ON NIELSEN'S THEOREM FOR 3-MANIFOLDS

YOKOHAMA MATHEMATICAL

JOURNAL VOL. 35, 1987

ON NIELSEN’S THEOREM FOR 3-MANIFOLDS By

WOLFGANG HEIL and JEFFREY L. TOLLEFSON (Received April 14, 1986)

\S 1. Introduction. A classical result of Nielsen [N] states that a map of a (closed, orientable) surface $F$ to itself such that the nth iterate is homotopic to the identity is homotopic to a homeomorphism whose nth iterate is equal to the identity. In this paper we prove an analogous theorem for irreducible, sufficiently large 3manifolds. Nielsen’s Theorem is a solution of a special case of the following realization problem: Let $M$ be a compact n-dimensional manifold, let : Homeo $(M)\rightarrow ff(M)$ be the homomorphism that assigns to each homeomorphism of $M$ its isotopy class in the homeotopy group of $M$ Let $H$ be a finite subgroup of . Does there exist a subgroup $G$ of Homeo $(M)$ such that $\phi|G$ is an isomorphism of $G$ onto $H$ ? For the 2-dimensional case various partial results have been obtained (see the discussion in [Z]), and a full solution has been given by Kerckhoff [K]. For 3-dimensional Seifert fiber spaces the problem was solved in [H. T.] for the case when $H$ is the cyclic group of order 2, and subsequently in [Z. Z.] for an arbitrary finite group $H$. Using the Splitting Theorem of Jaco-Shalen and Johannson and Thurston’s work on hyperbolic manifolds, we obtained in [H. T. II] the solution for the case that $M$ is an orientable Haken manifold and $H=Z_{2}$ . B. Zimmermann following the same approach, has generalized them ethods of [Z. Z.] to obtain the solution in this case for an aribtrary finite group $H$. The approach of Zieschang-Zimmermann differs drastically from ours, since the former use the theory of crystollographic groups on the universal cover of $M$ while we use methods from the topology of 3-manifolds. Still another proof for the case of Seifert fiber spaces has been given by W. Neumann and F. Raymond in an unpublished manuscript. In the present paper we generalize the methods of [H. T.] and [H. T. II] to obtain a proof for the case when $H$ is a finite cyclic group and $M$ is an orientable Haken manifold. Our proof proceeds according to the following outline. First we obtain a relative version of the Nielsen Theorem for the case that $M$ is a sufficiently large Seifert fiber space. $g$

$g^{n}$

$\phi$

.

$\ovalbox{\tt\small REJECT}(M)$

W. HEIL AND J. L. TOLLEFSON

2

Theorem 1. If is a map of a compact, orientable, irreducible Seifert fiber then there is a homeomospace $M$ with nonempty boundary such that rphism of $M$ such that $h\simeq gre1\partial M$ and $h‘‘=1$ . $g$

$g^{n}\simeq 1re1\partial M$

$h$

such that afterwards The idea of the proof of Theorem 1 is to deform $M,\hat{A}=\cup g^{\ell}(A)$ is a system of disjoint annuli and for an essential annulus $A$ in Then is homotopic to the identity by a homotopy that is constant on we consider restricted to the components splitting $M$ along the annuli of and use an induction argument to show that can be deof $M$ split along to the desired homeomorphism . formed Next we consider the case that $M$ is a (possibly closed) Seifert fiber space. $g$

$\partial M\cup\hat{A}$

$g^{n}$

$\hat{A}$

.

$g$

$g$

$\hat{A}$

$h$

$re1\partial M$

Theorem 2. Let $M$ be a $comPact$ , orientable, irreducible Seifert fiber sPace $M$ to itself which contains an incompressible fibered torus and let be a map of is homotoPic to the identity. Then is homotoPic to a homeomorPhism such that with $h^{n}=1$ if and only if Obs $(Z_{n}, \pi_{1}(M),$ ) $=0$ $g$

$g$

$g^{n}$

$\Psi$

$h$

.

The obstruction that appears in the statement of Theorem 2 is interpreted geometrically in Lemma 6. We show that if it vanishes then we can find an rel $T$ . essential torus $T$ in $M$ and a deformation of such that afterwards Then we proceed as in the proof of Theorem 1. The next step is to prove a relative version of the Nielsen Theorem for certain simple manifolds (Thm 3), by using the absolute version given by $M$ is Thurston’s and Mostow’s work. Finally, to treat the general case when a closed Haken manifold we use the Splitting Theorem to split $M$ along incompressible tori into simple 3-manifolds and Seifert fiber spaces and deform such by a homotopy that is constant on the splitting tori, and that afterwards obtain the following result. $g^{n}\simeq 1$

$g$

$g$

$g^{n}\simeq 1$

Theorem 4. Let $M$ be a closed Haken manifold that is not a Seifert fiber is homotoffic to the is a map of $M$ to itself such that space. $SuPPose$ that $h^{n}=1$ . is homotoPic to a homeomorphism with identity. Then $g^{n}$

$g$

$h$

$g$

$F$ $M$ We will work in the PL-category throughout this paper. A surface in . The terms iris usually assumed to be Properly embedded, $i.e$ . reducible, parallel, essential, sufficiently large, are standard (see [H] or [J]) and are also defined in [H. T., \S 1]. Similarly the terms Seifert ffier space, fibered solid torus, orbit surface, fiber-Preserving, are well-known (see $e.g$ . [S. T.]) and , is also denoted by are recalled in [H. T., \S 1]. A homotopy where $G_{0}=G|M\times 0$ and $G_{1}=G|M\times 1$ . The notation $G:f\simeq g$ rel $F$ is used to $i.e$ . $G(x, t)=f(x)=$ mean that the homotopy from to is constant on . then we let $g(x)$ for $x\in F$ and all . If $g:M\rightarrow M$ and $F\cap\partial M=\partial F$

$G_{t}$

$G:M\times I\rightarrow M$

$G$

$t\in I$

$f$

$F,$

$g$

$A\subset M$

$\hat{A}=\forall^{g^{i}(A)}$

3

ON NIELSEN’S THEOREM FOR 3-MANIFOLDS

\S 2. Periodic maps on surfaces and liftings of isotopies. be a compact surface. An essential arc in $F$ is a proper arc that is not homotopic . An essential curve in $F$ is a simple closed to an arc on noncontractible curve.

Let

$F$

$re1\partial F$

$\partial F$

.

Assume $F$ is then there is an essential arc in $F$ such . If different form then there is an disjoint arcs. If that is a union of $F$ (i) is a union of pairwise disjoint essential curve in such that either closed curves or (ii) the orbit surface $F/f$ is a 2-sphere with three branch Points, intersects each transversally and each compOnent of $F-\{\cup f^{\ell}(c)\}$ is an open disk.

Lemma 1. Let $S^{2},$

$f:F\rightarrow F$

$D^{g},$

$P^{t}$

$\{\cup f^{\ell}(c)\}$

be a

homeomorPhism of periOd

$n\geq 2$

$c$

$\partial F\neq\emptyset$

$\partial F=\emptyset$

$p\dot{\alpha}rwlse$

$\{\cup f^{\ell}(c)\}$

$c$

$\alpha mple$

$f^{\ell}(c)$

$c$

Proof. Let $p:F\rightarrow F/f$ be projection onto the orbit surface and let $B\subset F/f$ we can lift an esbe the set of branch points. If $F/f$ differs from ) that misses the zero ) or an essential curve (if sential arc (if $B$ dimensional components of , to obtain the desired . (a) If $F/f=D^{2}$ , observe that l-dimensional components of $B$ can only occur in , there are either no such components or at least two. and since $B$ In the former case consists of at least two points and we find an essential by lifting an arc on $F/f$ that separates points of $B$ . In the latter case arc we get by lifting an arc joining two l-dimensional components of $B$ . (b) If $F/f=P^{2}$ there are at least two branchpoints (since otherwise $F=S^{g}$ or ). We obtain by lifting a simple closed curve on $F/f$ that bounds a disk containing two branchpoints. (c) If $F/f=S^{2}$ there are at least three branchpoints. If there are more than three, we obtain as in case (b). Thus assume that $B$ consists of three points. The surface $p^{-1}(F/f-U(B))$ , where $U$ is a regular neighborhood of $B$ , is not planar and thus contains an essential curve that is not homotopic to a product $F$ of boundary curves of . Now $p(d)$ is homotopic in $F/f-U(B)$ to a curve that is a product of two simple closed curves and that are generators of $\pi_{1}(F/f-U(B))$ . If $P^{-1}(a)$ is a simple closed curve, then we let this be . Otherwise is an arc starting at the basepoint of where (for some arc ) is a closed curve. is a simple closed curve and , at least one of Since or is not homotopic to a product of boundary curves of $F$. If this is true for we let $c=c_{1}$ . Otherwise we repeat this process using in place of . Eventually we must find a simple closed subloop of with the desired property. In either case is essential in $F$ (since $F$ is obtained from $p^{-1}(F/f-U(B))$ by filling in the boundary curves with disks) and $F/f-p(c)$ consists of three disks, each containing a branchpoint. $D^{2},$

$S^{2},$

$P^{2}$

$\partial F=\emptyset$

$\partial F/f\neq\emptyset$

$c$

$\partial D^{2}$

$F\neq S^{2},$

$D^{2}$

$c$

$c$

$P^{2}$

$c$

$c$

$d$

$\alpha$

$a$

$b$

$c$

$P^{-1}(\alpha)\simeq(wc_{1}w^{-1})c_{2}$

$\pi_{1}(F),$

$w$

$c_{2}=wc_{\theta}$

$c_{1}$

$p^{-1}(\alpha)\simeq d$

$wc_{1}w^{-1}$

$c_{s}$

$c_{2}$

$wc_{1}w^{-1}$

$c_{2}$

$c$

$p^{-1}(\alpha)$

$p^{-1}(\alpha)$

$c$

W. HEIL AND J. L. TOLLEFSON

4

has Period and if is a simPle Lemma 2. Let $F$ be a torus. If closed curve on $F$ with $f(a)$ homotopic to , then there is a simple closed curve is a system of simPle closed curves. homofofflc to such that $f:F\rightarrow F$

$a$

$n$

$\beta$

$a$

$\hat{\beta}=\cup f^{i}(\beta)$

$a$

is an Proof. Define a Riemannian metric on $F$ with respect to which isometry. Since in every homotopy class of simple closed curves there is a that is (freely) homogeodesic, we can choose a geodesic simple closed curve there is and then if topic to . Since $f$

$\beta$

$f^{\ell}(\beta)\neq f^{f}(\beta)$

$ f^{\ell}(\beta)\cap f^{j}(\beta)\neq\emptyset$

$f^{\ell}(\beta)\simeq f^{j}(\beta)$

$a$

. But and an arc on a union of an arc on we could then by replacing one of these arcs by an arc of shorter length in construct a simple closed curve of shorter length in the homotopy class of , a . or contradiction. Thus We now use Nielsen’s Theorem and the method of [H. T. II, Theorem 2] to a disk

$D$

on

with

$F$

$f^{f}(\beta)$

$f^{\ell}(\beta)$

$\partial D$

$D$

$a$

$ f^{\ell}(\beta)\cap f_{j}(\beta)=\emptyset$

$f^{\ell}(\beta)=f^{j}(\beta)$

derive the following relative version of Nielsen’s Theorem.

If

Proposition 3.

homeomorPhism

is a map such that such that $h^{n}=1$ .

$g:F\rightarrow F$

$h\simeq gre1\partial F$

$g^{n}\simeq 1$

rel

there is a

$\partial F$

Proof. Since induces an isomorphism on the fundamental group we can is a homeomorphism. assume that is an annulus, $F=S^{1}\times I$ . Case (1) Without loss of generality, we may assume that is standard on . Thus, for some fixed or , where suppose . Observe that the homoor $1-t$ . Let 1 denote the arc or 1, and by . Since is determined topy class of any map , for some integer , it follows that Clearly and determined by . with where $g$

$g$

$F$

$\partial F$

$g$

$a(z)=\overline{z}$

$g|\partial F(z, \epsilon)=(\alpha(z), \lambda(\epsilon))$

$\{1\}\times I$

$\lambda(t)=t$

$\{(a(1)e^{2\pi\ell q\ell}, \lambda(t))|0\leq t\leq 1\}re1\partial F$

implies

$h^{n}=1$

$q$

$\alpha,$

$g$

$q$

$h^{n}\simeq 1$

.

is not an annulus. Case By a homotopy of , constant (2)

$g\simeq hre1\partial F$

$\lambda$

$h(z, t)=(\alpha(z)e^{8\pi iqt}, \lambda(t))$

$ g(l)\simeq$

$g|\partial F\cup l$

$g;F\rightarrow F$

$re1\partial F$

$re1\partial F$

$\epsilon=0$

$z_{0}\in S^{1},$

$zz_{0}$

$F$

$g$

on

,

$\partial F$

we may assume that there is a

such that $g(U)=U$ and the homotopy $G:g^{n}\simeq 1$ carries $U$ neighborhood . By Nielsen’s Theorem [N] there is a to itself at each stage. Let . By Baer’s Theorem [Z], and homeomorphism of to . Extend this isotopy to an isotopy of $F$ constant on is isotopic to maps of are and rel , such that get a map $G$ $G(U\times I)=U$ $h‘‘=h|U$ . Let . homotopy by a with rel period , and . rel : By case (1) it suffices to show that there is a homotopy is connected (otherwise the construction aPplies To see this, assume that , and to each component) and note that $h^{n}(x_{0})=x_{0}$ , for a basepoint represents an element that the trace of the cyclic homotopy , which is trivial. Hence we can assume by a 2-dimenof the center of $U=\partial F\times I$

$F^{\prime}=F\backslash \dot{U}$

$h^{\prime}$

$F^{\prime},$

$h^{\prime}\simeq g|F^{\prime}$

$h^{\prime}$

$(h^{\prime})^{n}=1$

$\partial F$

$g|F^{\prime}$

$g\simeq h$

$h;F\rightarrow F$

$n$

$h^{n}\simeq 1$

$\partial F$

$h|\partial F$

$h^{\prime}=h|F^{\prime}$

$\partial F$

$G_{1}$

$(h^{\prime})^{n}\simeq 1$

$\partial U$

$\partial F$

$x_{0}\in\partial F^{\prime}\subset\partial U$

$G|F^{\prime}\times I:h^{;n}\simeq 1$

$\tau$

$\pi_{1}(F^{\prime})$

ON NIELSEN’S THEOREM FOR 3-MANIFOLDS

5

sional version of lemma 7 that . Thus we now have is periodic $G^{\prime}=G|U\times I$ and is a homotopy rel with . The homotopy is now deformed by a homotopy to a desired homotopy by $H|U\times I\times\{0\}=G_{0},$ $H(x, 0, t)=(h^{\prime})^{n}(x)$ , and $H$ is constant on $\tau=x_{0}$

$h^{\prime}|\partial U$

$(h^{\prime})^{n}\simeq 1$

$(\{x_{0}\}\cup\partial F)$

$G^{\prime}=G_{0}$

$G_{t}^{\prime\prime}(\partial F^{\prime})\subset\partial F^{\prime}$

$H:U\times I\times I\rightarrow U$

$G_{1}$

$\partial F\times I\times I\cup U\times\{1\}\times I\cup\partial F^{\prime}\times I\times\{1\}\cup\{x_{0}\}\times I\times I$

Thus

is defined on the boundary of the 3-cell extended over this 3-cell. Now $H$ is defined on $H$

over

.

$(\partial F^{\prime}-\{x_{0}\})\times I\times I$

and can be and ex-

$U\times I\times\{0\}\cup\partial(U\times I)\times I$

to give the desired $G_{1}=H|U\times I\times\{1\}$ . The next proposition is a lifting theorem for Seifert fiber spaces. For such a space $M$ let be the projection onto its orbit surface, let $E$ be the set of exceptional fibers and $E^{*}=p(E)$ .

tends

$U\times I\times I$

$p:M\rightarrow M^{*}$

Proposition 4. Let $M$ be a sufficiently large Seifert fiber sPace. Let a map isotopic to the identity rel . Then there is a map $h:M$ that is isotoPic to the identity rel such that $ph=h^{*}p$ . $h^{*}:$ $M^{*}$

$\rightarrow M^{*}be$

$(\partial M^{*}\cup E^{*})\ovalbox{\tt\small REJECT}$

$\rightarrow M$

$(\partial M\cup E)$

Proof. Case (a) :proof is by induction on the length of a : $The$ $M$ hierachy for defined by fibered essential annuli reducing $M$ to solid tori. If $M^{*}=D^{g}$ and the assertion is trivial. If $M^{*}=D^{g}$ and consists of one Point, let be an arc from a point on to and proceed in a way. similar to what follows. In any other case there is an essential arc in such that and are in general position. Then is isotopic to rel and $A=P^{-1}(l)$ is an essential annulus isotopic rel by a fiber to preserving isotopy that is constant on the fibers of $A\cap P^{-1}(h^{*}(l))$ . Thus there is a fiber preserving isotopy $k:M\rightarrow M$, constant on , with $k^{*}h^{*}(l)=l$ . Now there is a fiber preserving isotopy $j:M\rightarrow M$, constant on , with $j^{*}k^{*}h^{*}|U(l)$ the identity on a regular neighborhood $U$ of . But then (by the argument below in case $(b))j^{*}k^{*}h^{*}$ is isotopic to the identity by an isotopy constant on . By induction there is a fiber preserving with isotopic to the identity and rel . Extend over $U$ by the identity to get $f:M\rightarrow M$. Then induces . (b) Case : First suppose $M$ contains an incompressible fibered torus $T$ . We wish to apply the argument of case (a) to $l=p(T)$ and we obtain fiber preserving isotopies and , constant on $E$ , with the identity on and isotopic to the identity by a homotopy , rel . To proceed with the proof in case (a), we wish to replace by a homotopy that is constant on . If we pick the basepoint on , the trace commutes with in . Thus, if is not a torus, it is homotopic to a $DoWer$ of 1 and can be $d_{P}fnrmer1\eta parJQ\cap rb+af\mathfrak{t}ar\tau xrarl\{\circ\Gamma/J\vee\Gamma 1-J$ $\partial M\neq\emptyset$

$ E^{*}=\emptyset$

$E^{*}$

$l$

$\partial M^{*}$

$E^{*}$

$l$

$l$

$h^{*}(l)$

$l$

$M*\backslash E^{*}$

$h^{*}(l)$

$(\partial M\cup E)$

$(\partial M^{*}\cup E^{*})$

$P^{-1}(h^{*}(l))$

$\partial M\cup E$

$\partial M\cup E$

$l$

$\partial M^{*}\cup E^{*}\cup U(l)$

$f:M\backslash \dot{U}(A)$

$f^{*}|(M^{*}\backslash U^{Q}(l))=j^{*}k^{*}h^{*}|(M*\backslash \mathring{U}(l))$

$\rightarrow M\backslash \mathring{U}(A)\circ$

$(\partial(M\backslash U(A))\cup E)$

$k^{-1}j^{-1}f$

$f$

$f$

$h^{*}$

$\partial M=\emptyset$

$k$

$j$

$j^{*}k^{*}h^{*}|l$

$j^{*}k^{*}h^{*}$

$G:M^{*}\times I\rightarrow M^{*}$

$l$

$E^{*}$

$G$

$E^{*}\cup l$

$x_{0}$

$l$

$\pi_{1}(M^{*}\backslash E^{*})$

$G$

$M^{*}\backslash E^{*}$

$l$

$\tau=G(x_{0}\times I)$

W. HEIL AND J. L. TOLLEFSON

6

we can assume that is constant on . and then by a further deformation of transversely is a torus we let be a simple closed curve that meets If and by an isotoPy. fixed on is isotopic to in one point. Then is the identity on clearly this isotopy lifts. Thus we can assume that . is isotopic to the identity by an isotopy constant on . But now and have If $M$ does not contain an incompressible fibered torus, we $M$ we can assume that consists of 3 points. By an isotopy of of rel $U(E^{*})$ . By case (a) there is a lift : over $U$ by the identity that is (fiber) isotopic to the identity rel . Extend of . to obtain a lift $l$

$G$

$G$

$l$

$t$

$M^{*}\backslash E^{*}$

$l$

$t$

$j^{*}k^{*}h^{*}(t)$

$j^{*}k^{*}h^{*}|t$

$t\cup l$

$j^{*}k^{*}h^{*}$

$t$

$M^{*}=S^{\epsilon}$

$h^{*}\simeq 1$

$E^{*}$

$h^{\prime}$

$h^{\prime}$

$\partial U$

$h$

Corollary 5. $\rightarrow M$

be a

$h^{\prime*}=h^{*}|(M^{*}\backslash U)$

$M\backslash p^{-1}(U)\rightarrow M\backslash p^{-1}(U)$

$h^{*}$

Let

$M$

homeomorPhism

be a sufficiently large Seifert such that the diagram

fiber space and

let $f:M$

$p_{NN}MM\downarrow\underline{f^{*}}\downarrow p\underline{f}$

is isotopic to is iso topic rel $(\partial MUE)$ to

commutes. If that

$h$

rel

$h^{*}$

$f^{*}$

$(\partial M^{*}\cup E^{*})$

then there is a

lift

$h$

of

$h^{*}$

such

.

$f$

Proof. Apply proposition 4 to get a lift

$g$

of

$(f^{*})^{-1}h^{*}$

and let

$h=fg$

.

\S 3. Making homotopies constant on surfaces. $g:M\rightarrow M$ of a Haken 3-manifold and In this section, given a homeomorphism $F$ in $M$ such a homotopy $G:g^{n}\simeq 1$ rel , and an essential annulus or torus tori, we would like to deform that $P=\cup g^{\ell}(F)$ is a union of disjoint annuli or $(\partial MuP)$ . and $G$ so that afterwards $G:g^{n}\simeq 1$ rel of $M$ be normalized by a homotopy such that a basepoint Let $\Psi:Z_{n}$ of $\pi=\pi_{1}(M, x_{0})$ . Let is fixed by . Then induces an automorphism Recall that a necessary condition for to $\rightarrow Out\pi=Aut\pi/Inn\pi$ be $\Psi(k)=[g_{*}^{k}]$ . has of period is that the abstract kernel be homotopic to a map an extension (see [H. T.] \S 2 or [C. R.]). $\partial M$

$g$

$X_{0}$

$g:M\rightarrow M$

$g$

$g_{*}$

$g$

$g$

$(Z_{n}, \pi, \Psi)$

$h$

$n$

, and let $\tau(t)=G_{t}(x_{0})$ be the trace of the Lemma 6. Suppose center $G$ can be chosen such homotoPy $G:g^{n}\simeq 1$ . Then Obs $(Z_{n}, \pi, \Psi)=0$ if and only if . that $(\pi)\simeq Z$

$ g_{*}(\tau)=\tau$

Proof. If Obs $(Z_{n}, \pi, \Psi)=0$ there is an extension

$E$

of the abstract kernel

ON NIELSEN’S THEOREM FOR

7

$3\cdot MANIFOLDS$

$1\rightarrow\pi-E\rightarrow Z_{n}\rightarrow 1$ $\downarrow$

$ 1\rightarrow$

$\downarrow\mu$

$\downarrow\Psi$

Inn $\pi\rightarrow Aut\pi\rightarrow Out\pi\rightarrow 1$

and we can choose $e\in E$ such that $\mu(e)=g_{*}$ (where is the inner auto$a^{-1}()a)$ morphism . Then . Thus there exists $c\in center(\pi)$ such $g_{*}(\tau c)=\mu(e)(\tau c)=\mu(e)(e^{n})=e^{n}=\tau c$ that . It follows that Since $M$ admits an action of with center generated by a principal orbit, there is a cyclic $H:1\simeq 1$ homotopy with trace (namely $H(x, t)=\alpha(t)x$ , where is a path in $(S^{1},1)$ with ). Composing $G$ with for the evaluation map : this cyclic homotopy yields a homotopy with trace . Conversely, suppose , and . We define $\phi(S)=g_{*}(0\leq s1$ . The isotopy induces an isotopy of rel and rel and we can apply Theorem 3 to deform each $g|M_{i}$ and $g|N_{f}$ by a homotopy that is constant on $F$ to homeomorphisms and such that and . This gives the desired deformation of to a homeomorphism with $h^{n}=1$ . $G$

$(g|M_{\ell})^{n}\simeq 1$

$\partial M_{i}$

$(g|N_{j})^{n}\simeq 1$

$\partial N_{f}$

$h_{\ell}$

$1|M_{\ell}$

$h_{j}^{n}=1|N_{f}$

$h_{j}$

$h_{\ell}^{n}=$

$g$

$h$

Fig. 1

W. HEIL AND J. L. TOLLEFSON

20

References [C. R]

[C. V] [He]

P. E. Conner and F. Raymond: Manifolds with few periodic homeomorphisms, Proc. 2nd Conference on Compact Transformation Groups, Lecture Notes in Mathematics, 299 (1970), 1-74. Charlap and Vasquez: Amer. J. Math. 95 (1973), 471-494. W. Heil: Almost sufficiently large Seifert fiber spaces, Michigan Math. (1973),

[H. T.]

J. 20

217-223.

W. Heil and J. Tollefson: Deforming homotopy involutions to involutions,

logy 17 (1978), [H. T. II] W. Heil and

349-365. J. Tollefson:

$\tau_{0}p_{\theta-}$

Deforming homotopy involutions to involutions, II,

Topology (1982), [H] [J]

J.

Hempel: 3-manifolds, Ann.

of Math. Studies 86, Princeton Univ.

Press,

1976.

W. Jaco: Lectures on 3.manifold Topology, CBMS series 43, Amer. Math. Soc. (1980).

K. Johannson: Homotopy equivalences of 3.manifolds with boundaries, Lecture Notes in Math. 761, Springer (1979). [M] A. Miller: Thesis, University of Connecticut, 1981. [McL] S. MacLane: Homology, Springer (1963). [N] J. Nielsen: Abbildungsklassen endlicher Ordnung, Acta Math. 75 (1943), 23-115. [K] Kerckhoff: The Nielsen Realization Problem, Bull. Amer. Math. Soc. 2 (1980),

[Jo]

452-453.

J. Math. 102

[S]

P. Scott: A new proof of the annulus and torus theorems, Amer.

[S. T.] [T]

H. Seifert and W. Threlfall: A Textbook of Topology, Academic Press 1980. W. Thurston: The geometry and Topology of 3.manifolds. Lecture notes Princeton

(1980),

241-277.

University. [To]

J. Tollefson: Involutions on 323-352.

sufficiently large 3.manifolds, TopOlOgy20 (1981),

F. Waldhausen: On irreducible 3-manifolds which are sufficiently large, Ann. Math. 87 (1968), 56-88. [Z] H. Zieschang: Finite groups of mappingclasses of surfaces. Springer Lecture Notes 875 (1981). [Z. Z.] H. Zieschang and B. Zimmermann: Endliche Gruppen von Abbildungsklassen gefaserter 3-Mannigfaltigkeiten, Math. Ann. 240 (1979), 41-62. [Zi] B. Zimmermann: Das Nielsensche Realisierugsproblem fur hinreichend grosse Mannigfaltigkeiten. Math. Z. 180 (1982), 349-359.

[W]

Department of Mathematics Florida State University Tallabassee, FL 32306-3027 USA