we rotate $F$ as above, we rotate $D$ as well. 
We obtain an immersed 3-sphere $X$. 

Take $f$ so that $f(S^3)$=$X$. 

This completes the proof of Lemma 5.1.1. 


{\bf Proof of Lemma 5.1.2 }
Take $B^5, F, A,$ and $D$ as above. 

Put $G$=
$\{(x,y,z,v,t)\vert$ $x=0,y=0, z\geqq0, v=0, t\leqq0 \}.$  


%In Figure 5.1 
In Figure 10, 11 
we suppose: 
If, in $F$, we rotate $D$ around $G$ by any angle, then $D\cap A$ does not change. 

As we rotate $F$ as above, 
we rotate $D$ around $A$ so that 
in $F$ we rotate $D$ around $G$ one time. 
We obtain an immersed 3-sphere $X$. 

Take $f$ so that $f(S^3)$=$X$.     

This completes the proof of Lemma 5.1.2. 



{\bf Proof of Lemma 5.1.3 }
Take $B^5, F, A, D, $ and $G$ as above. 


%In Figure 5.1 
In Figure 10, 11 
we suppose: 
If we rotate $D$ around $G$ half time, 
then the resultant $D$ coincides with the original $D$. 


As we rotate $F$ as above, 
we rotate $D$ around $A$ so that 
in $F$ we rotate $D$ around $G$ half time. 
We obtain an immersed 3-sphere $X$. 

Take $f$ so that $f(S^3)$=$X$. 

This completes the proof of the proof of Lemma 5.1.3. 




{\bf Lemma 5.2.1 } 
 {\it 
Let $L$ be a 2-component 1-link whose linking number is even. 
There exists a self-transverse immersion 
$g:S^3\i \natural^l S^2\x B^3$, 
for a non-negative integer $l$,  with the following properties. 

(1)
The singular point set (in $\natural^l S^2\x B^3$) is a single circle $C$.  

(2)
$g^{-1}(C)$ in $S^3$ is equivalent to $L$. 

(3)
$g(\overline{S^3-N(L)})$ $\subset \partial(\natural^l S^2\x B^3)=$
$\sharp^l S^2\x S^2$  
and 
 $g(\mathrm{Int}N(L))$ $\subset$ $\mathrm{Int}(\natural^l S^2\x B^3)$, where 
 $N({L})$ is a tubular neighborhood of $L$ in $S^3$.  

Here, we suppose that $\natural^0 S^2\x B^3$ means $B^5$.  
}


{\bf Lemma 5.2.2 }  
 {\it 
Let $L$ be a 2-component 1-link whose linking number is odd. 
There exists a self-transverse immersion 
$g:S^3\i \natural^l S^2\x B^3$, 
for a non-negative integer $l$,  
satisfying with the properties (1), (2) and (3) in Lemma 5.2.1. 
}



{\bf Lemma 5.2.3 } 
 {\it 
Let $L$ be a 1-knot.  
There exists a self-transverse immersion  
\nl$g:S^3\i \natural^l S^2\x B^3$, 
for a non-negative integer $l$,  
satisfying with the properties (1), (2) and (3) in Lemma 5.2.1. 
}


We prove:  

{\bf Claim 5.2.4} 
 {\it 
Lemma 5.2.1, 5.2.2, and 5.2.3 imply Theorem 1.3. 
}


There is an embedding 
$h:\natural^l S^2\x B^3\e S^5$. 
Then $h\circ g$ is an immersion in Theorem 1.3. 
This completes the proof. 


In order to prove Lemma 5.2.1, 5.2.2, and 5.2.3, 
we review the $\sharp$-moves.  
%See  [MN] and [Kw] for detail.
See  [14] and [7] for detail.  



{\bf Definition }  ([14]) %( [MN])  
Two 1-links are  {\it $\sharp$-move equivalent}
 if one is obtained from the other 
by a sequence of  $\sharp$-moves. 
See 
%Figure 5.2 
Figure 12 
for an illustrations of the $\sharp$-move. 
The  $\sharp$-move is different from the pass-move by the orientation.  

%$$\mathrm{Figure 5.2}$$

$$\mathrm{Figure 12}$$



In each 3-ball in 
%Figure 5.2 
Figure 12 
there are four arcs. 
Each of four arcs may belong to different components of the 1-link. 
We do not assume the four arcs belong to one component of the 1-link. 





The following proposition 5.3 is proved in the appendix of [14].%[MN].  



{\bf Proposition 5.3 } ([14])%( [MN] )
 {\it 
(1)
Let $L$ be a 2-component link. 
Then $L$ is  $\sharp$-move  equivalent to the trivial link 
if and only if the  linking number is even.  

(2)
Let $L$ be a 2-component link. 
Then $L$ is  $\sharp$-move  equivalent to the Hopf link   
if and only if the  linking number is odd.  

(3)
Any 1-knot is  $\sharp$-move equivalent to the trivial knot.
}


We have:  


{\bf Proposition 5.4 } ([14])%( [ MN ] )  
 {\it 
Let $L_1$ and $L_2$ be 1-links. 
Suppose that $L_1$ is $\sharp$-move equivalent to  $L_2$. 
 Take $L_1$ in $S^3$. 
Then there exists a disjoint union of 3-balls $B^3$ in $S^3$ 
such that $L_1\cap B^3$ is as in 
%Figure 5.2  
Figure 12  
and that  the $\sharp$-moves in all $B^3$ change $L_1$ to $L_2$.  
} 



Let $L_1$ and $L_2$ be 1-links. 
Suppose $L_1$ is $\sharp$-move equivalent to $L_2$. 
Take $L_1$ in $S^3_1$.  
Take a disjoint union of 3-balls $B^3$ in $S^3$ as in Proposition 5.3.     
Take a 1-link ($Y_{1}$, $Y_{2}$) in each $B^3$ as in 
%Figure 5.3.   
Figure 13.
Suppose that $Y_i$ bounds a 2-disc $B^2_i$ as in 
%Figure 5.3.   
Figure 13.   

%$$\mathrm{Figure 5.3}$$ 

$$\mathrm{Figure 13}$$ 


By considering the Kirby moves of framed links (in [10]), 
it is easy to prove:  


{\bf Lemma 5.5 }  
 {\it 
We carry out 0-framed surgeries on all $Y_{1}$ and all $Y_{2}$.    
After these surgeries,  we have the following.  
(1)
each $B^3$ becomes a 3-ball again,  
(2)
$S^3$ becomes  a  3-sphere  again. 
(3) $L_1$ in the old sphere $S^3$ changes to $L_2$ in the new 3-sphere. 
}


We go back to the proof of Lemma 5.2.1, 5.2.2, 5.2.3. 


 

{\bf Proof of Lemma 5.2.1 ( resp. 5.2.2, 5.2.3 )} 

By Proposition  5.3, 
a sequence of $\sharp$-moves changes $L$ into 
the trivial 2-component 1-link 
( resp. the Hopf link, the trivial 1-knot ). 
 

As in Lemma  5.1.1 ( resp. 5.1.2, 5.1.3 ),  take $f:S^3\i B^5$. 
See the 1-link $f^{-1}(C)$ in $S^3$. 
It is equivalent to the 1-link $L$.  
As in Proposition 5.4, take 3-balls $B^3$. 
As in Lemma 5.5, take $Y_1$ and $Y_2$ in $B^3$.  
Suppose that $f(Y_1)$ and $f(Y_2)$ are in $\partial B^5$.

Attach 4-dimensional 2-handles $h^2$ to 
$f(S^3-N(L))$  in $\partial B^5$ along $f(Y_i)$ with 0-framing.  
Here, 0-framing means the following. 
When attaching the 4-dimensional 2-handles $h^2$ to $f(S^3-N(L))$,  
we can attach 4-dimensinal 2-handles to $S^3$ naturally. 
These attaching maps are 0-framing. 

When attaching the 4-dimensional 2-handles $h^2$ to $f(S^3-N(L))$,  
we can attach 5-dimensional 2-handles  $h^2\x[-1,1]$ 
to $B^5$ along $f(Y_i)$ naturally.  
Of course the attached parts are in  $\partial B^5$. 
We obtained a 5-manifold. Call it $M$. 


Put $P=$ $f(S^3)-(f(S^3)\cap\partial h^2)$. 
Put $Q=$ $\partial h^2-(f(S^3)\cap\partial h^2)$. 
Put $\Sigma=P\cup Q$. 
Then $\Sigma$ is an immersed 3-sphere in $M$. 

We prove Lemma 5.6. 
Before the proof of Lemma 5.6, we prove Lemma 5.7. 


{\bf Lemma  5.6 }  
 {\it 
Let $M$ be as above. 
$M$=$\natural^l S^2\x B^3$, for a non-negative integer $l$. 
}

{\bf Lemma  5.7 }  
 {\it 
Lemma 5.6 implies Lemma 5.2.1, 5.2.2 and 5.2.3.  
}

{\bf Proof of Lemma 5.7.  }
Take a self-transverse immersion  $g:S^3\i M=\natural^l S^2\x B^3$
so that  $g(S^3)=\Sigma$.  



{\bf Proof of Lemma 5.6.}
Recall:
$J=$ (one 5-dimensional 0-handle)$\cup$(one 5-dimensional 2-handle) is 
$S^2\times B^3$ or $S^2\tilde\times B^3$. 
 $J$ is $S^2\times B^3$ 
if and only if the attaching map of the 2-handle is spin preserving diffeomorphism map. 

It suffices to prove that 
the attaching maps of the 5-dimensional 2-handles are 
spin-preserving diffeomorphism maps. 

We give a spin structure on $f(S^3-N(L))$ 
from the unique one on $\partial B^5$.  

We give a spin structure on $f(B^2_i)\cap\partial B^5$ 
from the spin structure on $f(S^3-N(L))$.  

We give a spin structure $\xi$ on $f(Y_i)$
 from the spin structure on $f(B^2_i)\cap\partial B^5$.  

Put $S^1_a\amalg S^1_b$= $f(B^2_i)\cap\partial N(L)$.  
We give a spin structure $\alpha$ (resp. $\beta$)  
on $S^1_a$ (resp.  $S^1_b$ ) 
 from the spin structure 
on $f(B^2_i)\cap\partial B^5$.  
By the construction,  $\alpha$ and  $\beta$  are the $S^1_{Lie}$ spin structure. 


Since $(f(Y_i), \xi)$ is spin cobordant to 

$(S^1_a,$ the $S^1_{Lie}$ spin structure $)$ $\amalg$ 
$(S^1_b,$ the $S^1_{Lie}$ spin structure $)$,   

$\xi$ is  the $S^1_{bd}$ spin structure. 


%See [Ki] 
See [10] 
for the $S^1_{bd}$ spin structure and the $S^1_{Lie}$ spin structure.  

This completes the proof of Lemma 5.6. 

This completes the proof of Lemma 5.2.1 ( resp. 5.2.2, 5.2.3 ).   
By Claim 5.2.4, Theorem 1.3 holds. 




\np


\noindent{\large{\bf Appendix. The proof of  Proposition 2.1 and 2.2 }}

Firstly we prove Proposition 2.1. 

Let $L=(K_1,...,K_m)$ be a 1-link. 
Let $F$ be a Seifert surface. 
Let $K_i\x[0,1] (\subset F)$ be a collar neighborhood of $K_i$ in $F$. 
Let  $K_i\x\{0\}$=$K_i$. 
Then lk$(K_i, L-K_i)$=lk$(K_i\x\{0\}, K_i\x\{1\})$. 

%See [Ki] 
See [10] 
for spin structures.   

Let $\varepsilon^2$ be the trivial bundle over $S^1$. 
Let $(e^1_p, e^2_p)$ (for each $p\in S^1$) denote the trivialization. 
Let $e^0_p$ (for each $p\in S^1$) denote the trivialization on $TS^1$.  
Spin structures on $S^1$ are defined by spin structures on 
$TS^1\oplus\varepsilon^2$. 
%[Ki] 
[10]
defines that the spin structure defined by $(e^0_p, e^1_p, e^2_p)$
is the {\it $S^1_{Lie}$ spin structure}. 
The other is the {\it $S^1_{bd}$ spin structure}. 


Let $(M, \sigma)$ and $(N,\tau)$ be spin manifolds. 
Let $f:M\rightarrow N$ be an orientation preserving diffeomorphism. 
$f$ is called a {\it spin preserving diffeomorphism}  
if  
$df\oplus id:TM\oplus\varepsilon^p  \rightarrow TN\oplus\varepsilon^p $
carries 
$\sigma$ to $\tau$, 
$id$ carries the trivialization to the trivialization.



We give $F$ a spin structure $\alpha$ induced from 
the unique spin structure $\beta$ on $S^3$. 
We give $K_i$ a spin structure $\gamma$ induced from $\alpha$.  
Let $K_i\x D^2$ be the tubular neighborhood of $K_i$ in $S^3$. 
Regard  $K_i\x D^2$ as the product $D^2$ bundle over $K_i$. 
We give $K_i\x D^2$ a trivialization such that 
$e^1_p$ is in  $K_i\x[0,1]$ and that 
$e^2_p$ is perpendicular to $K_i\x[0,1]$ for each $p\in K_i$.
Then we can regard $\alpha$ as a spin structure on 
the trivialized bundle $TK_i\oplus (K_i\x[0,1])$.  


Let $G$ be any Seifert surface of $K_i$. 
Let $g$ be a spin structure on $G$ 
induced from the unique one $S^3$. 
Let $\mu$ be a spin structure on $K_i$ induced from $(G,g)$. 
Since $\partial(G, g)=(K_i, \mu)$, $[(K_i, \mu)]=0$.  
We give $K_i\x D^2$ a trivialization such that 
$g^1_p$ is in $G$ and $g^2_p$ is perpendicular to $G$. 
Then the spin structure on $K_i$ induced by 
the framing $(e^0_p, g^1_p, g^2_p)$ is $\mu$. 


The homotopy class of the framing $(e^0_p, e^1_p, e^2_p)$ coincides with 
the homotopy class of the framing $(e^0_p, g^1_p, g^2_p)$ 
if and only if 
lk$(K_i\x\{0\}, K_i\x\{1\})$ is even.  


Therefore Proposition 2.1 holds. 


Next we prove Proposition 2.2. 

Under the above condition, 
suppose $L$ is a proper 1-link. 

Let $\hat F$ be the closed oriented surface $F\cup(\cup^{m}_{i=1}D^2)$.
Since $\gamma$ is the $S^1_{Lie}$ spin structure, 
$\alpha$ extends to $\hat F$, say $\hat\alpha$.  
Let $a_1,...,a_g,b_1,...,b_g$ be circles in $F$ representing 
symplectic basis of 
%$H_1(\hat F;\Bbb Z)$. 
$H_1(\hat F; Z)$. 
Let $a_i\x[-1,1]$, $b_i\x[-1,1]$ be the tubular neighborhood in $F$. 
Put mod 2 lk ($a_i\x\{-1\}, a_i\x\{1\}$)=$x_i$  
and mod 2 lk ($b_i\x\{-1\}, b_i\x\{1\}$)=$y_i$. 
Recall Arf$L=$$\Sigma x_i\cdot y_i$. 


We give a spin structure $\sigma_i$, (resp. $\tau_i$) 
on $a_i$ (resp. $b_i$). 
Then [($a_i, \sigma_i$)]$\in\Omega_1^{spin}=$ $x_i$ and 
 [($b_i, \tau_i$)]$\in\Omega_1^{spin}=$ $y_i$. 
By P.36 of [10],
%[Ki], 
[($\hat F, \hat\alpha$)]$\in\Omega_2^{spin}$ is $\Sigma x_i\cdot y_i$ .   


Hence Proposition 2.2 holds. 




\np
\footnotesize{\begin{thebibliography}{900}


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 S. Cappell and J. Shaneson: 
The codimension two placement problem and homology equivalent manifolds 
{\it Ann. of Math.}, 99, 271-348, 1972 


\bibitem{ 2} %CO
T. Cochran and K. Orr:  Not all links are concordant to boundary links 
 {\it Ann. of Math. }, 138, 519--554, 1993  

\bibitem{3} %F
R. Fox: Some problems in knot theory  
 {\it Top. 3-manifolds. Proc. 1961 Top. Inst. Georgia }, 168-176


\bibitem{ 4} %GM
F. Gonzalez-Accuna and J. Montesinos:  
Embedding knots in trivial knots 
 {\it Bull. London. Math. Soc.}, 14, 238-240, 1982 



\bibitem{5} %Kf1
L. Kauffman: Formal knot theory
 {\it Princeton University Press Math. Notes}, vol.30, 1983


 \bibitem{ 6} %Kf2
L. Kauffman:  On knots
 {\it  Ann. of Math. Stud. }, vol. 115, 1987




\bibitem{ 7} %Kw
A. Kawauchi edit.: Knot theory (In Japanese)
 {\it Springer Verlag Tokyo}, 1990
\newline
 A survey on knot theory (English translation),  
 {\it Birkh\"auser }, 1996


\bibitem{ 8} %Ke
M. Kervaire: Les noeudes de dimensions sup\'ereures 
 {\it Bull.Soc.Math.France},  93, 225-271, 1965  


\bibitem{9} %KW
 M. Kervaire and C. Weber: 
 A survey of multidimensional knots. In: Knot theory
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vol. 685, 61-134, 1965 


\bibitem{ 10 }%Ki
R. Kirby:  The topology of 4-manifolds 
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vol. 1374, 1989


 
\bibitem{ 11 }%L
J. Levine: Knot cobordism in codimension two
 {\it  Comment. Math. Helv.},  44, 229-244, 1969

 
\bibitem{ 12} %Ma
 T. Matumoto: 
 On a weakly unknotted 2-sphere in a simply connected 4-manifold
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\bibitem{ 13 }%Mi
J. W. Milnor:  Singular points of complex hypersurfaces
 {\it  Ann. of Math. Stud.}, vol. 61, 1968


 
\bibitem{ 14} %MN
 H. Murakami and Y. Nakanishi: 
  On a certain move generating link-homology 
 {\it Math. Ann.}, 284,  75-89, 1989 


 
\bibitem{15} %N
 W. Neuman: Cyclic suspension of knots and periodicity 
 of signature for singularities 
 {\it Bull. Amer. Math. Soc.},  80, 977-981, 1974



\bibitem{ 16 }%O1
 E. Ogasa:  Link cobordism and the intersection of slice discs
 {\it Bull. London. Math. Soc} 
 31, 1999, 1-8

\bibitem{ 17} %O2 
 E. Ogasa: 
 Intersectional pairs of $n$-knots, local moves of $n$-knots and 
 their associated invariants of $n$-knots
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\bibitem{ 18} %O3 
E. Ogasa: Some properties of ordinary sense slice links:
Some answers to the problem (26) of Fox 
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\bibitem{  19}  %O4      
 E. Ogasa:  
The intersection of three spheres in a sphere and a new application of 
the Sato-Levine invariant  
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\bibitem{ 20} 
  M. Saito:  On the unoriented Sato-Levine invariant
 {\it J. Knot Theory Ramifications}, 2, 335-358, 1993
 



{\bf Acknowledgments.}
 Theorem 1.1. and 1.3 are answers to Prof. Tsuboi's questions 
on the author's paper [16].
%[O1]. 
The author would like to thank Prof. Takashi Tsuboi for his advice. 

The author obtained an idea of a way of surgery of submanifolds 
in this paper  
from p.490 of [12]. 
%[Ma].       
Of course the author's way are different from his. 
After obtaining the idea, 
he got acquainted with Prof. Takao Matumoto and discussed with him.
The author would like to thank Prof. Takao Matumoto 
for the valuable discussion. 

The author obtained an idea of Problem 1.4 and that of [19] 
when he talked about [17] with Dr. Yoshitake Hashimoto.  
The author would like to thank Dr. Yoshitake Hashimoto 
for the valuable discussion. 

The author would like to thank Prof. Charles Livingston for his interest 
in Problem 1.4.  

\end{thebibliography} 



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\special{fp}%
% LINE 2 0 3 2
% 2 6820 3760 6390 3450
% 
\special{pn 8}%
\special{pa 6820 3360}%
\special{pa 6390 3050}%
\special{fp}%
% LINE 2 0 3 3
% 2 6820 3760 6390 3450
% 
\special{pn 8}%
\special{pa 6820 3360}%
\special{pa 6390 3050}%
\special{fp}%
% LINE 2 0 3 4
% 2 7370 2120 6940 1810
% 
\special{pn 8}%
\special{pa 7370 1720}%
\special{pa 6940 1410}%
\special{fp}%
% LINE 2 2 3 0
% 2 6760 3610 6260 3620
% 
\special{pn 8}%
\special{pa 6760 3210}%
\special{pa 6260 3220}%
\special{dt 0.045}%
\special{pa 6260 3220}%
\special{pa 6261 3220}%
\special{dt 0.045}%
% SPLINE 2 2 3 0
% 4 6460 3520 6390 3550 6390 3620 6390 3630
% 
\special{pn 8}%
\special{pa 6460 3120}%
\special{pa 6424 3128}%
\special{pa 6396 3142}%
\special{pa 6386 3167}%
\special{pa 6388 3201}%
\special{pa 6390 3230}%
\special{sp -0.045}%
% STR 2 0 3 0
% 3 6330 4980 6330 5080 2 0
% $0<u<1$
\put(63.3000,-46.8000){\makebox(0,0)[lb]{$0<u<1$}}%
% STR 2 0 3 0
% 3 70 5110 70 5210 2 0
% Figure 2
\put(0.7000,-48.1000){\makebox(0,0)[lb]{Figure 2}}%
% VECTOR 2 0 3 0
% 2 1450 2700 1450 390
% 
\special{pn 8}%
\special{pa 1450 2300}%
\special{pa 1450 -10}%
\special{fp}%
\special{sh 1}%
\special{pa 1450 -10}%
\special{pa 1430 57}%
\special{pa 1450 43}%
\special{pa 1470 57}%
\special{pa 1450 -10}%
\special{fp}%
\end{picture}%
}





\newpage
%\rotatebox[origin=c]{90}{\input {4.2.naka.tex}}
\rotatebox[origin=c]{90}{
%WinTpicVersion2.15
\unitlength 0.1in
\begin{picture}(51.40,46.00)(1.10,-45.90)
% POLYGON 2 0 3 0
% 6 320 3940 1130 1520 4130 1530 3320 3930 3320 3940 320 3940
% 
\special{pn 8}%
\special{pa 320 3540}%
\special{pa 1130 1120}%
\special{pa 4130 1130}%
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% LINE 2 0 3 0
% 2 1460 4030 1460 4840
% 
\special{pn 8}%
\special{pa 1460 3630}%
\special{pa 1460 4440}%
\special{fp}%
% CIRCLE 2 0 0 0
% 4 3210 1920 3270 1970 3270 1970 3270 1970
% 
\special{pn 8}%
\special{sh 0.600}%
\special{ar 3210 1520 78 78  0.0000000 6.2831853}%
% CIRCLE 2 0 0 0
% 4 2660 3530 2720 3580 2720 3580 2720 3580
% 
\special{pn 8}%
\special{sh 0.600}%
\special{ar 2660 3130 78 78  0.0000000 6.2831853}%
% STR 2 0 3 0
% 3 1040 660 1040 760 2 0
% $v$
\put(10.4000,-3.6000){\makebox(0,0)[lb]{$v$}}%
% BOX 2 5 2 0
% 2 1410 2710 1490 840
% 
\special{pn 8}%
\special{sh 0}%
\special{pa 1410 2310}%
\special{pa 1490 2310}%
\special{pa 1490 440}%
\special{pa 1410 440}%
\special{pa 1410 2310}%
\special{ip}%
% VECTOR 2 0 3 0
% 2 1450 2700 1450 390
% 
\special{pn 8}%
\special{pa 1450 2300}%
\special{pa 1450 -10}%
\special{fp}%
\special{sh 1}%
\special{pa 1450 -10}%
\special{pa 1430 57}%
\special{pa 1450 43}%
\special{pa 1470 57}%
\special{pa 1450 -10}%
\special{fp}%
% LINE 2 0 3 0
% 2 3170 1570 2640 3190
% 
\special{pn 8}%
\special{pa 3170 1170}%
\special{pa 2640 2790}%
\special{fp}%
% LINE 2 0 3 0
% 2 2650 3150 2650 3510
% 
\special{pn 8}%
\special{pa 2650 2750}%
\special{pa 2650 3110}%
\special{fp}%
% LINE 2 0 3 0
% 2 3180 1560 3200 1890
% 
\special{pn 8}%
\special{pa 3180 1160}%
\special{pa 3200 1490}%
\special{fp}%
% LINE 2 2 3 0
% 2 3200 1910 3200 2250
% 
\special{pn 8}%
\special{pa 3200 1510}%
\special{pa 3200 1850}%
\special{dt 0.045}%
\special{pa 3200 1850}%
\special{pa 3200 1849}%
\special{dt 0.045}%
% LINE 2 2 3 0
% 2 2660 3520 2660 3840
% 
\special{pn 8}%
\special{pa 2660 3120}%
\special{pa 2660 3440}%
\special{dt 0.045}%
\special{pa 2660 3440}%
\special{pa 2660 3439}%
\special{dt 0.045}%
% LINE 2 2 3 0
% 2 3210 2240 2680 3830
% 
\special{pn 8}%
\special{pa 3210 1840}%
\special{pa 2680 3430}%
\special{dt 0.045}%
\special{pa 2680 3430}%
\special{pa 2680 3429}%
\special{dt 0.045}%
% LINE 2 0 3 0
% 2 3200 1930 2670 3530
% 
\special{pn 8}%
\special{pa 3200 1530}%
\special{pa 2670 3130}%
\special{fp}%
% VECTOR 2 1 3 0
% 2 3710 3700 3260 1980
% 
\special{pn 8}%
\special{pa 3710 3300}%
\special{pa 3260 1580}%
\special{da 0.070}%
\special{sh 1}%
\special{pa 3260 1580}%
\special{pa 3258 1650}%
\special{pa 3273 1632}%
\special{pa 3296 1639}%
\special{pa 3260 1580}%
\special{fp}%
% VECTOR 2 1 3 0
% 2 3550 3900 2720 3560
% 
\special{pn 8}%
\special{pa 3550 3500}%
\special{pa 2720 3160}%
\special{da 0.070}%
\special{sh 1}%
\special{pa 2720 3160}%
\special{pa 2774 3204}%
\special{pa 2769 3180}%
\special{pa 2789 3167}%
\special{pa 2720 3160}%
\special{fp}%
% STR 2 0 3 0
% 3 3610 3880 3610 3980 2 0
% $L_2$
\put(36.1000,-35.8000){\makebox(0,0)[lb]{$L_2$}}%
% VECTOR 2 1 3 0
% 2 4090 2490 3080 2320
% 
\special{pn 8}%
\special{pa 4090 2090}%
\special{pa 3080 1920}%
\special{da 0.070}%
\special{sh 1}%
\special{pa 3080 1920}%
\special{pa 3142 1951}%
\special{pa 3133 1929}%
\special{pa 3149 1911}%
\special{pa 3080 1920}%
\special{fp}%
% STR 2 0 3 0
% 3 4070 2610 4070 2710 2 0
% $F$
\put(40.7000,-23.1000){\makebox(0,0)[lb]{$F$}}%
% STR 2 0 3 0
% 3 110 2280 110 2380 2 0
% $g(S^3_2)$
\put(1.1000,-19.8000){\makebox(0,0)[lb]{$g(S^3_2)$}}%
% STR 2 0 3 0
% 3 2600 4900 2600 5000 2 0
% $u=1$
\put(26.0000,-46.0000){\makebox(0,0)[lb]{$u=1$}}%
% STR 2 0 3 0
% 3 5250 5060 5250 5160 2 0
% Figure 3
\put(52.5000,-47.6000){\makebox(0,0)[lb]{Figure 3}}%
% LINE 2 2 3 0
% 2 1030 1850 1340 1850
% 
\special{pn 8}%
\special{pa 1030 1450}%
\special{pa 1340 1450}%
\special{dt 0.045}%
\special{pa 1340 1450}%
\special{pa 1339 1450}%
\special{dt 0.045}%
% LINE 2 2 3 0
% 2 920 2200 1320 2200
% 
\special{pn 8}%
\special{pa 920 1800}%
\special{pa 1320 1800}%
\special{dt 0.045}%
\special{pa 1320 1800}%
\special{pa 1319 1800}%
\special{dt 0.045}%
% LINE 2 2 3 0
% 2 700 2860 3420 2870
% 
\special{pn 8}%
\special{pa 700 2460}%
\special{pa 3420 2470}%
\special{dt 0.045}%
\special{pa 3420 2470}%
\special{pa 3419 2470}%
\special{dt 0.045}%
% LINE 2 2 3 0
% 2 660 3260 3450 3260
% 
\special{pn 8}%
\special{pa 660 2860}%
\special{pa 3450 2860}%
\special{dt 0.045}%
\special{pa 3450 2860}%
\special{pa 3449 2860}%
\special{dt 0.045}%
% LINE 2 2 3 0
% 2 530 3680 3260 3690
% 
\special{pn 8}%
\special{pa 530 3280}%
\special{pa 3260 3290}%
\special{dt 0.045}%
\special{pa 3260 3290}%
\special{pa 3259 3290}%
\special{dt 0.045}%
% LINE 2 2 3 0
% 2 1600 1850 3910 1860
% 
\special{pn 8}%
\special{pa 1600 1450}%
\special{pa 3910 1460}%
\special{dt 0.045}%
\special{pa 3910 1460}%
\special{pa 3909 1460}%
\special{dt 0.045}%
% LINE 2 2 3 0
% 2 1590 2200 3830 2200
% 
\special{pn 8}%
\special{pa 1590 1800}%
\special{pa 3830 1800}%
\special{dt 0.045}%
\special{pa 3830 1800}%
\special{pa 3829 1800}%
\special{dt 0.045}%
% LINE 2 2 3 0
% 2 890 2540 1310 2540
% 
\special{pn 8}%
\special{pa 890 2140}%
\special{pa 1310 2140}%
\special{dt 0.045}%
\special{pa 1310 2140}%
\special{pa 1309 2140}%
\special{dt 0.045}%
% LINE 2 2 3 0
% 2 1600 2540 3620 2540
% 
\special{pn 8}%
\special{pa 1600 2140}%
\special{pa 3620 2140}%
\special{dt 0.045}%
\special{pa 3620 2140}%
\special{pa 3619 2140}%
\special{dt 0.045}%
\end{picture}%
}


\newpage
%\rotatebox[origin=c]{90}{\input {4.2.migi.tex}}
\rotatebox[origin=c]{90}{
%WinTpicVersion2.15
\unitlength 0.1in
\begin{picture}(46.95,46.20)(10.30,-47.60)
% VECTOR 2 0 3 0
% 2 5720 4840 5720 560
% 
\special{pn 8}%
\special{pa 5720 4440}%
\special{pa 5720 160}%
\special{fp}%
\special{sh 1}%
\special{pa 5720 160}%
\special{pa 5700 227}%
\special{pa 5720 213}%
\special{pa 5740 227}%
\special{pa 5720 160}%
\special{fp}%
% STR 2 0 3 1
% 3 5160 610 5160 710 2 0
% $v$
\put(51.6000,-3.1000){\makebox(0,0)[lb]{$v$}}%
% POLYGON 2 0 1 2
% 6 6940 1610 6950 2280 6410 3920 6400 3230 6400 3230 6940 1610
% 
\special{pn 8}%
\special{sh 0.300}%
\special{pa 6940 1210}%
\special{pa 6950 1880}%
\special{pa 6410 3520}%
\special{pa 6400 2830}%
\special{pa 6400 2830}%
\special{pa 6940 1210}%
\special{fp}%
% STR 2 0 3 3
% 3 6190 4960 6190 5060 2 0
% $u=\frac{\pi}{2}$
\put(61.9000,-46.6000){\makebox(0,0)[lb]{$u=\frac{\pi}{2}$}}%
% VECTOR 2 0 3 0
% 2 1590 4840 1590 560
% 
\special{pn 8}%
\special{pa 1590 4440}%
\special{pa 1590 160}%
\special{fp}%
\special{sh 1}%
\special{pa 1590 160}%
\special{pa 1570 227}%
\special{pa 1590 213}%
\special{pa 1610 227}%
\special{pa 1590 160}%
\special{fp}%
% STR 2 0 3 0
% 3 1030 610 1030 710 2 0
% $v$
\put(10.3000,-3.1000){\makebox(0,0)[lb]{$v$}}%
% LINE 2 0 3 0
% 2 2270 3250 2800 1630
% 
\special{pn 8}%
\special{pa 2270 2850}%
\special{pa 2800 1230}%
\special{fp}%
% LINE 2 0 3 0
% 2 2350 3830 2880 2180
% 
\special{pn 8}%
\special{pa 2350 3430}%
\special{pa 2880 1780}%
\special{fp}%
% LINE 2 0 3 0
% 2 2360 3820 2280 3210
% 
\special{pn 8}%
\special{pa 2360 3420}%
\special{pa 2280 2810}%
\special{fp}%
% LINE 2 0 3 0
% 2 2880 2190 2820 1620
% 
\special{pn 8}%
\special{pa 2880 1790}%
\special{pa 2820 1220}%
\special{fp}%
% LINE 2 2 3 0
% 2 2390 3530 2150 3530
% 
\special{pn 8}%
\special{pa 2390 3130}%
\special{pa 2150 3130}%
\special{dt 0.045}%
\special{pa 2150 3130}%
\special{pa 2151 3130}%
\special{dt 0.045}%
% STR 2 0 3 0
% 3 2020 4870 2020 4970 2 0
% $1<u<\frac{\pi}{2}$
\put(20.2000,-45.7000){\makebox(0,0)[lb]{$1<u<\frac{\pi}{2}$}}%
% STR 2 0 3 0
% 3 3870 5230 3870 5330 2 0
% Figure 4
\put(38.7000,-49.3000){\makebox(0,0)[lb]{Figure 4}}%
\end{picture}%
}



\newpage
%\rotatebox[origin=c]{90}{\input {4.3.tex}}
\rotatebox[origin=c]{90}{
%WinTpicVersion2.15
\unitlength 0.1in
\begin{picture}(70.60,32.30)(4.00,-40.00)
% CIRCLE 2 0 3 0
% 4 1830 2680 2790 3740 2260 3280 2260 3280
% 
\special{pn 8}%
\special{ar 1830 2280 1430 1430  0.0000000 6.2831853}%
% VECTOR 2 0 3 0
% 2 720 3580 2530 1460
% 
\special{pn 8}%
\special{pa 720 3180}%
\special{pa 2530 1060}%
\special{fp}%
\special{sh 1}%
\special{pa 2530 1060}%
\special{pa 2472 1098}%
\special{pa 2495 1101}%
\special{pa 2502 1124}%
\special{pa 2530 1060}%
\special{fp}%
% VECTOR 2 0 3 0
% 2 2930 1780 1120 3910
% 
\special{pn 8}%
\special{pa 2930 1380}%
\special{pa 1120 3510}%
\special{fp}%
\special{sh 1}%
\special{pa 1120 3510}%
\special{pa 1178 3472}%
\special{pa 1155 3469}%
\special{pa 1148 3446}%
\special{pa 1120 3510}%
\special{fp}%
% LINE 2 0 3 0
% 2 2130 2600 1840 2390
% 
\special{pn 8}%
\special{pa 2130 2200}%
\special{pa 1840 1990}%
\special{fp}%
% LINE 2 0 3 0
% 2 1820 2980 1540 2740
% 
\special{pn 8}%
\special{pa 1820 2580}%
\special{pa 1540 2340}%
\special{fp}%
% VECTOR 2 0 3 0
% 2 4230 2890 3680 2892
% 
\special{pn 8}%
\special{pa 4230 2490}%
\special{pa 3680 2492}%
\special{fp}%
\special{sh 1}%
\special{pa 3680 2492}%
\special{pa 3747 2512}%
\special{pa 3733 2492}%
\special{pa 3747 2472}%
\special{pa 3680 2492}%
\special{fp}%
% VECTOR 2 0 3 0
% 2 3700 2660 4280 2660
% 
\special{pn 8}%
\special{pa 3700 2260}%
\special{pa 4280 2260}%
\special{fp}%
\special{sh 1}%
\special{pa 4280 2260}%
\special{pa 4213 2240}%
\special{pa 4227 2260}%
\special{pa 4213 2280}%
\special{pa 4280 2260}%
\special{fp}%
% STR 2 0 3 0
% 3 3600 3300 3600 3400 2 0
% pass-move
\put(36.0000,-30.0000){\makebox(0,0)[lb]{pass-move}}%
% STR 2 0 3 0
% 3 3530 4470 3530 4570 2 0
% Figure 5
\put(35.3000,-41.7000){\makebox(0,0)[lb]{Figure 5}}%
% CIRCLE 2 0 3 0
% 4 6030 2600 4967 3557 5428 3029 5428 3029
% 
\special{pn 8}%
\special{ar 6030 2200 1430 1430  0.0000000 6.2831853}%
% LINE 2 0 3 0
% 2 5730 2589 5970 2310
% 
\special{pn 8}%
\special{pa 5730 2189}%
\special{pa 5970 1910}%
\special{fp}%
% VECTOR 2 0 3 0
% 2 6020 3030 5380 3870
% 
\special{pn 8}%
\special{pa 6020 2630}%
\special{pa 5380 3470}%
\special{fp}%
\special{sh 1}%
\special{pa 5380 3470}%
\special{pa 5436 3429}%
\special{pa 5412 3428}%
\special{pa 5404 3405}%
\special{pa 5380 3470}%
\special{fp}%
% LINE 2 0 3 0
% 2 5640 2740 4960 3530
% 
\special{pn 8}%
\special{pa 5640 2340}%
\special{pa 4960 3130}%
\special{fp}%
% VECTOR 2 0 3 0
% 2 6100 2170 6730 1370
% 
\special{pn 8}%
\special{pa 6100 1770}%
\special{pa 6730 970}%
\special{fp}%
\special{sh 1}%
\special{pa 6730 970}%
\special{pa 6673 1010}%
\special{pa 6697 1012}%
\special{pa 6704 1035}%
\special{pa 6730 970}%
\special{fp}%
% LINE 2 0 3 0
% 2 6110 2930 6390 2590
% 
\special{pn 8}%
\special{pa 6110 2530}%
\special{pa 6390 2190}%
\special{fp}%
% LINE 2 0 3 0
% 2 6480 2510 7120 1700
% 
\special{pn 8}%
\special{pa 6480 2110}%
\special{pa 7120 1300}%
\special{fp}%
% VECTOR 2 0 3 0
% 2 2270 2720 3090 3350
% 
\special{pn 8}%
\special{pa 2270 2320}%
\special{pa 3090 2950}%
\special{fp}%
\special{sh 1}%
\special{pa 3090 2950}%
\special{pa 3049 2894}%
\special{pa 3048 2918}%
\special{pa 3025 2925}%
\special{pa 3090 2950}%
\special{fp}%
% LINE 2 0 3 0
% 2 1730 2270 920 1610
% 
\special{pn 8}%
\special{pa 1730 1870}%
\special{pa 920 1210}%
\special{fp}%
% LINE 2 0 3 0
% 2 2790 3730 1920 3070
% 
\special{pn 8}%
\special{pa 2790 3330}%
\special{pa 1920 2670}%
\special{fp}%
% VECTOR 2 0 3 0
% 2 1450 2650 600 1950
% 
\special{pn 8}%
\special{pa 1450 2250}%
\special{pa 600 1550}%
\special{fp}%
\special{sh 1}%
\special{pa 600 1550}%
\special{pa 639 1608}%
\special{pa 641 1584}%
\special{pa 664 1577}%
\special{pa 600 1550}%
\special{fp}%
% VECTOR 2 0 3 0
% 2 5190 1480 7300 3280
% 
\special{pn 8}%
\special{pa 5190 1080}%
\special{pa 7300 2880}%
\special{fp}%
\special{sh 1}%
\special{pa 7300 2880}%
\special{pa 7262 2822}%
\special{pa 7259 2845}%
\special{pa 7236 2852}%
\special{pa 7300 2880}%
\special{fp}%
% VECTOR 2 0 3 0
% 2 6940 3690 4790 1910
% 
\special{pn 8}%
\special{pa 6940 3290}%
\special{pa 4790 1510}%
\special{fp}%
\special{sh 1}%
\special{pa 4790 1510}%
\special{pa 4829 1568}%
\special{pa 4831 1544}%
\special{pa 4854 1537}%
\special{pa 4790 1510}%
\special{fp}%
\end{picture}%
}



\newpage

%\input {4.4.B.tex}
%WinTpicVersion2.15
\unitlength 0.1in
\begin{picture}(49.70,45.82)(4.60,-48.11)
% CIRCLE 2 0 3 0
% 4 2980 2920 1775 971 1775 971 1775 971
% 
\special{pn 8}%
\special{ar 2980 2520 2291 2291  0.0000000 6.2831853}%
% STR 2 0 3 0
% 3 5430 2440 5430 2540 2 0
% $B^3$
\put(54.3000,-21.4000){\makebox(0,0)[lb]{$B^3$}}%
% STR 2 0 3 0
% 3 460 4870 460 4970 2 0
% Figure 6
\put(4.6000,-45.7000){\makebox(0,0)[lb]{Figure 6}}%
% VECTOR 2 0 3 0
% 2 1120 4250 4530 1270
% 
\special{pn 8}%
\special{pa 1120 3850}%
\special{pa 4530 870}%
\special{fp}%
\special{sh 1}%
\special{pa 4530 870}%
\special{pa 4467 899}%
\special{pa 4490 905}%
\special{pa 4493 929}%
\special{pa 4530 870}%
\special{fp}%
% VECTOR 2 0 3 0
% 2 4870 1650 1460 4610
% 
\special{pn 8}%
\special{pa 4870 1250}%
\special{pa 1460 4210}%
\special{fp}%
\special{sh 1}%
\special{pa 1460 4210}%
\special{pa 1523 4181}%
\special{pa 1500 4175}%
\special{pa 1497 4151}%
\special{pa 1460 4210}%
\special{fp}%
% VECTOR 2 0 3 0
% 2 2560 2850 1290 1410
% 
\special{pn 8}%
\special{pa 2560 2450}%
\special{pa 1290 1010}%
\special{fp}%
\special{sh 1}%
\special{pa 1290 1010}%
\special{pa 1319 1073}%
\special{pa 1325 1050}%
\special{pa 1349 1047}%
\special{pa 1290 1010}%
\special{fp}%
% VECTOR 2 0 3 0
% 2 3390 3070 4650 4500
% 
\special{pn 8}%
\special{pa 3390 2670}%
\special{pa 4650 4100}%
\special{fp}%
\special{sh 1}%
\special{pa 4650 4100}%
\special{pa 4621 4037}%
\special{pa 4615 4060}%
\special{pa 4591 4063}%
\special{pa 4650 4100}%
\special{fp}%
% LINE 2 0 3 0
% 2 2940 2520 1660 1050
% 
\special{pn 8}%
\special{pa 2940 2120}%
\special{pa 1660 650}%
\special{fp}%
% LINE 2 0 3 0
% 2 3000 3370 4260 4830
% 
\special{pn 8}%
\special{pa 3000 2970}%
\special{pa 4260 4430}%
\special{fp}%
% LINE 2 2 3 0
% 2 1220 4370 4630 1350
% 
\special{pn 8}%
\special{pa 1220 3970}%
\special{pa 4630 950}%
\special{dt 0.045}%
\special{pa 4630 950}%
\special{pa 4630 950}%
\special{dt 0.045}%
% LINE 2 2 3 0
% 2 1340 4500 4740 1500
% 
\special{pn 8}%
\special{pa 1340 4100}%
\special{pa 4740 1100}%
\special{dt 0.045}%
\special{pa 4740 1100}%
\special{pa 4740 1100}%
\special{dt 0.045}%
% LINE 2 2 3 0
% 8 2820 2630 1550 1170 2710 2760 1440 1320 3300 3160 4490 4540 3170 3330 4360 4640
% 
\special{pn 8}%
\special{pa 2820 2230}%
\special{pa 1550 770}%
\special{dt 0.045}%
\special{pa 1550 770}%
\special{pa 1550 771}%
\special{dt 0.045}%
\special{pa 2710 2360}%
\special{pa 1440 920}%
\special{dt 0.045}%
\special{pa 1440 920}%
\special{pa 1440 920}%
\special{dt 0.045}%
\special{pa 3300 2760}%
\special{pa 4490 4140}%
\special{dt 0.045}%
\special{pa 4490 4140}%
\special{pa 4490 4139}%
\special{dt 0.045}%
\special{pa 3170 2930}%
\special{pa 4360 4240}%
\special{dt 0.045}%
\special{pa 4360 4240}%
\special{pa 4360 4240}%
\special{dt 0.045}%
\end{picture}%



The shaded part is $F\cap B^3$.



\newpage

%\input {4.5.B.tex}
%WinTpicVersion2.15
\unitlength 0.1in
\begin{picture}(45.98,45.82)(8.12,-46.25)
% LINE 2 0 3 0
% 2 3731 1414 3942 1593
% 
\special{pn 8}%
\special{pa 3731 1014}%
\special{pa 3942 1193}%
\special{fp}%
% SPLINE 2 0 3 0
% 10 4253 2923 4404 3042 4513 3062 4633 3022 4793 2882 4813 2692 4773 2492 4642 2282 4492 2092 4492 2092
% 
\special{pn 8}%
\special{pa 4253 2523}%
\special{pa 4276 2546}%
\special{pa 4299 2569}%
\special{pa 4323 2590}%
\special{pa 4349 2610}%
\special{pa 4375 2627}%
\special{pa 4404 2642}%
\special{pa 4434 2653}%
\special{pa 4466 2660}%
\special{pa 4498 2663}%
\special{pa 4530 2660}%
\special{pa 4560 2653}%
\special{pa 4591 2642}%
\special{pa 4620 2628}%
\special{pa 4650 2613}%
\special{pa 4679 2596}%
\special{pa 4707 2577}%
\special{pa 4733 2557}%
\special{pa 4756 2535}%
\special{pa 4776 2510}%
\special{pa 4792 2484}%
\special{pa 4804 2455}%
\special{pa 4811 2425}%
\special{pa 4815 2393}%
\special{pa 4816 2360}%
\special{pa 4815 2327}%
\special{pa 4813 2293}%
\special{pa 4810 2260}%
\special{pa 4806 2228}%
\special{pa 4801 2196}%
\special{pa 4795 2165}%
\special{pa 4787 2134}%
\special{pa 4778 2104}%
\special{pa 4766 2075}%
\special{pa 4752 2046}%
\special{pa 4737 2018}%
\special{pa 4720 1991}%
\special{pa 4703 1964}%
\special{pa 4684 1938}%
\special{pa 4665 1912}%
\special{pa 4645 1886}%
\special{pa 4625 1860}%
\special{pa 4605 1835}%
\special{pa 4586 1810}%
\special{pa 4566 1784}%
\special{pa 4546 1759}%
\special{pa 4526 1735}%
\special{pa 4506 1710}%
\special{pa 4492 1692}%
\special{sp}%
% SPLINE 2 0 3 0
% 9 3062 1505 3041 1374 3051 1304 3101 1193 3252 1144 3351 1134 3411 1163 3521 1234 3521 1234
% 
\special{pn 8}%
\special{pa 3062 1105}%
\special{pa 3054 1073}%
\special{pa 3047 1042}%
\special{pa 3043 1010}%
\special{pa 3041 979}%
\special{pa 3043 947}%
\special{pa 3049 915}%
\special{pa 3056 883}%
\special{pa 3066 851}%
\special{pa 3080 821}%
\special{pa 3098 796}%
\special{pa 3121 778}%
\special{pa 3149 766}%
\special{pa 3181 757}%
\special{pa 3214 751}%
\special{pa 3248 745}%
\special{pa 3281 738}%
\special{pa 3313 732}%
\special{pa 3344 733}%
\special{pa 3374 742}%
\special{pa 3402 758}%
\special{pa 3430 775}%
\special{pa 3457 792}%
\special{pa 3484 810}%
\special{pa 3510 827}%
\special{pa 3521 834}%
\special{sp}%
% STR 2 0 3 0
% 3 4410 3170 4410 3270 2 0
% $Y_2$
\put(44.1000,-28.7000){\makebox(0,0)[lb]{$Y_2$}}%
% STR 2 0 3 0
% 3 1430 2890 1430 2990 2 0
% $Y_1$
\put(14.3000,-25.9000){\makebox(0,0)[lb]{$Y_1$}}%
% STR 2 0 3 0
% 3 5410 4810 5410 4910 2 0
% Figure 7
\put(54.1000,-45.1000){\makebox(0,0)[lb]{Figure 7}}%
% STR 2 0 3 0
% 3 5010 680 5010 780 2 0
% $B^3$
\put(50.1000,-3.8000){\makebox(0,0)[lb]{$B^3$}}%
% VECTOR 2 0 3 0
% 2 2700 2640 1440 1210
% 
\special{pn 8}%
\special{pa 2700 2240}%
\special{pa 1440 810}%
\special{fp}%
\special{sh 1}%
\special{pa 1440 810}%
\special{pa 1469 873}%
\special{pa 1475 850}%
\special{pa 1499 847}%
\special{pa 1440 810}%
\special{fp}%
% VECTOR 2 0 3 0
% 2 3530 2920 4760 4300
% 
\special{pn 8}%
\special{pa 3530 2520}%
\special{pa 4760 3900}%
\special{fp}%
\special{sh 1}%
\special{pa 4760 3900}%
\special{pa 4731 3837}%
\special{pa 4725 3860}%
\special{pa 4701 3864}%
\special{pa 4760 3900}%
\special{fp}%
% LINE 2 2 3 0
% 2 3420 3010 4600 4350
% 
\special{pn 8}%
\special{pa 3420 2610}%
\special{pa 4600 3950}%
\special{dt 0.045}%
\special{pa 4600 3950}%
\special{pa 4600 3949}%
\special{dt 0.045}%
% LINE 2 2 3 0
% 2 4490 4480 3280 3110
% 
\special{pn 8}%
\special{pa 4490 4080}%
\special{pa 3280 2710}%
\special{dt 0.045}%
\special{pa 3280 2710}%
\special{pa 3280 2710}%
\special{dt 0.045}%
% LINE 2 2 3 0
% 2 2940 2440 1690 1030
% 
\special{pn 8}%
\special{pa 2940 2040}%
\special{pa 1690 630}%
\special{dt 0.045}%
\special{pa 1690 630}%
\special{pa 1690 630}%
\special{dt 0.045}%
% LINE 2 2 3 0
% 2 2810 2550 1590 1150
% 
\special{pn 8}%
\special{pa 2810 2150}%
\special{pa 1590 750}%
\special{dt 0.045}%
\special{pa 1590 750}%
\special{pa 1590 751}%
\special{dt 0.045}%
% CIRCLE 2 0 3 0
% 4 3103 2734 1896 787 1896 787 1896 787
% 
\special{pn 8}%
\special{ar 3103 2334 2291 2291  0.0000000 6.2831853}%
% VECTOR 2 0 3 0
% 2 1260 4070 4640 1100
% 
\special{pn 8}%
\special{pa 1260 3670}%
\special{pa 4640 700}%
\special{fp}%
\special{sh 1}%
\special{pa 4640 700}%
\special{pa 4577 729}%
\special{pa 4600 735}%
\special{pa 4603 759}%
\special{pa 4640 700}%
\special{fp}%
% VECTOR 2 0 3 0
% 2 4970 1470 1600 4430
% 
\special{pn 8}%
\special{pa 4970 1070}%
\special{pa 1600 4030}%
\special{fp}%
\special{sh 1}%
\special{pa 1600 4030}%
\special{pa 1663 4001}%
\special{pa 1640 3995}%
\special{pa 1637 3971}%
\special{pa 1600 4030}%
\special{fp}%
% LINE 2 0 3 0
% 2 3040 2360 1770 910
% 
\special{pn 8}%
\special{pa 3040 1960}%
\special{pa 1770 510}%
\special{fp}%
% LINE 2 0 3 0
% 2 3150 3200 4390 4630
% 
\special{pn 8}%
\special{pa 3150 2800}%
\special{pa 4390 4230}%
\special{fp}%
% LINE 2 2 3 0
% 2 1380 4190 4720 1230
% 
\special{pn 8}%
\special{pa 1380 3790}%
\special{pa 4720 830}%
\special{dt 0.045}%
\special{pa 4720 830}%
\special{pa 4720 830}%
\special{dt 0.045}%
% LINE 2 2 3 0
% 2 1500 4320 4820 1360
% 
\special{pn 8}%
\special{pa 1500 3920}%
\special{pa 4820 960}%
\special{dt 0.045}%
\special{pa 4820 960}%
\special{pa 4820 960}%
\special{dt 0.045}%
% POLYGON 2 5 2 0
% 5 3391 1843 3270 1963 4102 2815 4224 2695 3391 1843
% 
\special{pn 8}%
\special{sh 0}%
\special{pa 3391 1443}%
\special{pa 3270 1563}%
\special{pa 4102 2415}%
\special{pa 4224 2295}%
\special{pa 3391 1443}%
\special{ip}%
% POLYGON 2 5 2 0
% 5 3122 1923 3001 1803 2160 2646 2281 2766 3122 1923
% 
\special{pn 8}%
\special{sh 0}%
\special{pa 3122 1523}%
\special{pa 3001 1403}%
\special{pa 2160 2246}%
\special{pa 2281 2366}%
\special{pa 3122 1523}%
\special{ip}%
% SPLINE 2 0 3 0
% 35 2331 1425 2401 1375 2611 1165 2791 1044 3050 875 3141 824 3321 783 3471 774 3621 913 3681 1004 3722 1113 3721 1243 3671 1344 3592 1434 3442 1574 3362 1653 3262 1734 3202 1794 3112 1874 3012 1965 2563 2335 2523 2404 2473 2455 2443 2505 2432 2505 2243 2635 2063 2725 1934 2765 1823 2766 1693 2595 1643 2396 1642 2145 1712 1976 1832 1816 1832 1825
% 
\special{pn 8}%
\special{pa 2331 1025}%
\special{pa 2358 1007}%
\special{pa 2384 988}%
\special{pa 2409 969}%
\special{pa 2433 948}%
\special{pa 2456 925}%
\special{pa 2478 902}%
\special{pa 2500 879}%
\special{pa 2521 855}%
\special{pa 2543 831}%
\special{pa 2565 808}%
\special{pa 2588 786}%
\special{pa 2612 764}%
\special{pa 2637 744}%
\special{pa 2662 725}%
\special{pa 2689 708}%
\special{pa 2716 690}%
\special{pa 2743 673}%
\special{pa 2771 657}%
\special{pa 2798 640}%
\special{pa 2825 623}%
\special{pa 2852 605}%
\special{pa 2879 588}%
\special{pa 2905 570}%
\special{pa 2932 553}%
\special{pa 2958 535}%
\special{pa 2985 517}%
\special{pa 3012 500}%
\special{pa 3039 482}%
\special{pa 3066 465}%
\special{pa 3094 448}%
\special{pa 3122 433}%
\special{pa 3151 420}%
\special{pa 3180 410}%
\special{pa 3211 403}%
\special{pa 3242 396}%
\special{pa 3274 391}%
\special{pa 3307 386}%
\special{pa 3339 380}%
\special{pa 3373 374}%
\special{pa 3405 370}%
\special{pa 3437 369}%
\special{pa 3467 373}%
\special{pa 3495 383}%
\special{pa 3521 400}%
\special{pa 3546 421}%
\special{pa 3569 445}%
\special{pa 3590 471}%
\special{pa 3611 499}%
\special{pa 3630 526}%
\special{pa 3649 552}%
\special{pa 3666 579}%
\special{pa 3682 606}%
\special{pa 3696 634}%
\special{pa 3708 664}%
\special{pa 3718 695}%
\special{pa 3725 727}%
\special{pa 3728 759}%
\special{pa 3729 791}%
\special{pa 3725 823}%
\special{pa 3718 854}%
\special{pa 3706 884}%
\special{pa 3691 913}%
\special{pa 3674 940}%
\special{pa 3655 966}%
\special{pa 3634 990}%
\special{pa 3612 1014}%
\special{pa 3590 1036}%
\special{pa 3567 1058}%
\special{pa 3543 1080}%
\special{pa 3519 1102}%
\special{pa 3496 1123}%
\special{pa 3472 1145}%
\special{pa 3449 1167}%
\special{pa 3426 1190}%
\special{pa 3404 1213}%
\special{pa 3381 1235}%
\special{pa 3358 1257}%
\special{pa 3333 1277}%
\special{pa 3308 1296}%
\special{pa 3282 1316}%
\special{pa 3258 1337}%
\special{pa 3236 1360}%
\special{pa 3213 1383}%
\special{pa 3190 1405}%
\special{pa 3167 1427}%
\special{pa 3142 1447}%
\special{pa 3118 1468}%
\special{pa 3095 1490}%
\special{pa 3071 1512}%
\special{pa 3048 1534}%
\special{pa 3024 1555}%
\special{pa 2999 1575}%
\special{pa 2973 1595}%
\special{pa 2947 1614}%
\special{pa 2920 1632}%
\special{pa 2892 1650}%
\special{pa 2864 1668}%
\special{pa 2836 1685}%
\special{pa 2808 1703}%
\special{pa 2780 1720}%
\special{pa 2753 1738}%
\special{pa 2727 1757}%
\special{pa 2701 1776}%
\special{pa 2676 1796}%
\special{pa 2652 1818}%
\special{pa 2630 1840}%
\special{pa 2609 1864}%
\special{pa 2590 1889}%
\special{pa 2573 1916}%
\special{pa 2558 1945}%
\special{pa 2544 1974}%
\special{pa 2526 2000}%
\special{pa 2503 2021}%
\special{pa 2480 2044}%
\special{pa 2465 2076}%
\special{pa 2449 2103}%
\special{pa 2418 2105}%
\special{pa 2388 2112}%
\special{pa 2360 2127}%
\special{pa 2334 2146}%
\special{pa 2310 2169}%
\special{pa 2286 2194}%
\special{pa 2261 2218}%
\special{pa 2236 2241}%
\special{pa 2209 2260}%
\special{pa 2181 2276}%
\special{pa 2153 2290}%
\special{pa 2123 2303}%
\special{pa 2093 2314}%
\special{pa 2063 2325}%
\special{pa 2034 2336}%
\special{pa 2004 2346}%
\special{pa 1973 2355}%
\special{pa 1942 2363}%
\special{pa 1909 2370}%
\special{pa 1876 2373}%
\special{pa 1844 2371}%
\special{pa 1815 2363}%
\special{pa 1788 2348}%
\special{pa 1765 2327}%
\special{pa 1745 2301}%
\special{pa 1727 2272}%
\special{pa 1712 2241}%
\special{pa 1698 2208}%
\special{pa 1686 2177}%
\special{pa 1676 2146}%
\special{pa 1667 2115}%
\special{pa 1659 2084}%
\special{pa 1652 2054}%
\special{pa 1647 2022}%
\special{pa 1642 1990}%
\special{pa 1638 1958}%
\special{pa 1636 1924}%
\special{pa 1634 1890}%
\special{pa 1633 1857}%
\special{pa 1634 1825}%
\special{pa 1636 1793}%
\special{pa 1639 1764}%
\special{pa 1644 1736}%
\special{pa 1650 1711}%
\special{pa 1659 1685}%
\special{pa 1671 1658}%
\special{pa 1686 1627}%
\special{pa 1704 1592}%
\special{pa 1726 1550}%
\special{pa 1751 1504}%
\special{pa 1776 1460}%
\special{pa 1799 1425}%
\special{pa 1818 1404}%
\special{pa 1829 1403}%
\special{pa 1832 1425}%
\special{sp}%
% POLYGON 2 5 2 0
% 5 3171 1704 3172 1824 3331 1823 3332 1704 3171 1704
% 
\special{pn 8}%
\special{sh 0}%
\special{pa 3171 1304}%
\special{pa 3172 1424}%
\special{pa 3331 1423}%
\special{pa 3332 1304}%
\special{pa 3171 1304}%
\special{ip}%
% LINE 2 0 3 0
% 2 3062 1505 4263 2933
% 
\special{pn 8}%
\special{pa 3062 1105}%
\special{pa 4263 2533}%
\special{fp}%
\end{picture}%



\newpage
%\input {4.6.tex}


%WinTpicVersion2.15
\unitlength 0.1in
\begin{picture}(57.40,44.70)(1.80,-45.50)
% LINE 2 0 3 0
% 2 390 2790 4610 2790
% 
\special{pn 8}%
\special{pa 390 2390}%
\special{pa 4610 2390}%
\special{fp}%
% POLYLINE 2 0 3 0
% 5 1610 2800 1600 1400 3410 1410 3410 2790 3410 2790
% 
\special{pn 8}%
\special{pa 1610 2400}%
\special{pa 1600 1000}%
\special{pa 3410 1010}%
\special{pa 3410 2390}%
\special{pa 3410 2390}%
\special{fp}%
% POLYLINE 2 0 3 0
% 5 2410 2790 2400 4210 4210 4210 4200 2790 4200 2790
% 
\special{pn 8}%
\special{pa 2410 2390}%
\special{pa 2400 3810}%
\special{pa 4210 3810}%
\special{pa 4200 2390}%
\special{pa 4200 2390}%
\special{fp}%
% LINE 2 0 3 0
% 2 4610 2790 5190 2800
% 
\special{pn 8}%
\special{pa 4610 2390}%
\special{pa 5190 2400}%
\special{fp}%
% LINE 0 0 3 0
% 2 1200 2800 4610 2800
% 
\special{pn 20}%
\special{pa 1200 2400}%
\special{pa 4610 2400}%
\special{fp}%
% VECTOR 2 2 3 0
% 2 1030 3320 1300 2820
% 
\special{pn 8}%
\special{pa 1030 2920}%
\special{pa 1300 2420}%
\special{dt 0.045}%
\special{sh 1}%
\special{pa 1300 2420}%
\special{pa 1251 2469}%
\special{pa 1275 2467}%
\special{pa 1286 2488}%
\special{pa 1300 2420}%
\special{fp}%
% STR 2 0 3 0
% 3 560 3540 560 3640 2 0
% $B^3$
\put(5.6000,-32.4000){\makebox(0,0)[lb]{$B^3$}}%
% LINE 2 0 3 0
% 2 5010 610 5000 4950
% 
\special{pn 8}%
\special{pa 5010 210}%
\special{pa 5000 4550}%
\special{fp}%
% LINE 2 0 3 0
% 2 4850 2570 5750 4020
% 
\special{pn 8}%
\special{pa 4850 2170}%
\special{pa 5750 3620}%
\special{fp}%
% STR 2 0 3 0
% 3 180 2520 180 2620 2 0
% $R^3$
\put(1.8000,-22.2000){\makebox(0,0)[lb]{$R^3$}}%
% VECTOR 2 0 3 0
% 2 5010 660 5010 480
% 
\special{pn 8}%
\special{pa 5010 260}%
\special{pa 5010 80}%
\special{fp}%
\special{sh 1}%
\special{pa 5010 80}%
\special{pa 4990 147}%
\special{pa 5010 133}%
\special{pa 5030 147}%
\special{pa 5010 80}%
\special{fp}%
% VECTOR 2 0 3 0
% 2 630 2790 200 2790
% 
\special{pn 8}%
\special{pa 630 2390}%
\special{pa 200 2390}%
\special{fp}%
\special{sh 1}%
\special{pa 200 2390}%
\special{pa 267 2410}%
\special{pa 253 2390}%
\special{pa 267 2370}%
\special{pa 200 2390}%
\special{fp}%
% VECTOR 2 0 3 0
% 2 5650 3840 5830 4170
% 
\special{pn 8}%
\special{pa 5650 3440}%
\special{pa 5830 3770}%
\special{fp}%
\special{sh 1}%
\special{pa 5830 3770}%
\special{pa 5816 3702}%
\special{pa 5804 3723}%
\special{pa 5781 3721}%
\special{pa 5830 3770}%
\special{fp}%
% STR 2 0 3 0
% 3 5120 620 5120 720 2 0
% $v$
\put(51.2000,-3.2000){\makebox(0,0)[lb]{$v$}}%
% STR 2 0 3 0
% 3 5920 3890 5920 3990 2 0
% $u$
\put(59.2000,-35.9000){\makebox(0,0)[lb]{$u$}}%
% VECTOR 2 2 3 0
% 2 2310 2530 1630 2770
% 
\special{pn 8}%
\special{pa 2310 2130}%
\special{pa 1630 2370}%
\special{dt 0.045}%
\special{sh 1}%
\special{pa 1630 2370}%
\special{pa 1700 2367}%
\special{pa 1680 2352}%
\special{pa 1686 2329}%
\special{pa 1630 2370}%
\special{fp}%
% VECTOR 2 2 3 0
% 2 2770 2550 3400 2770
% 
\special{pn 8}%
\special{pa 2770 2150}%
\special{pa 3400 2370}%
\special{dt 0.045}%
\special{sh 1}%
\special{pa 3400 2370}%
\special{pa 3344 2329}%
\special{pa 3350 2352}%
\special{pa 3330 2367}%
\special{pa 3400 2370}%
\special{fp}%
% VECTOR 2 2 3 0
% 2 3210 3200 2460 2860
% 
\special{pn 8}%
\special{pa 3210 2800}%
\special{pa 2460 2460}%
\special{dt 0.045}%
\special{sh 1}%
\special{pa 2460 2460}%
\special{pa 2512 2506}%
\special{pa 2509 2482}%
\special{pa 2529 2469}%
\special{pa 2460 2460}%
\special{fp}%
% VECTOR 2 2 3 0
% 2 3600 3210 4150 2840
% 
\special{pn 8}%
\special{pa 3600 2810}%
\special{pa 4150 2440}%
\special{dt 0.045}%
\special{sh 1}%
\special{pa 4150 2440}%
\special{pa 4084 2461}%
\special{pa 4106 2470}%
\special{pa 4106 2494}%
\special{pa 4150 2440}%
\special{fp}%
% CIRCLE 2 0 0 0
% 4 1600 2820 1650 2850 1590 2830 1590 2830
% 
\special{pn 8}%
\special{sh 0.600}%
\special{ar 1600 2420 58 58  0.0000000 6.2831853}%
% CIRCLE 2 0 0 0
% 4 2400 2790 2450 2820 2390 2800 2390 2800
% 
\special{pn 8}%
\special{sh 0.600}%
\special{ar 2400 2390 58 58  0.0000000 6.2831853}%
% CIRCLE 2 0 0 0
% 4 3440 2810 3490 2840 3430 2820 3430 2820
% 
\special{pn 8}%
\special{sh 0.600}%
\special{ar 3440 2410 58 58  0.0000000 6.2831853}%
% CIRCLE 2 0 0 0
% 4 4190 2780 4240 2810 4180 2790 4180 2790
% 
\special{pn 8}%
\special{sh 0.600}%
\special{ar 4190 2380 58 58  0.0000000 6.2831853}%
% STR 2 0 3 0
% 3 2310 2380 2310 2480 2 0
% $Y_1$
\put(23.1000,-20.8000){\makebox(0,0)[lb]{$Y_1$}}%
% STR 2 0 3 0
% 3 3170 3320 3170 3420 2 0
% $Y_2$
\put(31.7000,-30.2000){\makebox(0,0)[lb]{$Y_2$}}%
% STR 2 0 3 0
% 3 1630 1180 1630 1280 2 0
% $h'^2_1$
\put(16.3000,-8.8000){\makebox(0,0)[lb]{$h'^2_1$}}%
% STR 2 0 3 0
% 3 3820 4390 3820 4490 2 0
% $h'^2_2$
\put(38.2000,-40.9000){\makebox(0,0)[lb]{$h'^2_2$}}%
% STR 2 0 3 0
% 3 800 4900 800 5000 2 0
% Figure 8
\put(8.0000,-46.0000){\makebox(0,0)[lb]{Figure 8}}%
\end{picture}%



 
\newpage
%\input {4.7.tex}

%WinTpicVersion2.15
\unitlength 0.1in
\begin{picture}(44.66,33.27)(6.09,-34.90)
% CIRCLE 2 0 3 0
% 4 1960 1970 2950 2890 2950 2890 2950 2890
% 
\special{pn 8}%
\special{ar 1960 1570 1351 1351  0.0000000 6.2831853}%
% CIRCLE 2 0 3 0
% 4 3724 1914 4718 2829 4718 2829 4718 2829
% 
\special{pn 8}%
\special{ar 3724 1514 1351 1351  0.0000000 6.2831853}%
% BOX 2 5 2 0
% 2 2700 790 2900 1050
% 
\special{pn 8}%
\special{sh 0}%
\special{pa 2700 390}%
\special{pa 2900 390}%
\special{pa 2900 650}%
\special{pa 2700 650}%
\special{pa 2700 390}%
\special{ip}%
% BOX 2 5 2 0
% 2 2770 2850 2970 3110
% 
\special{pn 8}%
\special{sh 0}%
\special{pa 2770 2450}%
\special{pa 2970 2450}%
\special{pa 2970 2710}%
\special{pa 2770 2710}%
\special{pa 2770 2450}%
\special{ip}%
% SPLINE 2 0 3 0
% 4 2690 850 2790 900 2850 960 2900 1000
% 
\special{pn 8}%
\special{pa 2690 450}%
\special{pa 2720 462}%
\special{pa 2750 475}%
\special{pa 2777 491}%
\special{pa 2802 511}%
\special{pa 2824 534}%
\special{pa 2847 557}%
\special{pa 2871 578}%
\special{pa 2896 597}%
\special{pa 2900 600}%
\special{sp}%
% SPLINE 2 0 3 0
% 5 2770 2870 2840 2940 2870 2980 2940 3010 2970 3040
% 
\special{pn 8}%
\special{pa 2770 2470}%
\special{pa 2795 2491}%
\special{pa 2818 2513}%
\special{pa 2838 2537}%
\special{pa 2856 2565}%
\special{pa 2879 2586}%
\special{pa 2909 2597}%
\special{pa 2939 2609}%
\special{pa 2962 2631}%
\special{pa 2970 2640}%
\special{sp}%
% STR 2 0 3 0
% 3 830 3380 830 3480 2 0
% 0
\put(8.3000,-30.8000){\makebox(0,0)[lb]{0}}%
% CIRCLE 2 0 0 0
% 4 4600 2960 4640 3020 4610 2990 4610 2990
% 
\special{pn 8}%
\special{sh 0.600}%
\special{ar 4600 2560 72 72  0.0000000 6.2831853}%
% STR 2 0 3 0
% 3 2480 3960 2480 4060 2 0
% Figure 9
\put(24.8000,-36.6000){\makebox(0,0)[lb]{Figure 9}}%
\end{picture}%




\newpage
%\input {5.1.ue.tex}

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% 
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% LINE 2 0 3 3
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% LINE 2 2 3 11
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% LINE 2 0 3 12
% 2 5360 3630 5360 3910
% 
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% VECTOR 2 0 3 13
% 2 5360 1030 5360 430
% 
\special{pn 8}%
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% LINE 2 0 3 14
% 2 4700 4280 4790 4340
% 
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% 
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% LINE 2 0 3 16
% 2 3330 1980 2200 1980
% 
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% VECTOR 2 0 3 17
% 2 6550 1980 7880 1980
% 
\special{pn 8}%
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% LINE 2 2 3 18
% 2 4510 2500 4750 2500
% 
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% LINE 2 2 3 19
% 2 4500 2220 4730 2220
% 
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% LINE 2 2 3 20
% 2 5490 1730 5480 3120
% 
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% LINE 2 2 3 21
% 2 4940 1350 4950 1730
% 
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% LINE 2 2 3 22
% 2 4240 3010 4240 3490
% 
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% LINE 2 2 3 23
% 2 4240 1360 4230 1830
% 
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% LINE 2 0 3 24
% 2 4680 2840 4570 2760
% 
\special{pn 8}%
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% LINE 2 0 3 25
% 2 4570 1980 4680 2060
% 
\special{pn 8}%
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% BOX 2 5 2 26
% 2 4541 2021 4601 2101
% 
\special{pn 8}%
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% LINE 2 2 3 27
% 2 5840 2660 6480 2530
% 
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% LINE 2 2 3 28
% 2 5690 1820 6300 1640
% 
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% LINE 2 2 3 29
% 6 5170 1310 5650 1150 5590 1180 5170 1300 5520 1570 6090 1360
% 
\special{pn 8}%
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% LINE 2 0 3 30
% 2 4360 1240 5170 1040
% 
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% LINE 2 0 3 31
% 2 4581 1981 4671 2041
% 
\special{pn 8}%
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% CIRCLE 2 5 3 32
% 4 2200 4070 2460 4140 2410 4150 2410 4150
% 
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% CIRCLE 2 0 3 33
% 4 4911 2391 5261 2751 5231 2731 5231 2731
% 
\special{pn 8}%
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% LINE 2 0 3 34
% 2 5030 3550 5840 3350
% 
\special{pn 8}%
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% SPLINE 2 0 3 35
% 3 5540 3150 5440 3260 5080 3530
% 
\special{pn 8}%
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% LINE 2 2 3 36
% 2 4940 2970 4930 3490
% 
\special{pn 8}%
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% BOX 2 5 2 37
% 2 4571 1951 4671 2071
% 
\special{pn 8}%
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% CIRCLE 2 0 3 38
% 4 4311 2401 4661 2761 4631 2741 4631 2741
% 
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% BOX 2 5 2 39
% 2 4571 2731 4671 2851
% 
\special{pn 8}%
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% LINE 2 2 3 40
% 2 3670 2780 3670 1820
% 
\special{pn 8}%
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% LINE 2 2 3 41
% 2 3670 2950 3670 3080
% 
\special{pn 8}%
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% LINE 2 0 3 42
% 2 4571 2761 4701 2851
% 
\special{pn 8}%
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% CIRCLE 2 0 3 43
% 4 4591 2421 5521 3181 5501 3161 5581 3251
% 
\special{pn 8}%
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% BOX 2 5 2 44
% 2 3470 2920 3550 3000
% 
\special{pn 8}%
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% BOX 2 5 2 45
% 2 3920 2710 4000 2790
% 
\special{pn 8}%
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% BOX 2 5 2 46
% 2 4380 2480 4460 2560
% 
\special{pn 8}%
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% VECTOR 2 0 3 47
% 2 4600 2430 2790 3300
% 
\special{pn 8}%
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\special{sh 1}%
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% BOX 2 5 2 48
% 2 4690 4250 4790 4370
% 
\special{pn 8}%
\special{sh 0}%
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% CIRCLE 2 0 3 49
% 4 4430 4700 4780 5060 4750 5040 4750 5040
% 
\special{pn 8}%
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% BOX 2 5 2 50
% 2 4690 5030 4790 5150
% 
\special{pn 8}%
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% LINE 2 0 3 51
% 2 4690 5060 4820 5150
% 
\special{pn 8}%
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\vskip4cm
Figure 10



\newpage
%\input {5.1.sita.tex}

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% CIRCLE 2 0 3 1
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% STR 2 0 3 0
% 3 2000 900 2000 1000 2 0
% $t=-1$
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% VECTOR 2 2 3 0
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% SPLINE 2 2 3 0
% 7 4530 1750 4570 1880 4660 1960 4750 2010 4800 2030 4940 2050 4960 2050
% 
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% STR 2 0 3 0
% 3 2000 4500 2000 4600 2 0
% $t=-2$
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% CIRCLE 2 0 1 0
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% CIRCLE 2 0 1 0
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% STR 2 0 3 0
% 3 5100 2100 5100 2200 2 0
% $p=(1,0,0,0,-1)$
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\vskip4cm
Figure 11




\newpage
%\rotatebox[origin=c]{90}{\input {5.2.tex}}
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% VECTOR 2 0 3 0
% 2 1430 2670 620 1940
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% LINE 2 0 3 0
% 2 2780 3750 1910 3080
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% Figure 12
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% VECTOR 2 0 3 0
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% 
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% VECTOR 2 0 3 0
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% LINE 2 0 3 0
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\end{document}








math0003089.extracted_bib
\begin{thebibliography}{900}


\bibitem{ 1} %CS
 S. Cappell and J. Shaneson: 
The codimension two placement problem and homology equivalent manifolds 
{\it Ann. of Math.}, 99, 271-348, 1972 


\bibitem{ 2} %CO
T. Cochran and K. Orr:  Not all links are concordant to boundary links 
 {\it Ann. of Math. }, 138, 519--554, 1993  

\bibitem{3} %F
R. Fox: Some problems in knot theory  
 {\it Top. 3-manifolds. Proc. 1961 Top. Inst. Georgia }, 168-176


\bibitem{ 4} %GM
F. Gonzalez-Accuna and J. Montesinos:  
Embedding knots in trivial knots 
 {\it Bull. London. Math. Soc.}, 14, 238-240, 1982 



\bibitem{5} %Kf1
L. Kauffman: Formal knot theory
 {\it Princeton University Press Math. Notes}, vol.30, 1983


 \bibitem{ 6} %Kf2
L. Kauffman:  On knots
 {\it  Ann. of Math. Stud. }, vol. 115, 1987




\bibitem{ 7} %Kw
A. Kawauchi edit.: Knot theory (In Japanese)
 {\it Springer Verlag Tokyo}, 1990
\newline
 A survey on knot theory (English translation),  
 {\it Birkh\"auser }, 1996


\bibitem{ 8} %Ke
M. Kervaire: Les noeudes de dimensions sup\'ereures 
 {\it Bull.Soc.Math.France},  93, 225-271, 1965  


\bibitem{9} %KW
 M. Kervaire and C. Weber: 
 A survey of multidimensional knots. In: Knot theory
 {\it  Lecture Notes in Math  (Springer Verlag)}
vol. 685, 61-134, 1965 


\bibitem{ 10 }%Ki
R. Kirby:  The topology of 4-manifolds 
 {\it Lecture Notes in Math  (Springer Verlag) }
vol. 1374, 1989


 
\bibitem{ 11 }%L
J. Levine: Knot cobordism in codimension two
 {\it  Comment. Math. Helv.},  44, 229-244, 1969

 
\bibitem{ 12} %Ma
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\bibitem{ 13 }%Mi
J. W. Milnor:  Singular points of complex hypersurfaces
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\bibitem{ 14} %MN
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\bibitem{15} %N
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\bibitem{ 16 }%O1
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\bibitem{  19}  %O4      
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{\bf Acknowledgments.}
 Theorem 1.1. and 1.3 are answers to Prof. Tsuboi's questions 
on the author's paper [16].
%[O1]. 
The author would like to thank Prof. Takashi Tsuboi for his advice. 

The author obtained an idea of a way of surgery of submanifolds 
in this paper  
from p.490 of [12]. 
%[Ma].       
Of course the author's way are different from his. 
After obtaining the idea, 
he got acquainted with Prof. Takao Matumoto and discussed with him.
The author would like to thank Prof. Takao Matumoto 
for the valuable discussion. 

The author obtained an idea of Problem 1.4 and that of [19] 
when he talked about [17] with Dr. Yoshitake Hashimoto.  
The author would like to thank Dr. Yoshitake Hashimoto 
for the valuable discussion. 

The author would like to thank Prof. Charles Livingston for his interest 
in Problem 1.4.  

\end{thebibliography}