数理解析研究所講究録1022 巻 1997 年 166-178

166

The realizations of the amalgamated freeof $-orbifold fundamental groups$\mathrm{p}\mathrm{r}o\mathrm{d}\mathrm{U}\mathrm{c}\mathrm{t}_{8}$Aichi University of Education, Yoshihiro Takeuchi (竹内 義浩)
Faculty of Science, Shizuoka University, Misako Yokoyama (横山 美佐子)
Abstract.
We introduce a kind of generalized orbifolds called “orbifold compositions,” and study on their topology, and extensions and deformations of maps
between them. As the main goal, we show the theorem which yields the geometrealizations of amalgamated free products 3-orbifold fundamental groups.
For the details or the HNN extension case, see [T-Y 3].
$\mathrm{r}\mathrm{i}\mathrm{c}$\S 1. Preliminaries on orbifolds. An
is called a manifold covering if
A covering :orbifold $M$ is good if the universal covering of $M$ is a manifold covering andbad otherwise.
Let $M$ be a 3-orbifold and $F$ a connected 2-suborbifold which is either. We say that is compressible
properly embedded in $M$ or contained in
in $M$ if one of the following conditions is satisfied.
(i) $F$ is a spherical orbifold which bounds a ballic orbifold in $M$ , oror there is a discal 2is a discal orbifold and either(ii). $F\cap G=\partial F=\partial G$
and a ballic 3-suborbifold
suborbifold
and $\partial B=F\cup G$ , ordoes not(iii) there is a discal orbifold $D\subset M$ with $D\cap F=\partial D$ andbound any discal orbifolds in $F$ .
Otherwise, $F$ is incompressible. By [K-S] and [M-Y 1], the ballic orbifold
bounded by $F$ in (i) is the cone on . Hence, a locally orientable 3-orbifold’s. A 3-orbifold $M$ is irreducible if there aredoes not include compressible
no incompressible spherical 2-suborbifolds in $M$ .
Throughout this paper, all orbifolds are connected unless otherwise stated.
At first, we review three theorems in [T-Y 2].
$p$$\Sigma\tilde{M}=\phi$$\tilde{M}arrow M$$F$$\partial M$$F\subset\partial M$$F$$B\subset M\mathrm{s}.\mathrm{t}$$G\subset\partial M$$\partial D$$F$$\mathrm{R}P^{2}$

Theoren 1.1. (The Loop Theorem [T-Y 2, 6.4]) Let $M$ be a good
. If
3-orbifold with boundaries. Let $F$ be a connected 2-suborbifold in properly, then there exists a discal 2-suborbifold$\partial M$$\mathrm{K}\mathrm{e}\mathrm{r}(\pi_{1}(F)arrow\pi_{1}(M))\neq 1$

$D$

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embedded in $Ms.t$ .
in $F$ .

$\partial D\subset F$

and$\partial D$

does not bound any discal 2-suborbifold

Theorem 1.2. (Dehn’s Lemma [T-Y 2, 6.5]) Let $M$ be a good 3-orbifoldwith boundaries. Let be a simple closed curve in $\partial M-\Sigma Ms.t$. the order
in $\pi_{1}(M)$ . Then there exists a discal suborbifold $D^{2}(n)$ properlyisof.
embedded in $M$ with$\gamma$$n$$[\gamma]$$\partial D^{\mathit{2}}(n)=\gamma$

Theorem 1.3. (The Sphere Theorem [T-Y 2, 6.7]) Let $M$ be a good
, thenbe the universal cover of M. If
3-orbifold. Let :, where isinthere exists a spherical suborbifold in $Ms.t$ .
any component of $p^{-1}(S)$ .
$p$$\tilde{M}arrow M$$\pi_{2}(\tilde{M})\neq 0$$S$$[\tilde{S}]\neq 0$$\tilde{S}$$\pi_{2}(\tilde{M})$The next corollary is derived directly from 1.3.
Let $M$ be a good 3-orbifold.
Corollary 1.4.
any manifold coveringof $M_{j}\pi_{2}(\tilde{M})=0$ .

If $M$ is irreducible, then for$\tilde{M}$\S $. Orbifold compositions

From now on, we assume that all orbifolds are good, connected, and locallyorientable, unless otherwise stated.
-orbifolds,
be countable sets,Let$(j\in J)$ (n-l)-orbifolds. Let.
be orbi-maps:
are monic where $j\in J,$ $i(j, \epsilon)\in I,$ $\epsilon=0,1$ . Then we call $X=$$[0,1],$arean -dimens ional orbifold composition. The maps
is called a component ofcalled the attaching maps of $X$ . Each . oris defined to bein
X. The equivalence
generated by

Deflnition 3.1.

$I,$

$J$

$f_{\dot{j}}^{\epsilon}$$X_{i}(i\in I)\mathrm{n}$$\mathrm{Y}_{\mathrm{j}}$$\mathrm{s}.\mathrm{t}$$\mathrm{Y}_{j}\mathrm{x}\epsilonarrow X_{1()}.j,\epsilon$$(f_{j}^{\epsilon})_{\mathrm{r}}$$(X\dot{‘},$$f_{j}^{e})_{\in I,jJ,\epsilon=0}.\cdot\in,1$$n$$\mathrm{Y}_{j}\mathrm{x}$$f_{j}^{\epsilon}$$\mathrm{Y}_{j}\mathrm{x}[0,1]$$X_{1}$$\mathrm{r}\mathrm{e}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\sim$$(y, \epsilon)\sim f_{j}(y)\neg$$\coprod_{\mathrm{i}\in I,j\in J}(|X.\cdot|\cup|\mathrm{Y}_{j}|\mathrm{x}[0,1])$,$\epsilon=0,1$,$y\in|\mathrm{Y}_{\mathrm{j}}|$We call the identified spaceof $X$ , denoted by $|X|$ , and call the identified space$[0,1]))\}/\sim$ the singular set of $X$ , denoted by.

,$j\in J$.

$\mathrm{U}_{2\in I},j\in J(|X.\cdot|\cup|\mathrm{Y}_{j}|\mathrm{x}[0,1])/\sim \mathrm{t}\mathrm{h}\mathrm{e}$underlying space$\{(\bigcup_{\in I^{\Sigma x}}.\cdot.\cdot)\cup(\bigcup_{\mathrm{j}\in f}\Sigma(\mathrm{Y}j\mathrm{x}$$\Sigma X$

From now on, we assume that the underlying space $|X|$ is connected. Noteare embedded in $|X|$ .
andthat$|X_{\mathrm{s}}|$$|\mathrm{Y}_{j}\mathrm{x}(0,1)|$

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Let $X=$
Deflnition 3.2.
composition. Define the identified space $C(X)$ by$(X_{\dot{\mathrm{c}}}, \mathrm{Y}_{j}\mathrm{x}[0,1],f_{j}^{\epsilon})_{2\in}I,j\in J,\epsilon=0,1$$\exists i\in I$$\mathrm{s}.\mathrm{t}$$x\approx y\Leftrightarrow\{$$\exists j\in J,$ $\exists \mathrm{t}\in[0,1]$$\mathrm{s}.\mathrm{t}$.

.

$x,$ $y\in|X.\cdot|/\sim$$x,$be an orbifold$|X|/\approx \mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}$, or$y\in|\mathrm{Y}_{j}\mathrm{x}t|/\sim$.

, the associated 1, and each, each
We call $C(X)$ , eachcomplex, a vertex orbifold, an edge orbifold of $X$ , and the core ofrespectively.
$\mathrm{Y}_{j}\mathrm{x}[0,1]$$X,\cdot$$\mathrm{Y}_{\mathrm{j}}\mathrm{x}\frac{1}{2}$$\mathrm{Y}_{j}\mathrm{x}[0,1])$

An isomorphism of orbifold compositions is a map which is componentwise
isomorphism and commutes with attaching maps.

and $X’=$
be orbifold compositions. We say that $X’$ is a,
. after changing
covering of $X$ if there exist a set of maps
the orientations of $[0,1]’ s$ if necessary, the following (1) $\sim(3)$ hold.
. $\in K$ .
is a covering map (of orbifolds) from $X’$ to(1) Each’ where,tois a covering map (of orbifolds) fromAnd eachwhere $p_{j}\in L$ .
.
and(2) For: $|X’|arrow|X|$ which is naturally induced by(3) The continuous map
is onto and induces the usual covering map from $|X’|-p^{-1}(\Sigma X)$
to $|X|-\Sigma X$ .
We call the above map a covering map from $X’$ to $X$ .

Let

Definition 3.4.

$X=(X_{\mathrm{k})}\mathrm{Y}_{\iota}\mathrm{x}[0,1], f_{Z}^{\epsilon})_{k\in\kappa,\ell}\in L,\epsilon=0,1$$(X_{i’ j}’\mathrm{Y}’\mathrm{x}[0,1], f_{j}^{\prime^{\mathrm{g}}})_{\in I,j\in f\mathrm{g}=0,1}.\cdot$$\{\varphi_{2}\cdot, \psi_{j}\}_{1\in I_{\mathit{3}\in}\int}.,\mathrm{s}.\mathrm{t}$$k_{1}$$X_{\mathrm{k}}:$$\varphi_{i}$$\mathrm{Y}_{j}’\mathrm{x}[0,1]$$\psi_{j}$$\epsilon=0,1,$$\forall j$$\mathrm{Y}_{\ell_{j}}\mathrm{x}[0,1]$$\varphi_{i(j,)}\mathrm{g}f^{\prime^{\epsilon}}\circ j=f_{\mathit{1}_{i}}^{\epsilon}\circ(\psi_{j}|\mathrm{Y}_{\mathrm{j}}’\mathrm{x}\epsilon)$$p$$\{\varphi|.)\psi_{j}\}_{\in I,j\in}.\cdot f$$p$

the above definition, if each component $X’$ is the universal
Remark 3.5.
, then for some base point $x_{0}\in|X|-\Sigma X$ , any pathcover of a component
there. Int $f\cap\Sigma X=\phi$ , and any pointwith the base point
areexists a unique lift of with the base point . This holds because$\ln$$X_{k_{1}}.$$\ell$$\tilde{x}_{0}\in p^{-1}(x_{0})_{l}$$x_{0}\mathrm{s}.\mathrm{t}$$f$$(f_{\mathit{1}}^{\epsilon})_{\mathrm{z}}$$\tilde{x}_{0}$monic.
Let $X$ be an orbifold composition, $x_{0}\in|X|-\Sigma X$ a baseDefinition 3.6.
. Int $f\cap\Sigma X=\phi$ , and $p:\tilde{X}arrow X$ any
point a path with the base point. Suppose there is a covering :. Fix any point. each component of is the universal cover of a component of . Fix any. By Remark 3.5 there exists a unique lift to of withpoint$\ell$$\mathrm{c}\mathrm{o}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{i}’ \mathrm{n}\mathrm{g}$$x_{0}\mathrm{s}.\mathrm{t}$$\hat{X}arrow\tilde{X}$$\tilde{x}_{0}\in p^{-1}(x_{0})$$\hat{p}$$\tilde{X}$$\hat{X}$$\mathrm{s}.\mathrm{t}$$\hat{\ell}$$\hat{x}_{0}\in\hat{p}^{-1}(\tilde{x}_{0})$$\hat{X}$$\ell$

169
the base point. Then we can determine a lift of uniquely by putting, which is called the canonical lift of with the base point .
$\tilde{f}$$f$$\hat{x}_{0}$$\tilde{f}=\hat{p}\circ\hat{f}$$\ell$$\tilde{x}_{0}$Definition 3.7.
Let $X’,$ $X$ be orbifold compositions, and : $X’arrow X$ acovering. We define the deck transformation group
of by
$p$

$\mathrm{A}\mathrm{u}\mathrm{t}(X’,p)$$\mathrm{A}\mathrm{u}\mathrm{t}(X’,p)=${ : $X’arrow X’|h$ is an isomorphism$h$$\mathrm{s}.\mathrm{t}$$p$. $p\mathrm{o}h=p$ }.

Definition 3.8.
Let , $X$ be orbifold compositions, and :acovering. We say that is a universal covering if for any covering : $X’arrow X$ ,there exists a covering :.
.
$\tilde{X}$$\tilde{X}arrow X$

$p$

$p’$

$p$

$\tilde{X}arrow X’\mathrm{s}.\mathrm{t}$$p=p^{\mathit{1}}\mathrm{o}q$$q$

Lemma 3.9. For any orbifold composition $X$ , there exists a unique universal
covering $p:\tilde{X}arrow X$ .

Proof.

See [T-Y 3].

We sometimes denote an orbifold composition or a good orbifold $X$ by$(\tilde{X},p, |X|)$
where :
is the universal covering, and $|X|$ is the underlying
space of $X$ . A good orbifold is considered as a special case of an orbifold
composition.
$p$$\tilde{X}arrow X$$X$ beProposition 3.10.
Let
orbifold compositions and :covering. If the restriction of to each component of is universal andis a tree; then the covering $p:\tilde{X}arrow X$ is universal.
$\tilde{X},$$p$$\tilde{X}$$\tilde{X}arrow X$

a$p$$C(\tilde{X})$Definition 3.11. Let $X=$
base point $x_{0}\in|X|-\Sigma X$ . Put

be an orbifold composition with the

Proof.

See [T-Y 3].
$(\tilde{X}, p, |X|)$$\Omega(\tilde{X}, x_{0})=${ a continuous map :$\tilde{\alpha}|$$\tilde{\alpha}$$[0,1]arrow\tilde{X}$with$p(\tilde{\alpha}(0))=p(\tilde{\alpha}(1))=x_{0}$}.

For any two elements
is equivalent to , denoted by, ifthere exists an element
and
.
The
an equivalence relation anda group with the
product defined by$\tilde{\alpha},\tilde{\beta}\in\Omega(\tilde{X}, x_{\mathit{0}}),\tilde{\alpha}$$\sqrt{}^{\sim}$$\tau\in \mathrm{A}\mathrm{u}\mathrm{t}(\tilde{X},p)_{\mathrm{S}.\mathrm{t}}.\tilde{\alpha}(0)=\tau(\sqrt{}^{\sim}(0))$$\mathrm{r}\mathrm{e}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{i}_{0}\mathrm{n}\sim \mathrm{i}\mathrm{s}$$\Omega(\tilde{X}, x_{0})/\sim \mathrm{i}\mathrm{s}$$[\tilde{\alpha}]\cdot[\sqrt]=[\vee\tilde{\alpha}\cdot\rho(\sqrt\sim)]$$\tilde{\alpha}\sim\sqrt{}^{\sim}$$\tilde{\alpha}(1)=\tau(\sqrt{}^{\sim}(1))$

170. The groupis the element .
where. Note that theis called the fundamental group of $X$ and denoted byis isomorphic to the deck transformation groupfundamental groupwhich is. By the symbol , we mean the element of.
corresponding to$\mathrm{s}.\mathrm{t}$$\rho\in \mathrm{A}\mathrm{u}\mathrm{t}(\tilde{X}p))$$\Omega(\tilde{X}, X_{0})/\sim$$\rho(\sqrt{}^{\sim}(0))=\tilde{\alpha}(1)$$\pi_{1}(X, x_{0})$$\pi_{1}(X, x_{0})$$\mathrm{A}\mathrm{u}\mathrm{t}(\tilde{X}, p)$$\mathrm{A}\mathrm{u}\mathrm{t}(\tilde{X},p)$$\sigma_{A}$$\sigma\in\pi_{1}(X, x_{0})$Let $X=(\tilde{X},p, |X|)$ andDeflnition 3.12.

compositions (or orbifolds). By an orbi-map
andof continuous maps,(i), there exists(ii) for each(iii) there exists:$Xarrow \mathrm{Y}$$\tilde{f}:\tilde{X}arrow\tilde{\mathrm{Y}}$$\overline{f}:|X|arrow|\mathrm{Y}|$$(\overline{f},\tilde{f})$$f$be orbifold, we mean the pairsatisfying$\mathrm{Y}=(\tilde{\mathrm{Y}}, q, |\mathrm{Y}|)$$\overline{f}\circ p=q\mathrm{o}\tilde{f}$$\tau\in \mathrm{A}\mathrm{u}\mathrm{t}(\tilde{\mathrm{Y}}, q)\mathrm{s}.\mathrm{t}.\tilde{f}\circ\sigma=\tau 0\tilde{f}$$\sigma\in \mathrm{A}\mathrm{u}\mathrm{t}(\tilde{X},p)$$x\in|X|-\Sigma x_{\mathrm{s}}.\mathrm{t}.\overline{f}(x)\in|\mathrm{Y}|-\Sigma \mathrm{Y}$Let $X=$:
compositions, andorbi-map, there exists a point:
induced homomorphism.
byDeflnition 3.13.

$f$

.

.

be orbifold
an orbi-map. By the definition of an. Then theof is naturally defined$(\tilde{X}, p, |X|)$$Xarrow \mathrm{Y}$$f=(\overline{f},\tilde{f})$,and$\mathrm{Y}=(\tilde{\mathrm{Y}}, q, |\mathrm{Y}|)$$x\in|X|-\Sigma X\mathrm{s}.\mathrm{t}.\overline{f}(x)\in|\mathrm{Y}|-\Sigma \mathrm{Y}$$\pi_{1}(X, x)arrow\pi_{1}(\mathrm{Y},\overline{f}(x))$$f$$f_{l}([\tilde{\alpha}])=[\tilde{f}\circ\tilde{\alpha}]$For an orbi-map and a covering between orbifold compositions we candefine the notions of -equivalence, orbi-homotopy, and lifting as well as those
for an orbi-map and a covering between orbifolds. We derive the relations
among fundament al groups, coverings, and liftings similar to those for orbifolds.
See [Ta 2] for the orbifold case.
The next proposition can be shown in a way similar to one in [Prop. 2.2of Ta 2].
$\mathrm{C}$

Proposition 3.14.
sitions, andhave that$f=(\overline{f},\tilde{f})$Let $X=$:$(\tilde{X}, p, |X|),$$Xarrow \mathrm{Y}$$\mathrm{Y}=(\tilde{\mathrm{Y}}, q, |\mathrm{Y}|)$an orbi-map. Then for

be orbifold compo] $\in\pi_{1}(X, x)$ , we$\eta_{\tilde{\alpha}}$.

$\tilde{f}0[\tilde{\alpha}]_{A}=(f_{l}([\tilde{\alpha}]))_{A}0\tilde{f}$\S 4. The tree constructions of the universal coverings
one of edge orbifold compoLet be an orbifold composition andare two disjoint orbifold compositions
nents of . Suppose thatandare mapped into
and attaching orbi-maps fromand. We construct the universal covering,:
described as$X$$\mathrm{Y}\mathrm{x}[0,1]$$X-\mathrm{Y}\mathrm{x}(0,1)$$X$$X^{0}$$\mathrm{Y}\mathrm{x}\epsilon$$X^{1}$}$f^{\epsilon}$$\mathrm{Y}\mathrm{x}\epsilonarrow X^{\mathrm{g}}$$\epsilon=0,1$$X^{\mathrm{g}}$171of an orbifold composition $X$ by the “tree construction”, and show that thefundamental group $\pi_{1}(X)$ of $X$ is the free product ofandwith$\epsilon=0,1$
the amalgamated subgroups.
$\epsilon=0,1$ , andLet:
be the universal
coverings. Put
andleft coset representative systembyof, which includes the identity ), $\epsilon=0,1$ . A group is definedas the free product ofandwith the amalgamated subgroups
andunder the map, denoted by$\pi_{1}(X^{1})$$\pi_{1}(X^{\mathit{0}})$$f^{\epsilon}.\pi_{1}(\mathrm{Y}\mathrm{x}\epsilon),$$\tilde{X}^{\epsilon}arrow X^{\epsilon},$$q:\tilde{\mathrm{Y}}\mathrm{x}[0,1]arrow \mathrm{Y}\mathrm{x}[0,1]$$p^{\epsilon}$$H^{\epsilon}=f^{\epsilon}.\pi_{1}(\mathrm{Y}\mathrm{x}\epsilon)$$A^{\epsilon}=(\mathrm{a}$$H^{\epsilon}$$\pi_{1}(X^{\epsilon})$$\pi_{1}(X^{0})$$H^{0}$$G$$e$$H^{1}$$\pi_{1}(X^{1})$$f_{l}^{1}\mathrm{o}(f^{0}.)^{-1}$)$G=\{\pi_{1}(x^{\mathrm{Q}})*\pi 1(x1)|H^{0}=H^{1}, f^{1}.\mathrm{o}(f^{\mathit{0}}.)^{-1}\}$

And three subsets $K,$

$K^{0},$$K^{1}$

ofare defined by$G$$K=\{e,a_{1}a_{2}\cdots a_{m}|a.\cdot\neq e,$ $a|$.

$a_{2},$$a_{+1}.\cdot$.

,$\in A^{0}\cup A^{1}$

are not both in$A^{0}$

or both in$A^{1}.$}$K^{0}=\{e, a_{1}a_{2}\cdots a_{m}\in K|a_{m}\in A^{1}\}$$K^{1}=\{e, a_{1}a_{2}\cdots a_{m}\in K|a_{m}\in A^{0}\}$.

For each, prepare a copy
, and the identity mapof:Note that there are
equivalent classes of,. And for each, prepare a copyof
, and the identity map:. Let$\epsilon=0,1$ .
:be structure maps ofThen we can define
structure maps:
naturally. Put. Define the projections
:
and
:
by
and
)’
, respectively. Note thatandare theuniversal coverings. Furthermore, it is easy to see thatis a tree. Henceby 3.10,:is the universal
A$k\in K^{\epsilon}$$\tilde{X}^{\epsilon}$$\tilde{X}_{\mathrm{k}}^{\epsilon}$$id_{k}^{e}$$\tilde{X}_{k}^{\epsilon}arrow\tilde{X}^{\mathrm{g}}$$\mathrm{m}\mathrm{o}\mathrm{d} (H^{\epsilon})_{A}$$\epsilon=0,1$$(k, a)\in K^{0_{\mathrm{X}}}A^{0}$$\tilde{\mathrm{Y}}\mathrm{x}[0,1]$$\tilde{f}^{\epsilon}$$\mathrm{A}\mathrm{u}\mathrm{t}(\tilde{X}^{\mathrm{g}},p\epsilon)\tilde{f}^{\mathrm{g}}(\tilde{\mathrm{Y}}\mathrm{x}\epsilon)$$\# A^{\epsilon}$$\tilde{\mathrm{Y}}_{(\mathrm{k},a)}\mathrm{X}[0,1]$$\tilde{\mathrm{Y}}_{(k,\mathrm{n})}\mathrm{x}[0,1]arrow\tilde{\mathrm{Y}}\mathrm{x}[0,1]$$id_{(k,a)}$$\tilde{\mathrm{Y}}\mathrm{x}\epsilonarrow\tilde{X}^{\epsilon}$$f^{\epsilon},$$\tilde{f_{()}}^{\epsilon_{\mathrm{k},a}}$$\tilde{\mathrm{Y}}_{(k,a)}\mathrm{x}\epsilonarrow\tilde{X}_{h}^{\epsilon}$$\tilde{X}=(\tilde{X}_{k\ell(k}^{0},\tilde{x}^{1},\tilde{\mathrm{Y}},a)\mathrm{x}$$[0,1],\tilde{f(}k,a)’(k\tilde{f}^{1},q)0)_{k\in^{\kappa^{0}},\in q}\mathit{1}\kappa^{1},\in A^{0}$$q_{(h,a)}$$\tilde{\mathrm{Y}}_{(h,a)}\mathrm{x}[0,1]arrow \mathrm{Y}\mathrm{x}[0,1]$$k\in K^{\epsilon}\epsilon)=0,1,$$\tilde{X}_{k}^{\epsilon}arrow X^{\epsilon}$$p_{k}^{\epsilon}$$q_{(h,\alpha)}=q\mathrm{o}id_{(}h,a$$p_{k}^{\epsilon}=p^{\epsilon}\circ id_{\mathrm{t}}^{\epsilon}$$(h, a)\in K^{0}\mathrm{x}A^{0}$$p_{\mathrm{k}}^{\epsilon}$$C(\tilde{X})$$p= \bigcup_{k\in}K$‘$\mathrm{g}0,1,(h_{0},)\epsilon K0\mathrm{x}$$0(p_{k}^{\epsilon}\cup q_{(h_{*}a)})$$\tilde{X}arrow X$

covering.

Lemma 4.1.

$\pi_{1}(X, x_{0})\underline{\simeq}$

G.

Proof.

See [T-Y 3].

\S 5. Extensions and constructions of orbi-maps$q_{(h,a)}$172

Deflnition 5.1.

Let $X$ be an orbifold composition. Define

is a discal 2-orbifold, is an orbi-map},{ :$O_{2}(X)=$ {: $Sarrow X|S$ is a spherical 2-orbifold, is an orbi-map},$O_{1}(X)=$$f$$\partial Darrow X|D$

$f$

$f$

$f$

$O_{9}(X)=\{f$: $DBarrow X|’DB$ is the double of a ballic 3-orbifold
is an orbi-map}.

$B$

,

$f$

We call : $\partial Darrow X\in O_{1}(X)$ trivial if there exists an orbi-map : $Darrow X$. $g|\partial D=f$ , and call $O_{1}(X)$ trivial if any element of $O_{1}(X)$ is trivial. We.
call : $Sarrow X\in O_{2}(X)$ trivial if there exists an orbi-map :$g|S=f$ , where $c*S$ is the cone on , and call $O_{2}(X)$ trivial if any element of$O_{2}(X)$ is trivial. We define the trivialities of $O_{3}(X)$ similarly.
of $X$ inherits the triviNote that if $o_{i}(X)$ is trivial, then any covering
ality.
$g$$f$$\mathrm{s}.\mathrm{t}$

$g$

$f$

$c*Sarrow X\mathrm{s}.\mathrm{t}$$S$$\tilde{X}$be a compact 2-orbifold and $X$ be an orbifold
Let
Proposition 5.2.
composition. If $O_{1}(X)$ is trivial, then for any homomorphism : $\pi_{1}(F, y)arrow$.
$\pi_{1}(X, x)$ , there exists an orbi-map: $(F, y)arrow(X, x)s.t$.
$F$

$f$

$f$

.

$\varphi$$=\varphi$-Int $U(\Sigma F)$ , where $U(\Sigma F)$ is the small regular neighborto $X$ associated with .
. We construct an (orbi-) map fromhood of.D.)
Since $O_{1}(X)$ is trivial, it is extendable to the desired orbi-map. (Proof.

Let$F_{\mathit{0}}=F$$F_{0}$$\Sigma F$$\varphi$$\mathrm{Q}.\mathrm{E}$The following propositions 5.3 and 5.4 are proved similarly.

Let $M$ be a compact 3-orbifold and $X$ an orbifold comProposition 5.3.
position . t. $O_{1}(X)$ and $O_{2}(X)$ are trivial. Then for any homomorphism: $\pi_{1}(M, x)arrow\pi_{1}(X, y),$ ’ there exists an orbi-map : $(M, x)arrow(X, y)s.t$ .
$s$

.

$f$

$\varphi$$f$$=\varphi$.

Proposition 5.4. Let $M$ be a 3-orbifold and $X$ be an orbifold composition. t. $O_{3}(X)$ is trivial. If $f,$ : $Marrow X$ are C- equivalent orbi-maps, then andare orbi-homotopic.
$s$

$f$

$g$

$g$

The following lemmas 5.5, 5.6, and 5.7 give sufficient conditions whichenable us to extend certain orbi-maps.

Lemma 5.5.

Let $X$ be an orbifold composition,$D$a discal 2-orbifold, and

173

:

an orbi-map. If Fix$([f]_{A})\neq\phi$ , thenorbi-map fromto $X$ .

$f$$\partial Darrow X$

is extendable to an

$f$

$D$

Proof.

Let : $D^{2}arrow D$ be the universal covering. Take a point.
We can construct the structure map of the desired orbi-map by mapping thecone point of
to and performing the skeletonwise and equivariant extension. (Q.E.D.)$q$$x\in \mathrm{F}\mathrm{i}\mathrm{x}([f]_{A})$$D^{2}$$x$

Let be a spherical 2-orbifold and :the universal covering. Let
be an element ofandthe point of.
, where is thenormal loop aroundand is an integer. By the symbol, we mean the
$S$

$q$

$\pi_{1}(S)$$\tau$$\tilde{S}arrow S$$[\ell]^{k}=\tau$$\Sigma S\mathrm{s}.\mathrm{t}$$x_{\tau}$$k$$x_{\tau}$$f$$\mu(\ell)$.
$\mathrm{b}\mathrm{e}\mathrm{t}1\mathrm{o}\mathrm{C}\mathrm{a}_{\mathrm{h}\mathrm{e}}1\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}_{\mathrm{i}}\mathrm{a}\mathrm{p}_{0}\mathrm{n}\mathrm{t}\mathrm{o}11_{0}\mathrm{o}\mathrm{f}q^{-1}(\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{d}x,.\mathrm{S}.\mathrm{t}.

path around$\tilde{x}_{\tau}$\ell=m-1.(^{\ell}\Sigma s^{\mathrm{n}})_{\mathrm{S}}.\mathrm{t}.\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{l}\mathrm{f}\mathrm{t}\mathrm{o}\mathrm{f}\mu(f)\mathrm{f}_{\mathrm{o}11\mathrm{o}\mathrm{w}}\mathrm{n}\mathrm{g}\mathrm{a}\mathrm{f}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{t}\mathrm{h}\mathrm{e}1\mu)\cdot m_{\mathrm{i}},\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}m\mathrm{i}_{\mathrm{S}\mathrm{a}\mathrm{a}}\mathrm{P}\mathrm{t}\mathrm{h}.m\mathrm{i}\mathrm{f}\mathrm{t}\mathrm{o}\mathrm{f}-1\mathrm{i}\mathrm{L}\mathrm{e}\mathrm{t}_{\tilde{X}}\mathrm{s}’ \mathrm{a}$.

Lemma 5.6.
Let $X$ be an orbifold composition;a spherical 2-orbifo ,and : $Sarrow X$ an orbi-map. Suppose that there is a pointand for any, there is an interval
including andwhich iswhere
. If of the universal cover of $X$ is , thenfixed byis extendable to an orbi-map from the cone on to $X$ .
$S$

$ld$

$f$

$\tilde{d}\in \mathrm{F}\mathrm{i}\mathrm{x}(f.\pi(1S))_{A}$$\tau\in\pi_{1}(S)$$\sigma_{A;}$$\tilde{d}$$\tilde{f}(\tilde{x}_{\tau})$$\ell_{\sigma}$$\sigma=f_{4}(\tau)$$\tilde{X}$$\pi_{2}$$\mathit{0}$

$f$

$S$

Proof.

See [T-Y 3].

Lemma 5.7.

Let $X$ be an orbifofd composition, $B$ a ballic 3-orbifold, and: $DBarrow X$ an orbi-map. Suppose that there is a pointand for, there is an intervaf
including andwhich is$\sigma=f.(\tau)$
where. If and of the universal cover of $X$ is ,fixed bythen is extendable to an orbi-map from the cone on $DB$ to $X$ .
$f$$\tilde{d}\in \mathrm{F}\mathrm{i}\mathrm{x}(f.\pi_{1}(\partial B))A$$\forall\tau\in\pi_{1}(\partial B)$$\tilde{d}$$\tilde{f}(\tilde{x}_{\Gamma},)$$\ell_{\sigma}$$\tilde{X}$$\sigma_{A;}$$\pi_{2}$$\pi_{8}$$\mathit{0}$$f$Proof.

Similar to 5.6.

($\mathrm{Q}.\mathrm{E}$.D.)

Lemma 5.8.
Let $M$ be an irreducibfe 3-orbifold. Let : $\hat{M}arrow M$ be theuniversal covering andbe an orientation preserving element offinite order. Suppose that is non-compact, then the following (i), (ii) hold:(i) Fix
and is homeomorphic to an interval ( $i.e$ . homeomorphic toeither $[0,1],$ $[0,1)$ , or $(0,1))$ .
(ii) If $M$ is orientable, then $O_{1}(M)$ is trivial.
$p$$\sigma\in \mathrm{A}\mathrm{u}\mathrm{t}(\hat{M}p))$$\hat{M}$$(\sigma)\neq\phi$174

Proof.

(i) See [T-Y 3].

(ii) lt is obtained by (i) and 5.5.

$\hat{M}arrow M$
theLet $M$ be an irreducible 3-otbifold, and :Lemma 5.9.
, which is isomorphic
universal covering. Let be any bgroup ofa spherical 2-orbifold and all elements
to the orbifold fundamentalof preserve the orientation of M. Suppose that is non-compact, then thefollowing (i), (ii) hold:.
(i) Fix
are triviaI, $i=1,2,3$ .
(ii) If $M$ is orientable, then
$p$

$G$

$\mathrm{A}\mathrm{u}\mathrm{t}(\hat{M}, p)$$s\mathrm{u}$$S$$group_{\wedge}of$$\hat{M}$$G$$(c)\neq\emptyset$$O_{1}\cdot(M)’ s$

Proof.

(i) See [T-Y 3].

(ii) It is obtained by (i), 5.5, 5.6, 5.7, and 5.8.

be an orbifold compoLet $X=$
is an orientabfe, irreducible 3-orbifold, and is an orisition, where eachand are all $non- Compact_{\gamma}$
entable 2-orbifold. If the universal coverings of$i=1,2,3$ .
then $O_{i}(X)$ are

Proposition 5.10.

$(X^{e} , \mathrm{Y}\mathrm{x}[0,1], f^{\mathrm{g}})_{\epsilon=0,1}$$\mathrm{Y}$$X^{\epsilon}$$\mathrm{Y}$$X^{\epsilon}$$t_{\mathrm{f}\dot{8}}vial,$

Proof.

See [T-Y 3].

Let $X$ be an orbifold composition, and a core of an edge orbifold Yx $[0,1]$
of $X$ . When we consider each piece (or its closure) of $|X|-|F|$ , it naturally
admits the orbifold composition structure by restricting the structure of $X$ . We
(resp.
this situation, a component of typedenote it by $X-F$ , etc.
open) half-edge$\epsilon=0,1$ , appears, and is called a closed (resp.
orbifold of the orbifold composition. lterating this process, we can consider
an orbifold composition with several half-edge orbifolds. About new types oforbifold compositions described above, the same arguments and statements
hold as those in Sect. 3\sim 5.
$E$

$\ln$

$\mathrm{Y}\mathrm{x}[\epsilon, \frac{1}{2}]$$\mathrm{Y}\mathrm{x}[\epsilon, \frac{1}{2})),$\S 6. More on orbifold compositions
if
be an orbifold composition. An orbifold belongs to the set
Let
sat\‘isfies the following (i) or (ii):. is disjoint(i) is a boundary component of a vertex orbifold of
from any images of attaching maps of .
.
.
(ii) is the core of a closed half-edge of$X$$\delta X$$\mathrm{Y}$$\mathrm{Y}$$Xs.\mathrm{t}$$\mathrm{Y}$$\mathrm{Y}$$X$$X\mathrm{s}.\mathrm{t}$$\mathrm{Y}$Theorem 6.1. ($\mathrm{n}_{\mathrm{a}\mathrm{n}\mathrm{S}\mathrm{V}\mathrm{e}\mathrm{r}}\mathrm{s}\mathrm{a}\mathrm{h}\mathrm{t}\mathrm{y}$theorem)$\partial \mathrm{Y}=\emptyset$

Let $M$ be a compact and ori-

175

entable 3-orbifold, and $X$ a 3-orbifold composition with trivial $O_{1}.(X)ft,$ $i=2,3$ .
Suppose that there is an edge orbifold whose core is an orientable and nonspherical 2-orbifold $Fs.t$ . $O_{\mathrm{s}}(X-F)$ are trivial, $i=2,3$ . Then; for any
orbi-map : $Marrow X$ , there is an orbi-map : $Marrow Xs.t$ .
(i) is orbi-homotopic to ,(ii) each component of $g^{-1}(F)$ is a compact; properfy embedded, 2-sided,
incompressible 2-suborbifold in $M$ , and(iii) for properly chosen product neighborhoods $F\mathrm{x}[-1,1]$ of $F=F\mathrm{x}0$
in $X$ and $g^{-1}(F)\mathrm{x}[-1,1]$ of $g^{-1}(F)=g^{-1}(F)\mathrm{x}\mathrm{o}$ in
maps each fiber$x\mathrm{x}|[-1,1]|$ homeomorphically tothe fiber
for each $x\in|g^{-1}(F)|$where : $|M|arrow|X|$ is the underlying map of .
$f$

$g$

$f$

$g$

$M,\overline{g}$$\overline{g}(x)\mathrm{x}|[-1,1]|$$\overline{g}$Proof.

$g$See [T-Y 3].

Theorem 6.2. (I-bundle theorem) Let $M$ be a compact, orientable andirreducible 3-orbifold with boundaries, and $X$ be acomposition. Let: $(M, \partial M)arrow(X, \delta X)$ be an orbi-map $s.i$ .
is monic. Suppose there is a$arrow(|M|-\Sigma
M,|\partialM|)$path: (I, ), incompressible components,of, and a component
which
satisfy
the following
;
of

.

$f$

$\mathit{3}- orbifo\iota d$$f$$\partial I$$\alpha$$B_{0}$$\partial M$$C$$\delta X$$B_{1}$$(\mathrm{i})\sim(\mathrm{i}\mathrm{V})$

(i)
.
(ii)
.
(iii)
in $\pi_{1}(X)$ ,where is a lift of a to the universaf coverof $M$ and(iv) . (resp. ) includes(resp.
),
and $(f|B|.):B.\cdotarrow C$ is a covering, $i=0,1$ (possibly $B_{0}=B_{1}$ ).
Then $M$ is an -bundle over a closed 2-orbifold.
$\alpha(0)\neq\alpha(1)$$\overline{f}(\alpha(0))=\overline{f}(\alpha(1))\in|\delta X|-\Sigma X$$[\tilde{f}\circ\hat{\alpha}]=1$$\tilde{M}$$\hat{\alpha}$$C$$B_{1}$$\alpha(i)$$\overline{f}(\alpha(0))$$f=(\overline{f},\tilde{f})$.

$\mathrm{K}\mathrm{e}\mathrm{r}(\pi 1(C)arrow\pi_{1}(X))=1$,$I$

Proof.

See [T-Y 3].

Theorem 6.3. (Retraction theorem) Let $M$ be an orientable 3-orbifold
which is orbi-isomorphic to an -bundle over a closed 2-orbifold F. Let $X$ beacomposition with trivial $O_{2}(X)’S,$ $i=2,3$ . Let : $(M, \partial M)arrow$$(X, \delta X)$ be an orbi-map $s.t$ .
is not an orbi-embedding and $s.t$ . there is acomponent of$f(B)\subset C$ and $(f|B):Barrow$, for each component ofis an orbi-covering.
is orbi-homotopic to a path
If there is a point $x\in|F|-\Sigma F$ s. .
$Crel$
in.
, where : $Marrow F$ is the $fibrat\dot{f}on$ , then there isan orbi-homotopy : $Marrow Xs$ . t. $f_{0}=f,$
, and
.
$I$

$\mathit{3}- orb\dot{8}f_{\mathit{0}}ld$$f$$f|\partial M$

$C$

$B$

$\delta X$$\partial M,$

$C$

$t$

$\{x\}\mathrm{x}\partial I\ldots(6.3.1)$$f_{t}$$f|(\varphi^{-1}(x))$$\varphi$$f_{1}(M)\subset\delta X$$f_{t}|\partial M=f|\partial M$

176

See [T-Y 3].

Proof.

.

Remark 6.4. In 6.3, if : $\pi_{1}(M)arrow\pi_{1}(X)$ is an isomorphism and isentable; then the condition (6.3.1) stands. Furthermore, $M$ is orbi-isomorphic
is orbi-isomorphic to .
, andto the product -bundle over
$f$

$C$

$I$

$B_{0}$$or\dot{\mathrm{t}}-$$C$$B_{\mathit{0}}$\S 7. Main Theorem$\ln$this section, we assume that all free products with amalgamations are

non-trivial.
Let $M$ be a 3-orbifold with trivial $O_{1}(M)$ . Let be a closed,orientable, non-spherical 2-orbifold. Suppose $\pi_{1}(M)=(A_{1}*A_{2}|H_{1}=H_{2},$: $X$ $arrow M$ be the orbiand there is an isomorphism : $\pi_{1}(S)arrow H_{1}$ . Let
are trivial, $i=1,2$ .
$i=1,2$ . Note thatcovering associated with)(resp.
$i=1,2$ . Note thatPut). By 5.2, we can construct(resp.
tois an isomorphism fromand.
:orbi-maps: $Sarrow X_{1}$ and,. We call the orbifold composition $X=(X_{1},$,the orbifold composition associated with $\{A_{1^{*}}A_{2}|H_{1}=H_{2}, \varphi\}$ .
We also define the orbifold composition associated with $(A,$ $t|t^{-1}H_{1}t=H_{2},$similarly.
From 4.1 (resp. 4.2), it holds that $\pi_{1}(X)=\}\pi_{1}(X_{1})*\pi_{1}(X_{2})|h_{1}.\pi_{1}(S)=$).
} (resp. $\{\pi_{1}(X’),$ $t|t^{-1}h_{1*}\pi_{1}(S)t=h_{2},\pi_{1}(S),$Furthermore, we have the following proposition.

Deflnition 7.1.

$S$$\psi$$O_{1}(X\cdot)|$$A_{i},$$Sarrow X_{2}\mathrm{s}.\mathrm{t}$$h_{2}$$h_{1}$$\tilde{H}_{2}$$\tilde{H}_{1}$$\pi_{1}(S)$.

$(p_{2}.|\tilde{H}_{2})^{-1}0\varphi\circ\psi$$(p_{1}.|\tilde{H}_{1})^{-1_{\circ}}\psi$$\tilde{H}.\cdot=p^{-1}..(H.),$$h_{\mathit{2}}$.

$p_{\mathfrak{i}}$$\varphi\}$$h_{1}$.

$=(p_{11}|\tilde{H}_{1})^{-1}\circ\psi$$X_{2}$$=(p_{22}|\tilde{H}_{2})^{-1}0\varphi 0\psi$$h_{2})$$S\mathrm{x}[\mathrm{o}, 1]_{)}h1$$\varphi \mathrm{I}$$h_{2}.\pi_{1}(S),$$h_{2}$.

$h_{2}$$\mathrm{o}h_{1}^{-1}.$.

$\mathrm{o}h_{1}^{-1},\}$

be
Let $M$ be a 3-orbifold with trivial $O_{1}(M)$ . Let
Proposition 7.2.
pose $\pi_{1}(M)=(A_{1}*$.
a closed, orientable, and non-spherical$A_{2}|H_{1}=H_{2},$(resp. $(A, t|t^{-1}H_{1}t=H_{2}, \varphi\})$ and there is an isomor: $\pi_{1}(S)arrow H_{1}$ . Let $X$ be the orbifold composition associated withphism
$t|t^{-1}H_{1}t=H_{2},$). Then there is an$\{A_{1}*A_{2}|H_{1}=H_{2}, \varphi\}$ (resp.
isomorphism : $\pi_{1}(X)arrow\pi_{1}(M)s.t$.
(resp. $\Psi(\pi_{1}(X’))=A$ ).
(i)$S$$\mathit{2}- orbif_{ol}d$$Su\mathrm{p}$$\varphi\}$$\psi$$\{A,$$\varphi\}$$\Psi$.

$\Psi(\pi_{1}(X_{i}))=A_{\mathrm{s}},$

(ii)
(iii)

$\Psi(\tilde{H}_{\mathfrak{i}})=H$;$i=1,\mathit{2}$$i=l,$$\mathit{2}$(note that$\Psi\circ(h_{2}, \circ h_{1}^{-1}.)=\varphi\circ\Psi$$h\cdot\pi\cdot(|l|)S=\tilde{H}.,$).

.

Deflnition 7.3. Let $M$ be a 3-orbifold, and be a closed, orientable, andnon-spherical 2-orbifold. We say that algebraically splits $\pi_{1}(M)$ as an amal$S$

$S$

177

gamated free product if $\pi_{1}(M)$ is expressed as a free product with an amalgamation, $\{A_{1}*A_{2}|H_{1}=H_{2}, \varphi\}$ , and there is an isomorphism : $H_{1}arrow\pi_{1}(S)$ .
We say that the splitting above respects the peripheral structure of $M$ if
for each component of, some conjugate ofis contained in either, where is the inclusion orbi-map $Garrow M$ .
or$\Psi$$G$$A_{1}$$A_{2}$$\partial M$$\eta.\pi_{1}(G)$$\eta$

Proposition 7.4. Let $M$ be a compact, orientable; and irreducible 3-orbifold.
Let
be a closed, orientable, and non-spherical 2-orbifo . Suppose
alge$\pi_{1}(M)$braically splits
as an amalgamated free product $(A_{1}*A_{2}|H_{1}=H_{\mathit{2}},$and this splitting respects the peripheral structure of M. Let $X$ be the orbifold
composition associated with $\{A_{1}*A_{2}|H_{1}=H_{2}, \varphi\}$ . Then there is an orbi-map: $Marrow Xs.t$ .
is an isomorphism and $f(\partial M)\cap(S\mathrm{x}(0,1))=\phi$ .
$S$

$ld$

$S$

$\varphi\}$$f$$f_{l}$

Proof.

See [T-Y 3].

Definition 7.5.
Let
be a closed, properly embedded, 2-sided, incompressible, and separating 2-suborbifold in $M$ . Let
be the orbifolds
derived from $M$ by cutting open alongand: $Farrow M\dot{.},\dot{f}=1,2$ be theinclusion orbi-maps. Note that $\pi_{1}(M)$ is expressed as the amalgamated freeproduct. We say that$F$

$M_{1},$$F$$M_{2}$$\eta_{i}$$\{\pi_{1}(M_{1})*\pi_{1}(M_{2})|\eta_{1}.\pi_{1}(F)=\eta_{2}.\pi_{1}(F), \eta 21\circ\eta_{1}^{-1}.\}$$F$

geometrically realizes the algebraic splitting $\{A_{1}*A_{2}|H_{1}=H_{2}, \varphi\}$ of $\pi_{1}(M)$ if
there is an isomorphism :.
.
(i) $\Psi(\pi_{1}(Mi))=A,$ $)i=1,2$ .
(ii) $\Psi(\eta.\cdot.\pi_{1}(F\mathrm{x}i))=H.,$ $i=1,2$ .
(iii).
$\Psi$$\pi_{1}(M)arrow\pi_{1}(M)\mathrm{s}.\mathrm{t}$$\Psi \mathrm{o}(\eta_{2}.0\eta_{1\prime}^{-1})=\varphi 0\Psi$

Theorem 7.6.
Let $M$ be a compact, orientable, and irreducible 3-orbifold.
Let
be aorientable, and non-spherical 2-orbifold. Suppose
alge$\pi_{1}(M)$braicafly splits
as an amalgamated free product $(A_{1}*A_{2}|H_{1}=H_{2},$and this splitting respects the peripheral structure of M. Then there exists ageometric splitting realizing the algebraic splitting above.
$S$

$S$

$C\iota_{ose}d_{:}$$\varphi\}$Let us take an overview of the proof of the main theorem, to see how
effectively our preparations are used:(i) Recall that the fundamental group $\pi_{1}(M)$ of a 3-orbifold $M$ is decomposed as ( $A_{1}*A_{2}|H_{1}=H_{2}$ , . First we takeand the orbi-covering
associated with A. and construct an orbifold composition $X$ by attachingthem together where Section 4 and 5.2 essentially contribute to. This newly$\phi\}$$M_{\dot{\iota}}$$\cdot$$S\mathrm{x}$

$I$

178

space.
constructed space $X$ plays a role like as an(ii) Make an orbi-map : $Marrow X$ which induces an isomorphism from$\pi_{1}(M)$ to. At this time, we need theorems prepared in Sections 4 and5.
(iii) Each component of the inverse image of by is an incompressible
2-suborbifold by 6.1. We decrease the numbers of these components using 6.2
and 6.3 repeatedly. At last the inver se image turns to be only one component$F$
which actually realizes the decomposition of $\pi_{1}(M)$ .
For the details or the HNN extension case, see [T-Y 3].
$\mathrm{E}\mathrm{i}\mathrm{l}\mathrm{e}\mathrm{n}\mathrm{b}\mathrm{e}\Gamma \mathrm{g}-\mathrm{M}\mathrm{a}\mathrm{C}\mathrm{L}\mathrm{a}\mathrm{n}\mathrm{e}$$f$$\pi_{1}(X)$

$S$

$f$

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$R^{3}$$\mathrm{R}^{3}$$\mathrm{P}\mathrm{L}$$\cdot$