数理解析研究所講究録1182 巻 2001 年 89-102

89

A method of computations of fundamental groupsof 3-dimensional manifolds(高井 真希)Maki Takai

1Introduction.

By the theorem of Hilden-Montesinos (Hilden [7], Montesinos [9]), for every 3dimensional compact oriented manifold , there exists a topological branched cov$\mathrm{Y}$

ering
$h$

:

$\mathrm{Y}arrow S^{3}$, whose monodromy aroundof the 3-sphereof degree 3 branching at a knot
the knot is given only by transpositions.
to
We regard the knot
as a braid, for every knot (and link) is isotopic in, wherea braid. We may identify
is the discwithin the complex plane with the center and the radius . We may assume thatis contained in
as in Figure 1.
$S^{3}$$B_{h}$$S^{3}$$B_{h}$$S^{3}$$\partial(\overline{\Delta(\mathrm{o},a^{;})}\cross\Delta(\mathrm{O}, b’))$$\Delta(0, a’)$

$a’$

$0$

$\mathbb{C}$$\partial\overline{\Delta(0,a^{J})}\cross\Delta(0, b^{l})$$B_{h}$$(0\leq s\leq 2\pi)$Figure 1:
Let

$B$

be

$\mathrm{t}.\mathrm{h}.\mathrm{e}$cone over$B_{h}$

conneCtin.g every oint of$.\mathrm{p}$

Let
$f$

:

$B_{h}$

with the origin of$\mathbb{C}^{2}$.

$Xarrow\Delta(\mathrm{O}, a)\cross\Delta(0, b)$same monodromy
be the topological finite branched covering branching at(Such a branched covering exists by Fox completion (Fox [5]). In fact $X$ is aas$B\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}’ \mathrm{t}\mathrm{h}\mathrm{e}$

$h$

.

90
cone over Y.) Since $X$ is a topological cone over$\pi_{1}(X-\{_{X}\}, p_{0})\simeq\pi_{1}(\mathrm{Y}, p0)$,$\mathrm{Y}$,$(x=f^{-1}((0,0)))$.

Put$=f^{-1}(t\mathrm{X}\Delta(0, b))$$X_{t}$$=f|_{X_{t}}$$f_{t}$:,$X_{t}arrow t\mathrm{x}\Delta(0, b)$

.

, and
Then every $f_{t}(t\neq 0)$ is a finite branched covering of the disccan be regarded as a topological degenerating family of finite branched coveringsof discs: $f=\{f_{t}\}$ . Its topological type is determined by the pair$t\cross\Delta(\mathrm{O}, b)$$f$$(\Phi_{t}, \theta(\delta))$of the monodromy
But they must$\Phi_{t}$$\mathrm{S}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{s}\mathfrak{h}^{r}$,$(\delta:s\mapsto a’e^{is}, (0\leq S\leq 2\pi))$of (for a fixed $t\neq 0$ ) and the braid monodromy
the following equality (Namba [10]):$f_{t}$$\Phi_{t}0\theta(\delta)=\Phi t$

is regarded as an automorphism of
where
2).
Conversely, let$\Phi$:$\pi_{1}(\Delta(0, b)-$

of .
$f$

,

$\pi_{1}(t\cross\Delta(\mathrm{o}, b)-B_{f,q0})$$\theta(\delta)$$\theta(\delta)${ points},$n$(see Section$q\mathrm{o})arrow S_{d}$be a representation whose image is a transitive subgroup of the d-th symmetricgroup . Let be a braid which satisfies$S_{d}$$\sigma$$\Phi\circ\sigma=\Phi$We denote the -points by2.
Then$\{q_{1}, \ldots , q_{n}\}$$n$.

and let$\pi_{1}(\Delta(\mathrm{o}, b)-\{q_{1}, \ldots, qn\},$$\gamma_{1},$$\ldots$,$q0)=<\gamma 1,$$\gamma_{n}$$\ldots,$

is a free group. Put$A_{j}=\Phi(\gamma_{j})$be the lassos as in Figure$(j=1,2, \ldots, n)$$\gamma_{n}>$.

as
We regard the braid as a link which is contained in, we can construct a topological branched
in Figure 1. By the condition
covering$\partial\overline{\Delta(\mathrm{o},a’)}\mathrm{X}\Delta(0, b’)$$\sigma$$\Phi 0\sigma=\Phi$$h$:$\mathrm{Y}arrow\partial(\overline{\Delta(0,a’)}\cross\overline{\Delta(\mathrm{o},b’)})$

branching at the link whose monodromy is . More precisely, we can constructbranching at the linkof
a topological branched covering
at the part corresponding
whose monodromy is . We then attach solid tori topermutation
to the mutually prime cyclic decomposition of the$\Phi$$\sigma$$\mathrm{Y}’$$\Phi$$\partial\overline{\Delta(0,a)\prime}\cross\Delta(\mathrm{O}, b’)$$\mathrm{Y}’$$A_{\infty}=\Phi(\gamma n\ldots\gamma_{1})^{-}1=(An\ldots A1)-1$$\sigma$91$q_{0}$

Figure 2:

over. Then we get a 3-dimensional compact oriented manifold
and a topological finite branched covering$\partial\overline{\Delta(0,a)\prime}\cross\partial\overline{\Delta(0,b’)}$$\mathrm{Y}$:$h$$\mathrm{Y}arrow\partial(\overline{\Delta(0,a’)}\cross\overline{\Delta(\mathrm{o},b\prime)})$of the 3-sphere branching at the link a whose monodromy is .
We then construct the topological cone $X$ of as above and construct a topological finite branched covering$\Phi$$\mathrm{Y}$$f$:$Xarrow\Delta(\mathrm{o}, a)\mathrm{x}\Delta(\mathrm{o}, b)$such that$\Phi_{f}=\Phi$,$\theta(\delta)=\sigma$.

This is regarded as a topological degenerating family of finite branched coveringsof discs.
Thus to construct topological degenerating families of finite branched coveringof discs (hence to construct 3-dimensional compact oriented manifolds) is reduced
to find out the pair
as above such that.
$s$$\Phi 0\sigma=\Phi$$(\Phi, \sigma)$2

Monodromy of a branched covering of degree 3of the disc and its canonical forms.

Let $X$ and be Riemann surfaces and let :be a finite branched covering,that is, a surjective proper finite holomorphic mapping. A point of $X$ is called aramification point of if is not biholomorphic around . Its image $q=f(p)$ iscalled a branch point of . The set of all ramification points (resp. branch points) is(resp.
denoted by) and is called the ramification locus (resp. branch locus).
Then
$f$

$\mathrm{Y}$$Xarrow \mathrm{Y}$

$p$

$f$

$f$

$p$

$f$

$R_{f}$$B_{f}$$f$: $X-f-\iota(B_{f})arrow \mathrm{Y}-B_{f}$92is an unbranched covering, whose mapping degree is called the degree of and isdenoted by degf. (X, ) (or simply ) is called a finite branched covering of Y.
$f$

$f$

$f$

Deflnition 1. Two finite branched coverings,$f’:x’arrow \mathrm{Y}$$f:Xarrow \mathrm{Y}$are said to be isomorphic if there is a biholomo rphic mapping
following diagram commutative:$\psi$

which makes the.
$X \frac{\psi\backslash }{r}X’$$\downarrow f’$$f\downarrow$$\mathrm{Y}arrow\dot{.}d\mathrm{Y}$Deflnition 2. Two finite branched coverings,$f’:X’arrow \mathrm{Y}$$f:Xarrow \mathrm{Y}$are said to be equivalent (resp. topologically equivalent) if there are biholomorphic
preserving homeomorphisms) and which make themappings (resp.
following diagram commutative:$or\dot{\eta}entation$$\psi$$\varphi$$Xarrow\psi X’$$\downarrow f’$$f\downarrow$$\mathrm{Y}arrow^{\varphi}\mathrm{Y}$Let$B_{n}$

be the Artin braid group of$B_{n}=<\sigma_{1},$$\ldots,$$n$strings. Then$\{q_{1}, \ldots , q_{n}\}$is expressed as follows:$|\sigma_{i}\sigma_{i}+1\sigma*\cdot=\sigma_{i}+1\sigma i\sigma i+1$$\sigma n-1$$\sigma_{i}\sigma_{j}=\sigma_{j}\sigma_{i}$Let$B_{n}$,

for $|i-j|\geq 2>$ .

be a set of distinct points in C. The fundamental groupis the free group$n$$\pi_{1}(\mathbb{C}-\{q_{1}, \ldots , q_{n}\}, q_{0})$$\pi_{1}(\mathbb{C}-\{q1, \ldots, qn\}, q\mathrm{o})=<\gamma_{1},$$\ldots,$$\gamma_{n}>$as in Figure 2.
generated by the lassos
acts on this group as follows:
The braid group$\gamma_{1},$$\ldots,$$\gamma_{n}$$B_{n}$.
$\sigma_{i}(\gamma_{i})$$-$$=\gamma_{i}^{-1}.\gamma i+1\gamma i$$\sigma:(\gamma_{i+1})$$=’\gamma_{i}$$\sigma_{i}(\gamma_{j})$$=\gamma_{j}$$(j\neq i, i+1)$.

Note that this action is faithful (Birman [1]). A similar assertion holds if we replace.
by a disc
The following theorem is well known:$\mathbb{C}$$\Delta(0, b)$

93. For any homomorphism
whose image $Im\Phi$ is transitive, there exists a unique (upto isomorphisms) finite branched covering :such that

Theorem 1. Put$B=\{q_{1}, .\mathrm{Y}.

, q_{n}\}\subset \mathrm{P}^{1}=\mathbb{C}\cup\{\infty\}$$\Phi:\pi_{1}(\mathrm{P}1-B, q0)arrow S_{d}$$f$$B_{f}\subset B$,$Xarrow \mathrm{P}^{1}$$\Phi_{f}=\Phi$.

For the proof of Theorem 1, see Forster [4]. There is a higher dimensional analogyof the theorem (Grauert-Remmert [6]).

, $f’$ :Theorem 2. For two finite branched coverings :such, they are topologically equivalent if and only ifthat. Here the equality is that
there is a braid insuch thatas representation classes.
$X’arrow \mathrm{P}^{1}$$Xarrow \mathrm{P}^{1}$$f$$B_{f}=B_{f’}=\{q_{1}, \ldots , q_{n}\}\subset \mathbb{C}$$\sigma^{*}(\Phi_{f})=\Phi_{f}0\sigma=\Phi_{f’}$$B_{n}$$\sigma$

For the proof of Theorem 2, see Namba [10].

Remark. Theorem 2 holds even if
in C.

$\mathrm{P}^{1}$is replaced by the compelx plane$\mathbb{C}$or a disc

Every branched covering$f$of degree$d$: $Xarrow\Delta(0, b)$

can be extended to a branched covering$\hat{f}:\hat{X}arrow \mathrm{P}^{1}$of degree$d$in the following canonical manner: Put$B_{f}=\{q_{1}, \ldots, q_{n}\}$

,

where
is a lasso as in Figure 2. Let
Figure 3.
Then$\gamma_{j}$$(j=1, \ldots, n)$ ,$A_{j}=\Phi_{f}(\gamma_{j})$$\gamma_{\infty}$$\pi_{1}(\mathrm{P}^{1}-\{q1, \ldots, q_{n}, \infty\}, q\mathrm{o})=<\gamma 1,$

be the las so around the point

$.:$

.

$\gamma_{n},$$\gamma\infty|\gamma_{\infty}\gamma_{n}\cdots\gamma 1=1>$

$,$

Put
$A_{\infty}=(A_{n}\cdots A_{1})^{-1}$.

We define a homomorphism$\Phi:\pi_{1}(^{\mathrm{p}}1-\{q1, \ldots, q_{n}, \infty\}, q0)arrow S_{d}$by$\Phi(\gamma_{j})=A_{j}$$(j=1, \cdots n)$ ,$\Phi(\gamma_{\infty})=A_{\infty}$

Then the branched covering$\hat{f}:\hat{X}arrow \mathrm{P}^{1}$

corresponding to$\Phi$(see Theorem 1) is an extension of .
$f$.

$\infty$

.

as in

94

$q_{0}$

Figure 3:
Note that ifLet$A_{\infty}=1$, then$\hat{f}$does not branch at the point$f$$\infty$.

: $Xarrow\Delta(0, b)$of degree 3. Let $\gamma_{j}(j=1, \ldots , n)$ be thebe a branched covering of the discis a, n). Suppose that every( $j=1,$
lassos as in Figure 2. Puttransposition in the 3rd symmetric group . As above, we extend the covering tobe thewhich is denoted by the same notation for simplicity. Letthat ofand put
lasso around the point$\Delta(0, b)$$A_{j}$$A_{j}=\Phi_{f}(\gamma_{j})$$\ldots$$S_{3}$$f$$\mathrm{P}^{1}$$\gamma_{\infty}$$\infty$$A_{\infty}=(A_{n1}\ldots A)^{-}1=\Phi_{f(\gamma)}\infty$

as above. There are three cases:
Case 1.

$A_{\infty}=1$. In this case, the extended covering does not branch at$\infty$.

is a branch point, thatis a transposition. In this case, the point
Case 2.
with the ramification index is 2. Since we mayis there is a point overchange the monodromy with an equivalent representation , we may assume
that $A_{\infty}=(12)$ .
$\infty$$A_{\infty}$$\infty$

is a cyclic permutation. In this case, the point
Case 3.
We may assume that $A_{\infty}=(132)$ .
$A_{\infty}$$\infty$

is a branch point.

Under these assumptions, we have the following theorem:
Theorem 3. Under the above assumptions, the covering
to one of the following canonical forms: Arranging$A_{1},$

Case 1: (12), (12), (23), (23),$f$is topologically equivalent
in this order:$A_{2},$$\ldots,$$A_{n}$

95
Case 2: (12), (23), (23),

where

3

$g\iota s$

the genus of the Riemann surface $X$ .

Isotropy subgroups of the braid groups.

Let$\Phi:<\gamma_{1},$$\ldots,$$\gamma_{n}>arrow sd$

be a representation of the free group,
generators into the d-th
$Im\Phi$symmetric groupwhose image
is transitive.
By the discussion in Section 1, it is important to consider the braidsuch that $\Phi 0\sigma=\Phi$ , where the equality is not as representaion classes but is justas representaions. (The action of the braid on the free group,defined in Section 2.) Put$<\gamma_{1},$$\ldots$$\gamma_{n},>\mathrm{o}\mathrm{f}n$$S_{d}$$\sigma\in B_{n}$$\sigma$$<\gamma_{1},$$I(\Phi)=\{\sigma\in B|n\Phi 0\sigma=\Phi\}$$\ldots$$\gamma_{n}>\mathrm{i}\mathrm{s}$,the isotropy subgroup offor .
Since the number of representaions is finite (in fact is less thana subgroup ofof finite index.
Note that the following equality holds:$\Phi$$B_{n}$$\Phi$$(d!)^{n}$), $I(\Phi)$ is$B_{n}$$I(\Phi 0\tau)=\tau-1I(\Phi)\tau$.

Put$(j=1,2, \cdots, n)$$\Phi(\gamma_{j})=A_{j}$.

Now, let be the representation of the canonical forms as in Theorem 3.
For Case 2 and 3$A_{2}=\cdots=A_{n}=(23))$ , by the theorem ofBirman-Wajnryb (Birman-Wajnryb [2])is generated by the following elements:$\Phi$$(\mathrm{i}.\mathrm{e}, A_{1}=(12),$$I(\Phi)$$\sigma_{1}^{3},$$\sigma_{2}$

,

... ,

$\sigma_{n-1}$,.

$\sigma_{1}^{-1}\sigma_{2}^{-}\sigma-2\sigma_{2}\mathrm{r}3-1\sigma_{1}-\mathit{2}-\sigma_{2}\iota 1\sigma^{-}3\sigma_{43}\sigma\sigma_{2}\sigma_{1\mathit{2}3}2\sigma\sigma\sigma_{2}2\sigma_{1}(n\geq 5)$

The following theorem for Case 1
is the main result.

Theorem 4. For Case 1,$I(\Phi)$$\sigma_{2}^{3},$$\sigma_{1},$$\sigma_{3}$$(\mathrm{i}.\mathrm{e}, A_{1}=A_{2}=(12),$$A_{3}=\cdots=A_{n}=(23))$is generated by the following elements:, ... ,$\sigma_{n-1},$$\sigma_{2}^{-1}\sigma_{3}^{-}\sigma_{2}\sigma 2-11\sigma \mathit{2}\sigma 3\mathit{2}\sigma 2$,.

$\sigma_{2}^{-1}\sigma_{3}^{-}\sigma_{4}\sigma^{-}\sigma_{2}\sigma^{-}\sigma_{4}\sigma 1-231-231-15\sigma 4\sigma_{3}\sigma_{234}\mathit{2}\sigma\sigma\sigma_{3}2\sigma_{2}(n\geq 6)$

Remark. For Case 1, the generators of the isotropy subgroup
described in Birman-Wajnryb ([2]) but not of .
$B_{n}$$I(\Phi)$of$B_{n}(S^{\mathit{2}})$

are

96

4

Riemann pictures and symplectic basis for canonical forms.

In this section, we introduce a picture, (we call it a Riemann picture), which repre(see Namba-Takai [11]). We
sents a finite branched covering of a discexplain it by an example:
Let us consider Case 1 of genus 1.
be a branched:
Let $X$ be a Riemann surface of genus 1. Letof canonical form of Case 1. Putcovering of degree 3 with the monodromy. Let be a reference point. We take the lassos aroundas in Figure 2. We extend the covering to the branched covering of in a canonicalway as in Section 2. In this case, we have$\mathrm{t}\mathrm{o}\mathrm{p}\mathrm{o}1.\mathrm{o}\mathrm{g}\mathrm{i}_{\mathrm{C}\mathrm{a}}11\mathrm{y}$$f$$Xarrow \mathbb{C}$$\Phi$$B_{f}=\{q_{1}, q_{2}, \ldots , q_{6}\}$$q_{0}$$q_{j}$$\gamma_{j}$$\mathrm{P}^{1}$$\pi_{1}(^{\mathrm{p}_{-}^{1}}B_{f}, q\mathrm{o})=<\gamma_{1},$$\gamma 2,$$.,.,$$\gamma\epsilon,$$\gamma_{\infty}|\gamma\infty\gamma\epsilon\cdots\gamma 2\gamma 1=1>$, $A_{\infty}=id$
Consider the picture (Figure 4) in which the circle part of every lasso
2 is degenerated to the point :$A_{1}=A_{2}=(12)$,$(A_{j}=\Phi(\gamma_{j}))$$A_{\mathit{2}}=\cdots A_{6}=(23)$$\gamma_{j}$.
in Figure$q_{j}$$q_{0}$

Figure 4:

We then pull the picture in Figure 4 back over the covering and get the following
picture in Figure 5 which we call the Riemann picture of :In Figure 5, the point \copyright , \copyright , \copyright are the inverse images of the reference point, and, 6 andare the inverse images ofwhile the pointts 1,
respectively. Note that around every point \copyright , \copyright , \copyright , the pathes connecting to the, 6 and in this order are arranged clockwisely. On the other hand,points 1,, 6 and , the pathes connecting to the points \copyright , \copyright , \copyrightaround every point 1,
are arranged counterclockwisely in order to be compatible with the monodromy.
(We omit unramified points in the picture.)
The covering (X, ) can be topologically expressed by this picture.
Put$[1, 21][\infty, 11][1,12]$ ,,.
$[6, 23][\infty, 33][6,32]$ ,$\alpha=$$[3, 23][4,32]$ ,
$f$

$f$

...

$q_{0}$...

$s$$\infty$$\infty$...

$\infty$$f$$\xi_{3}$$=$$\xi_{2}$$=$$\xi_{1}$$=$$\beta=$$[\infty, 22]$$[5,23][4,32]$ .

$q_{1},$$\ldots$$q_{6}$$\infty$

97

Figure 5:
Here the notation $[6, 23]$ for example means the path in Figure 5 whose initialpoint is \copyright and the terminal point is \copyright passing through the branch point 6. Thenthese are loops with the initial point \copyright . We can observe the following relations:$\beta\alpha\beta^{-1}\alpha^{-}\xi_{3}1\xi_{2}\xi_{1}$$<\alpha,$

where the notation $<,$$>\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{n}\mathrm{s}$$\beta>$$=1$ ,$=$$1$,

the intersection number. We pull back the relation$\gamma_{\infty}\gamma_{6}\cdots\gamma_{1}=1$

over

$f$

and get the following three relations:$[\infty, 11][2,12][1,21]=1$ ,$[\infty, 22][6,23][5,32][4,23][3,32][2,21][1,12]=1$ ,$[\infty, 33][6,32][5,23][4,32][3,23]=1$.

The above relation$\beta\alpha\beta^{-1}\alpha^{-}\xi_{3}1\xi 2\xi_{1}=1$

can be induced from these three relations.
The Riemann picture of a general (X, ) is defined as in the above example, that
is, a pull-back over of the graph onof Figure 2 degenerated the circle part ofevery lasso to the branch point.
$f$

$f$

$\mathrm{P}^{1}$

Remark. 1. The Riemann picture is determined by (X, ) up to orientation
preserving homeomorphisms of $X$ .
2. As noted above, we can draw the Riemann picture of (X, ) even when only themonodromy
is given and (X, ) is not explicitely given.
3. In Namba [10], we have introduced another picture in order to express (X, )
$f$

$f$

$\Phi=\Phi_{f}$

$f$

$f$

98
topologically, which we called a Klein picture. Klein pictures and Riemann pictures
are dual in a sense. Klein pictures are useful to observe the degeneration of branchedcoverings, while Riemann picutres are useful to compute fundamental groups as will
be seen in Section 5.
We draw the Riemann pictures of the canonical forms in Theorem 3 (see Figuresof6, 7 and 8 for $g=3$ ), from which we easily find canonical generators
the fundamental group of $X$ such that$\{\alpha_{i}, \beta i, \xi_{i}\}$,Case 1:$\beta_{g}\alpha_{g}\beta_{g}^{-1-}\alpha_{g}1\beta g-^{\iota 1\beta_{g}^{-1}\alpha_{g1}^{-}}\alpha g--1-1\ldots\beta 1\alpha_{1}\beta 1-1\alpha_{1}-1\xi 3\xi_{\mathit{2}}\xi_{1}=1$

Case 2:$\beta_{g}\alpha_{g}\beta^{-1-}gg\alpha 1\beta_{g}-1\alpha g-1\beta_{g-1}^{-1}\alpha_{g1}^{-}-1\ldots\beta 1\alpha_{1}\beta^{-1}11\alpha^{-}1\xi 2\xi_{1}=1$

Case 3:$\beta_{g}\alpha_{g}\beta_{ggg1}-1-\alpha 1\beta_{g1g-}-\alpha 1\beta_{g-}^{-1}1\alpha--1\ldots\beta_{11}\alpha\beta^{-}11\alpha_{1}^{-}\xi 1=1$

Hereof $X$ :$\{\alpha_{i}, \beta_{i} (i=1, \ldots , g)\}$,is a symplectic basis in homology level of the extension$\hat{X}$$<\alpha_{i},$$\beta_{j}>=\delta_{ij}$$(i, j=1, \ldots,g)$ , where $<,$,$<\alpha_{i},$$>\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{n}\mathrm{s}$$\alpha_{j}>=0$,$<\beta_{i},$$\beta_{j}>=0$

the intersection number.

Figure 6: Case 1

99

Figure 7: Case 2$\bullet$Figure 8: Case 3

are deleted.

100

In fact we may take as follows:

Case 1:$\xi_{3}$$=$$[1, 2l][\infty, 11][1,12]$$\xi_{2}$$=$$[\infty, 22]$$\xi_{1}$$=$$[2g+4,23][\infty, 33][2g+4,32]$$=$$[3, 23][4,32]$$\beta_{1}$$=$$[5, 23][4,32]$$\alpha_{j}$$=-[2j+1,23][2j, 32]\cdots[3,23][2j, +2,32]$$\beta_{j}$$=$$[2j+3,23][2j+2,32]$$=$$[2g+1,23][2g, 32]\cdots[3,23][2g+2,32]$’.

$\alpha_{1}$$\alpha_{g}$$[2g+3,23][2g+2,32]$ .

$\beta_{\theta}$$=$$\xi_{\mathit{2}}$$=$$[\infty, 21][\infty, 12]$$\xi_{1}$$=$$[2g+3,23][\infty, 33]12g+3,32]$$=$$[2, 23][3,32]$$=$$[4,23][3,32]$$=$$[2j, 23][2j-1,32]\cdots[2,23][2j+1,32]$$=$$[2j+2,23][2j+1,32]$$=$$[2g, 23][2g-1,32]\cdots[2,23][2g+1,32]$

Case 2:$\alpha_{1}$$\beta_{1}$$\alpha_{j}$$\beta_{j}$$\alpha_{g}$$\beta_{g}$

$=$

$\xi$

$=$

$[2g+2,23][2g+1,32]$ .

Case 3:$\alpha_{1}$$\beta_{1}$$–[2^{:},$$\beta_{j}$$\alpha_{g}$$\beta_{g}$$231[3,32]$$=$$[4, 23]13,32]$$=$$[2j, 23][2j-1,32]\cdots[2,23][2j+1,32]$$=$$[2j+2,23][2j,+1,32]$$=$$[2g, 23][2g-1,32]\cdots[2,23][2g+1,32]$$’\lambda$$\alpha_{j}$$[\infty, 21][\infty, 13][\infty, 32]$$=$$[2g+2,23][2g+1,32]$ .

101

5Calculations of fundamental groups.
In this section, we compute fundamental groups of 3-dimensional compact oriented
manifolds using the local version of the theorem of Zariski-van Kampen (see Dimca[3], Matsuno [8] and the method of Reidemeister-Schreier (see Rolfsen [12]). One
can compute the fundamental group rigorously if one uses the Riemann picture. We
explain this using a concrete example:
Let us cons.ider Case 1 of genus 1 for simplicity. If we take the braid as$)$$\sigma$$\sigma=\sigma^{-1}\sigma^{-}\sigma^{-}\sigma 1\sigma \mathit{2}\sigma\sigma 23\mathit{2}21\mathit{2}\mathit{2}35\sigma\sigma_{4}\sigma 3\sigma 32$( induces a knot), then we have the equality$\sigma$$\Phi 0\sigma=\Phi$where is the monodromy of the canonical form. Hence we may construct a topological degenerating family$\Phi$$f$:$Xarrow\Delta(\mathrm{o}, a)\mathrm{x}\Delta(\mathrm{o}, b)$(see Section 1). Letof branched coverings of discs constructed from the pair$(j=1,\ldots,6)$
be the lassos as in Figure 2. The
be the branch locus of . Let
local version of the theorem of Zariski-van Kampen asserts that the fundamental
is generated by $\gamma_{j}(j=1, \ldots , 6)$ whose generatinggroup of$\sigma(\gamma_{j})=\gamma_{j}(j=1,.., , 6)$ . That is to say
relations are$(\Phi, \sigma)$$f$$B_{f}$$\gamma_{j}$$\Delta(0_{\backslash ,J}a)\cross\Delta(\mathrm{O}, b)-B_{f}$$\pi_{1}(\triangle(0, a)\cross\Delta(0, b)-B_{f},$$=<\gamma_{1},$$q\mathrm{o})=<\gamma 1,$$\ldots,\gamma_{6}|\sigma(\gamma j)=\gamma j(j=1, \ldots, 6)>$$\ldots,\gamma_{6}|(\sigma_{232}-12-1\sigma\sigma^{-}\sigma\sigma_{1}\sigma 23\sigma 2\sigma-1-125\sigma_{4}\sigma 3\sigma^{3})2\gamma_{j\gamma_{j}}=(j=1, \ldots,6)>$$\gamma^{-1}1\gamma 4\gamma 3\gamma 2\gamma_{3}-1\gamma 4\gamma-1-2\gamma_{3}-1\gamma_{4}-1\gamma_{5}^{-}\gamma_{\epsilon}\gamma 511-1$$=<\gamma_{1}-1-1’\ldots,\gamma_{6}|\gamma_{1}\gamma_{4}\gamma_{3}\gamma 2\gamma_{3}-1^{-1}\gamma_{4}\gamma 1\gamma_{1}^{-1}=1,$$\gamma_{1}\gamma_{\mathrm{s}_{1-1}}-1-\gamma_{6\gamma_{5}}\gamma 1\gamma 5\gamma 6\gamma 5\gamma_{4}\gamma 3\gamma_{\mathit{2}}\gamma 4\gamma 3\gamma_{2}-1\gamma 3-1\gamma_{4}-1\gamma 1\gamma_{\mathit{2}}^{-1}=1,$$\gamma_{1\gamma 4}-1-1-1-1-\gamma 31\gamma\gamma 3\gamma \mathit{2}\gamma_{3}-1-1-\gamma_{4}\gamma 214-1-1$$\gamma 1-1\gamma 4\gamma 3\gamma \mathit{2}\gamma_{3}\gamma_{4}\gamma_{3}\gamma_{4}\gamma 3\gamma_{2}\gamma_{3}$$\gamma_{5-1}\gamma_{6}\gamma_{5}\gamma 1\gamma 5\gamma_{\epsilon}\gamma 5\gamma 4\gamma 3\gamma_{2}\gamma_{4\gamma,-1}3\gamma \mathit{2}-1\gamma_{3}^{-}1-\gamma 41\gamma_{1}\gamma_{3}^{-1}=1,$$\gamma_{4}\gamma_{1}\gamma_{4}^{-1}=1,$$\gamma^{-1}1\gamma 4\gamma_{3}\gamma 2\gamma_{3}\gamma_{4}\gamma_{3}\gamma_{\mathit{2}}-1\gamma_{3}^{-}1-\gamma 41\gamma 1\gamma_{5}-1=1,$

Now, for fixed $t\neq 0$ , the restriction of$f_{t}$$f$$\gamma_{\mathrm{s}}\gamma_{6}^{-}=11>$.

is: $x_{t}arrow t\cross\Delta(0, b)$ .

is 1. We extend the covering toThis is a covering of degree 3 and the genus ofin the caconincal way as in Section 2 which is denoted
the branched covering ofby the same notation for simplicity.
Now the method of Reidemeister-Schreier says that the fundamental group $\pi_{1}(X-$$(x=f^{-1}((0,0)))$ is generated by these loops
and (see Section 4) and their generating relations are pull-back over of these of the fundamental
and .
, expressed by the generatorsgroup
We can carry this out observing the Riemann picture in Figure 5. The result is
as follows:,
where is the 3-dimensional compact oriented manifold on which $X$ is a cone (seeSection 1).
$X_{t}$$\mathrm{P}^{1}$$\{x\},$$p_{0}),$$\xi_{1},$$\xi_{2},$$\xi_{3},$$\beta$$\alpha$$f_{t}$$\pi_{1}(\Delta(0, a)\cross\Delta(\mathrm{O}, b)-B_{f,q0})$$\pi_{1}(\mathrm{Y}, p\mathrm{o})\simeq\pi_{1}(X-\{X\}, p\mathrm{o})=\{1\}$$\mathrm{Y}$$\xi_{1},$$\xi_{\mathit{2}},$$\xi_{3},$$\alpha$$\beta$102

References[1] J. S. Birman: Braids, Links, and Mapping Class Group, Ann. Math. Studies,
82, Princeton, (1974).

[2] J. S. Birman, B. Wajnryb: 3-Fold branched coverings and the mapping classgroup of a surface, Lect. Notes in Math., 1167, Springer, (1985), 24-46.
[3] A. Dimca : Singurarities and Topology of Hypersurfaces, Springer- Verlag,(1992).
[4] O. Forster: Riemannsche Fl\"achen, Springer-Verlag, (1977).
[5] R. H. Fox: Covering spaces with singularities, Lefschetz Symposium, Princeton
Univ. Press, (1957), 243-262.
[6] H. Grauert, R. Remmert: Komplexe R\"aume, Math. Ann., 136(1958),245-318.
[7] H. M. Hilden: Every closed orientable 3-manifold is a 3-fold branched covering
space of , Bull. Amer. Math. Soc., Vol. 80, No. 6 (1974), 1243-1244.
$S^{3}$[8] T. Matsuno : On a theorem of Zariski-Van Kampen type and its applications, Osaka J. Math. 32, 645-658, (1995).
[9] J. M. Montesinos : A representation of closed, orientable 3-manifolds as 3fold branched coverings of , Bull. Amer. Math. Soc., Vol.80, No. 5 (1974),$S^{3}$

845-846.
[10] M. Namba: Degenerationg families of meromorphic functions, Proc. Internat.
Conf. ”Geometry and Analysis in Several Complex Variables”, Kyoto Univ.
(1997), RIMS Kokyuroku, 1058 (1998), 77-94.
[11] M. Namba, M. Takai: Degenerating families of branched coverings, to appear.
[12] D. Rolfsen: Knots and Links, Publish or Perisch Inc., (1976).

Maki Takai
Depertment of Mathematics
Osaka University
Toyonaka City, 560-0043, JapanE–mail: takai@gaia.math.wani.osaka-u.ac.jp