Extending automorphisms of the genus-2 surface over the 3-sphere
aa r X i v : . [ m a t h . G T ] A p r EXTENDING AUTOMORPHISMS OF THE GENUS- SURFACEOVER THE -SPHERE KENTA FUNAYOSHI AND YUYA KODA
Abstract.
An automorphism f of a closed orientable surface Σ is said to be extend-able over the 3-sphere S if f extends to an automorphism of the pair ( S , Σ) withrespect to some embedding Σ ֒ → S . We prove that if an automorphism of a genus-2surface Σ is extendable over S , then f extends to an automorphism of the pair ( S , Σ)with respect to an embedding Σ ֒ → S such that Σ bounds genus-2 handlebodies onboth sides. The classification of essential annuli in the exterior of genus-2 handlebodiesembedded in S due to Ozawa and the second author plays a key role. : 57M60; 57S25 Keywords : mapping class group, handlebody, essential annulus, JSJ-decomposition
Introduction
Let Σ g be the closed orientable surface of genus g . An automorphism f of Σ g is saidto be null-cobordant if f extends to an automorphism of a compact orientable 3-manifold M with respect to an identification Σ g = ∂M . In Bonahon [1], it is proved that if f isan orientation-preserving, null-cobordant, periodic automorphism of Σ g , then the above M can be chosen to be a handlebody. It is also showed in the same paper that the sameconsequence holds for an arbitrary irreducible automorphism of Σ . These results givepartial answers to the problem: Given an automorphism f of Σ g that extends to anautomorphism of a certain 3-manifold M with ∂M = Σ g , find the “simplest” 3-manifoldamong such M for f . In fact, a handlebody of genus g is definitely the “simplest”3-manifold with the boundary Σ g .An automorphism f of Σ g is said to be extendable over S if f extends to an auto-morphism of the pair ( S , Σ g ) with respect to an embedding Σ g ֒ → S . In particular,if we can choose the above embedding Σ g ֒ → S to be standard (i.e. an embeddingsuch that Σ g bounds handlebodies on both sides), we say that f is standardly extendableover S . An automorphism of Σ g extendable over S is clearly null-cobordant, but theconverse is false. Indeed, the Dehn twist along a non-separating simple closed curveon Σ g is null-cobordant, but not extendable over S (see Section 3). As an analogy ofthe above-mentioned problem for null-cobordant automorphisms, the following problemnaturally arises: The second author is supported in part by JSPS KAKENHI Grant Numbers 15H03620, 17K05254,17H06463, and JST CREST Grant Number JPMJCR17J4.
Problem.
Given an automorphism f of Σ g that extends to an automorphism of the pair( S , Σ g ) with respect to an embedding Σ g ֒ → S , find the “simplest” such an embeddingΣ g ֒ → S for f .The best candidate for the “simplest” embedding Σ g ֒ → S is definitely a standardone. There is a series of studies by Guo-Wang-Wang [25], Guo-Wang-Wang-Zhang [9]and Wang-Wang-Zhang-Zimmermann [26, 27, 28] considering finite subgroups of theautomorphism group Homeo(Σ g ) of Σ g that extend to subgroups of Homeo( S , Σ g ) withrespect to some Σ g ֒ → S . A direct consequence of their results is that a periodicautomorphism of Σ that is extendable over S is actually standardly extendable over S . In the present paper, we show that we can remove the periodicity condition here.In fact, we prove the following: Theorem 3.3.
Let f be an automorphism of a closed orientable surface of genus two.If f is extendable over S , then f is standardly extendable over S . It is easily seen that the same fact is valid for a closed surface of genus less than twoas well, see Section 3. The case of higher genera remains open. The classification ofessential annuli in the exteriors of genus-two handlebodies embedded in S (see Section2) given by [19] plays a key role in our proof of Theorem 3.3.Throughout the paper, we will work in the piecewise linear category. Any surfacesin a 3-manifold are always assumed to be properly embedded, and their intersection istransverse and minimal up to isotopy. For convenience, we will not distinguish surfaces,compression bodies, e.t.c. from their isotopy classes in their notation. Nbd( Y ) willdenote a regular neighborhood of X , Cl( X ) the closure of X , and Int( X ) the interior of X for a subspace X of a space, where the ambient space will always be clear from thecontext. The number of components of X is denoted by X . Let M be a 3-manifold,and let L ⊂ M be a submanifold, or a graph. When L is 1 or 2-dimensional, we write E ( L ) = Cl( M \ Nbd( L )). When L is of 3-dimension, we write E ( L ) = Cl( M \ L ).1. Preliminaries
Let L , L , . . . , L n , R be possibly empty subspaces of a compact orientable 3-manifold M . We will denote by Homeo( M, L , L , . . . , L n rel R ) the group of automorphisms of M which map L i onto L i for any i = 1 , , . . . , n and which are the identity on R . The map-ping class group , denoted by MCG( M, L , L , . . . , L n rel R ), is defined to be the groupof isotopy classes of elements of Homeo( M, L , L , . . . , L n rel R ). When R = ∅ , we willdrop rel R . The “plus” subscripts, for instance in Homeo + ( M, L , L , . . . , L n rel R ) andMCG + ( M, L , L , . . . , L n rel R ), indicate the subgroups of Homeo( M, L , L , . . . , L n rel R )and MCG( M, L , L , . . . , L n rel R ), respectively, consisting of orientation-preserving au-tomorphisms (or their classes) of M .1.1. Handlebodies.
Let V be a handlebody. A simple closed curve c on ∂V is said tobe primitive if there exists a disk D properly embedded in V such that the two loops c and ∂D intersect transversely in a single point. Suppose that V is embedded in S sothat the exterior W := E ( V ) is also a handlebody. Then a disk D properly embeddedin V is said to be primitive if ∂D is primitive in W . XTENDING AUTOMORPHISMS OF THE GENUS-2 SURFACE OVER THE 3-SPHERE 3
Lemma 1.1.
Let V be a handlebody of genus two, and E be a separating essential diskin V . Let c be a simple closed curve on ∂V with ∂E ∩ c = ∅ and [ c ] = 1 ∈ π ( V ) . Thenthere exists a unique non-separating disk D in V with ∂D ∩ c = ∅ . Further, this disk D is disjoint from E .Proof. The disk E cuts V into two solid tori X and X . Without loss of generality, wecan assume that c ⊂ X . Then a meridian disk D in X with ∂D ⊂ ∂V is disjoint from c , and furthermore, disjoint from E .We show the uniqueness of D . Assume that there exists a non-separating disk D ′ in V that is disjoint from c and not isotopic to D . If D ∩ D ′ = ∅ , then D ′ is a meridiandisk of a solid torus V cut off by D . Since [ c ] = 1 ∈ π ( V ), D ′ intersects c . This is acontradiction. Suppose that D ∩ D ′ = ∅ . Then any outermost subdisk of D ′ cut off by D ∩ D ′ must intersect c , a contradiction again. (cid:3) Corollary 1.2.
Let V be a handlebody of genus two, and c be a primitive curve on ∂V .Then there exists a unique non-separating disk D in V with ∂D ∩ c = ∅ .Proof. Since c is primitive, there exists a disk D properly embedded in V such that thetwo loops c and ∂D intersect transversely in a single point. Then E := Cl( ∂ Nbd( c ∪ D ) \ ∂V ) is a separating disk in V disjoint from c . The assertion thus follows fromLemma 1.1. (cid:3) Lemma 1.3.
Let V be a handlebody of genus two, and E be an essential separating diskin V . Let X and X be the solid tori V cut off by E , and c i ( i = 1 , be an essentialsimple closed curve on ∂V ∩ ∂X i satisfying [ c i ] = 1 ∈ π ( V ) . Then E is the uniqueessential separating disk in V disjoint from c ∪ c .Proof. Let E ′ be an essential separating disk in V disjoint from c ∪ c . Let D i be ameridian disk of X i satisfying ∂D ⊂ ∂V . Let S be the 4-holed sphere ∂V cut off by ∂D ∪ ∂D . Denote by d + i and d − i ( i = 1 ,
2) the boundary circles of S coming from ∂D i . Then c i ∩ S consists of parallel arcs connecting d + i and d − i . It suffices to show that E ′ ∩ ( D ∪ D ) = ∅ . Assume for a contradiction that E ′ ∩ ( D ∪ D ) = ∅ . Let δ be anoutermost subdisk of E ′ cut off by E ′ ∩ ( D ∪ D ). Then α := ∂δ ∩ S is a separating arcin S . Without loss of generality, we can assume that the end points of α line in d +2 . SeeFigure 1. αβ βd +1 d +2 d − d − Figure 1.
The arc α and the arcs β . KENTA FUNAYOSHI AND YUYA KODA
Put β := ∂E ′ ∩ S . We note that β consists of essential arcs in S , and α ⊂ β . Then it iseasily seen that β ∩ d − ) + 2 ≤ β ∩ d +2 ), which is a contradiction. (cid:3) A pair (
V, k ) of a handlebody V and the union k = ∪ ni =1 k i of mutually disjoint, mutu-ally non-parallel, simple closed curves k , k , . . . , k n on ∂V is called a relative handlebody .A relative handlebody ( V, k ) is said to be boundary-irreducible if ∂V \ k is incompressiblein V . A surface S in V is said to be essential in ( V, k ) if S is an incompressible surfacedisjoint from k , and for every disk D ⊂ V such that D ∩ S = Cl( ∂D − ∂V ) is connectedand D ∩ k = ∅ , there is a disk D ′ ⊂ S with Cl( ∂D ′ − ∂S ) = D ∩ S and D ′ ∩ k = ∅ .Denote by A ( V, k ) the subgroup of the mapping class group MCG(
V, k ) generated byall twists along essential annuli in (
V, k ). Lemma 1.4 (Johannson [15], Remark 8.9) . Let ( V, k ) be a boundary-irreducible relativehandlebody. Then MCG(
V, k ) / A ( V, k ) is a finite group. Compression bodies.
Let V be a handlebody. A (possibly empty, possibly dis-connected) subgraph of a spine of V is called a subspine of V if it does not contain acontractible component. A compression body W is the exterior E (Γ) of a subspine Γ ofa handlebody V . The component ∂ + W = ∂V is called the exterior boundary of W , and ∂ − W = ∂W \ ∂ + W = ∂ Nbd(Γ) is called the interior boundary of W . We remark thatthe interior boundary is incompressible in W , see Bonahon [1].A compression body W with ∂ + W a closed orientable surface of genus two is one ofthe following (see Figure 2): (i) ∂ − W = ∅ ; (ii) ∂ − W is a torus; (iii) ∂ − W consists of twotori; (iv) ∂ − W is a closed orientable surface of genus two.(i) (ii) (iii) (iv) Figure 2.
The compression bodies with the exterior boundary a closedsurface of genus two.
Theorem 1.5 (Bonahon [1], Theorem 2.1) . For a compact, orientable, irreducible -manifold M with ∂M connected, there exists a unique compression body W in M suchthat ∂ + W = ∂M , and that the closure of M \ W is boundary irreducible. The compression body W in Theorem 1.5 is called the characteristic compression body of M . Remark.
In [1], Theorem 1 . W-decompositions.
Let M be a boundary-irreducible Haken 3-manifold. An es-sential annulus or torus S in M is said to be canonical if any other essential annulus ortorus in M can be isotoped to be disjoint from S . A maximal system { S , S , . . . , S n } ofpairwise disjoint, pairwise non-parallel canonical surfaces in M is called a W-system . Lemma 1.6 (Neumann-Swarup [22]) . The W-system of a boundary-irreducible Haken3-manifold is unique up to isotopy.
XTENDING AUTOMORPHISMS OF THE GENUS-2 SURFACE OVER THE 3-SPHERE 5
The result of cutting M off by a W-system is called a W-decomposition of M .Let M , M , . . . , M m be the result of performing the W-decomposition of M . We set ∂ M i := ∂M i ∩ ∂M and ∂ M i := ∂M i \ ∂ M i ( i = 1 , , . . . , m ). We say that ( M i , ∂ M i )(or simply M i ) is simple if any essential annulus or torus ( S, ∂S ) ⊂ ( M i , ∂ M i ) is parallelto ∂ M i . Lemma 1.7 (Neumann-Swarup [22]) . If M i is not simple, then M i is either a Seifertfibered space or an I -bundle. Essential annuli in the exterior of a genus-two handlebody in S Essential annuli in the exterior of a handlebody of genus two embedded in S playan important role in our paper. We first briefly review the classification of the essentialannuli in the exterior of a handlebody of genus two embedded in S obtained in [19].Type 1: Let Γ be a handcuff-graph embedded in S . Let S be a sphere in S thatintersects Γ in exactly one edge of Γ twice and transversely. Set V := Nbd(Γ).We call A := S \ Int( V ) a Type annulus for V ⊂ S . See Figure 3. We note thatif V ⊂ S admits an essential annulus of Type 1, then there exists an annuluscomponent B of ∂V cut off by ∂A , and A ∪ B forms an essential torus in E ( V ). A A
Figure 3.
Type 1 annuli.Type 2: Let Γ be a handcuff-graph embedded in S . Assume that one of the twoloops of Γ is a trivial knot bounding a disk D such that Int( D ) intersects Γ inan edge e once and transversely. Set V := Nbd(Γ) and A := D ∩ E ( V ). We call A a Type annulus for V ⊂ S . When e is a loop, we say that A is of Type - e is the cut-edge, we say that A is of Type -
2. See Figure 4.(i) (ii)
A A
Figure 4. (i) A Type 2-1 annulus, (i) A Type 2-2 annulus.Type 3: Let X be a solid torus embedded in S . Let A be an annulus properlyembedded in E ( X ) so that ∂A ∩ ∂X consists of parallel non-trivial simple closedcurves on ∂X . KENTA FUNAYOSHI AND YUYA KODA • Let τ be a properly embedded trivial simple arc in X with ∂τ ∩ ∂A = ∅ .Set V := X \ Int(Nbd( τ )). Then we call A a Type - annulus for V ⊂ S provided that, if ∂A bounds an essential disk in X , then any meridian diskof X has non-empty intersection with τ . We note that if V ⊂ S admits anessential annulus of Type 3-1, E ( V ) is not boundary-irreducible. • Suppose that ∂A does not bound an essential disk in X . Let τ be a properlyembedded simple arc in E ( X ) with τ ∩ A = ∅ . Set V := X ∪ Nbd( τ ). Thenwe call A a Type - annulus for V ⊂ S .Let X , X be two disjoint solid tori embedded in S . Assume that there existsan annulus A properly embedded in E ( X ⊔ X ) such that A ∩ ∂X i ( i = 1 ,
2) isa simple closed curve on ∂X i that does not bound a disk in X i . Let τ ⊂ E ( X ⊔ X ) \ A be a proper arc connecting ∂X and ∂X . Set V := X ∪ X ∪ Nbd( τ ).Then we call A a Type - annulus for V ⊂ S . An annulus A in the exterior ofa genus two handlebody V ⊂ S is called a Type annulus if it is one of Types3-1, 3-2 and 3-3 annuli. Figure 5 shows schematic pictures of Type 3 annuli.(i) (ii) (iii) X X X X τ τ τA A A A A Figure 5. (i) A Type 3-1 annulus, (ii) A Type 3-2 annulus, (iii)A Type 3-3 annulus.Type 4: Let K be a knot whose exterior E ( K ) contains an essential two-holed torus P such that • P cuts E ( K ) into two handlebodies V and W of genus two; and • ∂P consists of parallel non-integral slopes on ∂ Nbd( K ).Such a knot is called an Eudave-Mu˜noz knot . See Eudave-Mu˜noz [6] for moredetails of such knots. Set A := ∂ Nbd( K ) \ Int( ∂ Nbd( K ) ∩ ∂V ). • We call A a Type - annulus for V ⊂ S . • Let U ⊂ S be a knot or a two component link contained in W so that W \ Int(Nbd( U )) is a compression body. Let i : E ( U ) → S be a re-embedding such that E ( i ( E ( U ))) is not a solid torus or two solid tori. Thenwe call i ( A ) a Type - annulus for i ( V ) ⊂ S . We note that if V ⊂ S admits an essential annulus of Type 4-2, E ( V ) contains an essential torus.An annulus A in the exterior of a genus two handlebody V ⊂ S is called a Type annulus if it is one of Types 4-1 and 4-2. Figure 6 depicts schematic picture ofan essential annulus of Type 4. XTENDING AUTOMORPHISMS OF THE GENUS-2 SURFACE OVER THE 3-SPHERE 7
Nbd( K ) V a non-integral slope a non-integral slope A Figure 6.
A Type 4 annulus.
Theorem 2.1 ([19]) . Let V be a handlebody of genus two embedded in S . Then eachessential annulus in the exterior of V belongs to exactly one of the four Types listedabove. The following is straightforward.
Lemma 2.2.
Let V be a handlebody of genus two embedded in S so that E ( V ) isboundary-irreducible and atoroidal. Then E ( V ) does not admit Types , - or - annuli. Existence of an essential annulus A ⊂ E ( V ) sometimes obstructs the existence ofessential annuli of some other types disjoint from A . In the following part of this section,we will be concerned with this feature. Lemma 2.3.
Let V be a handlebody of genus two embedded in S such that E ( V ) isboundary-irreducible and atoroidal. If E ( V ) admits disjoint essential annuli A, A ′ ofType - , then A ′ is isotopic to A in E ( V ) .Proof. Let
A, A ′ ⊂ E ( V ) be disjoint essential annuli of Type 2-1. Set ∂A = a ⊔ a and ∂A ′ = a ′ ⊔ a ′ , where a and a ′ bound disks in V , and a and a ′ are primitive curveson ∂V . Since a is primitive, we have a = a ′ by Corollary 1.2. By slightly pushing theinterior of the annulus ˆ A = A ∪ A ′ into Int( E ( V )), ˆ A becomes an incompressible annulusin E ( V ) with ∂ ˆ A = a ∪ a ′ (see Figure 7). We will see that a is isotopic to a ′ on ∂V . Ifˆ A is parallel to ∂E ( V ), then clearly a is isotopic to a ′ . Suppose that ˆ A is not parallelto ∂E ( V ). By Lemma 1.4 of [19], ˆ A is essential in E ( V ). Recall that each of a and a ′ is primitive (and so non-separating) in ∂V . Since E ( V ) does not admit none of Types 1,3-1 and 4-2 annuli by Lemma 2.2, such an annulus ˆ A is one of Types 3-2, Type 3-3 and4-1 annuli. If ˆ A one of Types 3-2 and 4-1 annuli, the boundary circles of ˆ A is parallelon ∂V by the definitions. Suppose that ˆ A is a Type 3-3 annulus. By the definition of aType 3-3 annulus, there exists a separating essential disk E in VV V V Va a a ′ a a ′ AA ′ ˆ A Figure 7.
The annulus ˆ A . KENTA FUNAYOSHI AND YUYA KODA that cuts V off into two solid tori X and X with a ⊂ ∂X and a ′ ⊂ ∂X . Note that a ( a ′ , respectively) does not bound a disk in X ( X , respectively). By Lemma 1.1, themeridian disk D of X is the unique non-separating disk in V disjoint from a . Thus wehave ∂D = a . This implies that a ∩ a ′ = ∅ , a contradiction. Therefore, in any case wehave a = a ′ and a = a ′ . By moving A ′ by isotopy so that a = a ′ and slightly pushingit into Int( E ( V )), ˆ A becomes a torus in E ( V ). Since E ( V ) is atoroidal and irreducible, ˆ A bounds a solid torus X in E ( V ). Since a bounds a disk in V , a intersects the meridianof X once and transversely. In consequence A ′ can be isotoped to A through X . (cid:3) Lemma 2.4.
Let V be a handlebody of genus two embedded in S . If there exists anessential annulus A of Type - in E ( V ) , then E ( V ) does not admit an essential annulusof Type - disjoint from A .Proof. Let A ⊂ E ( V ) be an essential annulus A of Type 2-1. Suppose that there existsan essential Type 3-2 annulus A ′ ⊂ E ( V ). We will show that A ∩ A ′ = ∅ . Set ∂A = a ⊔ a and ∂A ′ = a ′ ⊔ a ′ , where a bounds a disk in V and a is a primitive curve on ∂V . Wenote that by the definition of a Type 3-2 annulus, a ′ and a ′ are parallel non-separatingcurves on ∂V . Suppose for a contradiction that A ∩ A ′ = ∅ . Let D be a disk in V with ∂D = a . By cutting V along D , we get an unknotted solid torus X in S . Then a , andso a ′ i ( i = 1 , ∂X . We note that A ′ is a boundary-parallelannulus in E ( X ). Now V is obtained by attaching a 1-handle Nbd( D ) to X , however,anyhow we attach such a 1-handle to X , A ′ becomes boundary-compressible in E ( V ).This contradicts the assumption that A ′ is essential in E ( V ). (cid:3) Lemma 2.5.
Let V be a handlebody of genus two embedded in S , and let A ⊂ E ( V ) be an essential annulus of Type - . Let X be a solid torus for A in the definition of aType - annulus. Then exactly one of the following (i) and (ii) holds: (i) X is a regular neighborhood of a torus knot or a cable knot, and A is the cablingannulus for X . (ii) A is an incompressible annulus in E ( X ) parallel to an annulus A ′ on ∂X . Thereexists and arc τ contained in the parallelism region between A and A ′ , with end-points on A ′ , such that V = X ∪ Nbd( τ ) .Proof. Set ∂A = a ⊔ a . We first show that A is incompressible in E ( X ). Supposethat there exists a compressing disk D for A . We note that ∂D is the core of A . If E ( X ) is not the solid torus, a i ( i = 1 ,
2) is a non-essential simple closed curve on ∂X .In this case each component of ∂A bounds a disk in X , so does in V . This contradictsthe assumption that A is a Type 3-2 annulus. Thus E ( X ) is the solid torus. If at leastone of a and a are non-essential on ∂X , we get a contradiction by the same reason asabove. Thus both a and a are the preferred longitudes of X . Then A is parallel toan annulus A ′ on ∂X by Lemma 3.1 of Kobayashi [17]. Let Y be the parallelism regionbetween A and A ′ . There exist a boundary-compressing disk D ′ for A in E ( X ). Wenote that D lies in E ( X ) − Y while D ′ lies in Y . Thus A remains incompressible orboundary-compressible even in E ( V ), which is a contradiction. The annulus A is thusincompressible in E ( X ).If A is essential in E ( X ), the the property (i) follows from Lemma 15.26 of Burde-Zieschang [2]. Suppose A is not essential in E ( X ). Then again by Kobayashi [17] in the XTENDING AUTOMORPHISMS OF THE GENUS-2 SURFACE OVER THE 3-SPHERE 9 case where E ( X ) is the solid torus, and by Lemma 15.18 of Burde-Zieschang [2] in theother case, A is parallel to ∂X . Hence the the property (ii) follows. (cid:3) Lemma 2.5 indicates that the boundary of a Type 3-2 annulus in the exterior of ahandlebody V in S consists of parallel primitive curves on ∂V . Lemma 2.6.
Let V be a handlebody of genus two embedded in S , and let A ⊂ E ( V ) bean essential annulus of Type - . Then, for A , the solid torus X in the definition of aType - annulus is uniquely determined.Proof. Set ∂A = a ⊔ a . Recall that a and a are parallel primitive curves on ∂V . ByCorollary 1.2, there exists a unique non-separating disk disjoint from a ∪ a . This impliesthat the solid torus X in the definition of a Type 3-2 annulus is uniquely determined.Indeed, X is obtained by cutting V off by that the non-separating disk. (cid:3) Let V be a handlebody of genus two embedded in S , and let A ⊂ E ( V ) be an essentialannulus of Type 4-1. The two circles ∂A are parallel on ∂V , thus cobound an annulus A ′ on ∂V . We note that the union A ∪ A ′ is the boundary of a regular neighborhoodNbd( K ) of an Eudave-Mu˜noz knot K . The annulus A cuts E ( V ) off into a solid torusNbd( K ) and a handlebody W of genus two. The core k A ′ of the annulus A ′ is an essentialsimple closed curve on V . Set P := Cl( ∂V \ A ′ ). P is a two-holed torus. (Recall Figure6). Lemma 2.7.
The relative handlebody ( V, k A ′ ) admits no non-separating essential annuli.Proof. We suppose for a contradiction that there exists a non-separating essential annulus B in ( V, k A ′ ). Set ∂A = a ⊔ a and ∂B = b ⊔ b . Let Y be the result of Dehn surgeryon K along the slope k A ′ . That is, Y is obtained by attaching a solid torus X to E ( K )along their boundaries so that k A ′ bounds a disk in X . Let D i ( i = 1 ,
2) be the disks in X bounded by a i . Then T := P ∪ ( D ∪ D ) is an essential torus in Y . By Proposition5.4 of Eudave-Mu˜noz [7] each component Y i ( i = 1 ,
2) of Y cut off by T is a Seifertfibered space with the base space a disk and two exceptional fibers. Without loss ofgenerality we can assume that V ⊂ Y and W ⊂ Y . Since B is non-separating in V ,so is in Y . This contradicts Theorem VI.34 of Jaco [11], which asserts that there is nonon-separating essential annuli in Y . (cid:3) Automorphisms extendable over S It is well known that every closed orientable 3-manifold M can be decomposed intotwo handlebodies V and W of the same genus g for some g ≥
0. That is, V ∪ W = M and V ∩ W = ∂V = ∂W = Σ g , a closed orientable surface of genus g . Such a decompositionis called a Heegaard splitting for the manifold M . The surface Σ g is called the Heegaardsurface of the splitting, and g is called the genus of the splitting. By Waldhausen [24]there exists a unique Heegaard splitting of a given genus for S up to isotopy. We saythat an embedding ι : Σ g ֒ → S is standard if the image ι (Σ g ) is a Heegaard surface in S . Definition.
Let f be an automorphism of a closed orientable surface Σ g . Then f is saidto be extendable over S if there exist an embedding ι : Σ g ֒ → S and an automorphism ˆ f : ( S , ι (Σ g )) → ( S , ι (Σ g )) satisfying ˆ f ◦ ι = ι ◦ f . In particular, if we can choose theabove ι to be standard, we say that f is standardly extendable over S .It is easy to see that the Dehn twist T c along a separating simple closed curve c on Σ g is standardly extendable over S . Indeed, there exists a standard embedding ι : Σ g ֒ → S such that ι ( c ) bounds disks on both sides. On the other hand, the Dehn twist T c ′ along anon-separating simple closed curve c ′ on Σ g is not extendable over S . This is because if T c ′ is extendable over S , there should exist a non-separating 2-sphere in S by Theorem3.1 below, which is a contradiction. Theorem 3.1 (McCullough [21]) . Let M be a compact orientable -manifold, and c , c , . . . , c n be mutually disjoint simple closed curves on ∂M . If ( powers of ) Dehn twistsalong c , c , . . . , c n extend to an automorphism of M , then for each i ∈ { , , . . . , n } ,either c i bounds a disk in M , or for some j = i , c i and c j cobound an incompressibleannulus in M . In [9], Guo-Wang-Wang-Zhang determined all periodic automorphisms of the closedsurface Σ of genus two that are extendable over S . They showed that among the twenty-one (conjugacy classes of) periodic maps of Σ , exactly thirteen maps are extendableover S . They described those maps explicitly, and as a direct consequence, we have thefollowing: Theorem 3.2 (Guo-Wang-Wang-Zhang [9]) . Let f be a periodic automorphism of aclosed surface of genus two. If f is extendable over S , then f is standardly extendableover S . The following is our main theorem:
Theorem 3.3.
Let f be an automorphism of a closed surface of genus two. If f isextendable over S , then f is standardly extendable over S . We remark that in [27], Wang-Wang-Zhang-Zimmermann found some finite, non-cyclic subgroups of Homeo(Σ g ) that extend to subgroups of Homeo( S , Σ g ) with respectto some Σ g ֒ → S , but do not extend to those of Homeo( S , Σ g ) with respect to anystandard embedding Σ g ֒ → S .The extendability (standardly extendability, respectively) over S is preserved underconjugation. Indeed, suppose that an automorphism f of Σ g extends to an automor-phism ˆ f of the pair ( S , ι (Σ g )) with respect to an embedding (a standard embedding,respectively) ι : Σ g ֒ → S . Then for any automorphism h of Σ g , f ′ := h − ◦ f ◦ h extendsto ˆ f with respect to the embedding (a standard embedding, respectively) ι ◦ h .The extendability and standardly extendability, respectively over S are preservedunder isotopy as well. In the remaining of the paper, we will not distinguish homeomor-phisms from their isotopy classes in their notation unless otherwise mentioned. Here werecall the cyclic Nielsen realization theorem: Theorem 3.4 (Nielsen [23]) . Every element f ∈ MCG(Σ g ) ( g ≥ of finite order k hasa representative φ ∈ Homeo(Σ g ) of order k . By this theorem, we have the mapping class version of Theorem 3.2 as follows:
XTENDING AUTOMORPHISMS OF THE GENUS-2 SURFACE OVER THE 3-SPHERE 11
Lemma 3.5.
Let f ∈ MCG(Σ ) be an element of finite order. If f is extendable over S , then f is standardly extendable over S .Remark. The same consequence of Theorem 3.3 holds for a closed surface Σ g with g ≤ is trivial. In fact, every automorphism of Σ is standardlyextendable over S by the Sch¨onflies Theorem. Suppose that an automorphism f ofΣ extends to an automorphism ˆ f of the pair ( S , ι (Σ )) with respect to an embedding ι : Σ ֒ → S . Note that at least one of the two components of S cut off by ι (Σ ) isa solid torus V . Then E ( V ) is the exterior of a knot. If E ( V ) is also a solid torus,there is nothing to prove. Suppose that E ( V ) is not a solid torus. We note that inthis case we have ˆ f ( V ) = V . Let µ and λ be the meridian and preferred longitude of V ⊂ S . By the uniqueness of µ and λ , the automorphism ˆ f preserves µ and λ . Thus,it is straightforward that f is standardly extendable over S .4. Proof of Theorem 3.3.
Let f be an automorphism of a closed surface Σ of genus two. Suppose that f extendsto an automorphism ˆ f of the pair ( S , ι (Σ)) with respect to an embedding ι : Σ ֒ → S .The surface ι (Σ) separates S into two compact 3-manifolds M and M , i.e. S = M ∪ ι (Σ) M . Suppose for a contradiction that f is not standardly extendable over S . Claim 1.
After changing ι ( and so ˆ f ) if necessary, we can assume that M is a handle-body and M is boundary-irreducible.Proof of Claim . By Theorem 1.5 there exist the characteristic compression bodies W and W of M and M , respectively. Then each of W and W is homeomorphic to oneof the four compression bodies shown in Figure 2.Since ι (Σ) is compressible in S , both ∂ − W and ∂ − W cannot be a closed surface ofgenus two. Both ∂ − W and ∂ − W cannot be empty as well, otherwise f is standardlyextendable over S , which contradict our assumption. If ∂ − W = ∅ ( ∂ − W = ∅ , respec-tively) and ∂ − W ( ∂ − W , respectively) is a closed surface of genus two, there is nothingto prove. Thus we only need to consider the case where at least one of ∂ − W and ∂ − W is a torus or the disjoint union of two tori.Suppose for example that each of ∂ − W and ∂ − W is a torus. Set T i := ∂ − W i = ∂W i \ ι (Σ) ( i = 1 , S = ( W ∪ M ) ∪ T Cl( S \ ( W ∪ M )). By theuniqueness (Theorem 1.5) of the characteristic compression bodies W and W of M and M , respectively, we may assume that ˆ f preserves W ∪ W setwise. Further, by thedefinition of a characteristic compression body, E ( W ∪ M ) = Cl( M \ W ) is boundary-irreducible. In particular, E ( W ∪ M ) is not a solid torus. Since any torus in S boundsa solid torus, W ∪ M is a solid torus. By the same reason, W ∪ M is a solid torus. Let λ and λ be the preferred longitudes of W ∪ M and W ∪ M , respectively. We attachtwo solid tori X and X to W ∪ W by homeomorphisms φ i : ∂X i → T i ( i = 1 ,
2) thatsend the meridian of ∂X i to λ i . Then the resulting manifold X ∪ φ ( W ∪ W ) ∪ φ X isagain the 3-sphere S and ι (Σ) is now a Heegaard surface. (This operation correspondsto a re-embedding of Σ into S ). See Figure 8. X X φ φ λ λ W W T T ι (Σ) Figure 8.
The manifold X ∪ φ ( W ∪ W ) ∪ φ X ∼ = S .By the uniqueness of the preferred longitude, ˆ f | W ∪ W preserves λ ⊔ λ setwise. Thereforeˆ f | W ∪ W extends to an automorphism of X ∪ φ ( W ∪ W ) ∪ φ X . This implies that f is standardly extendable over S , which is a contradiction.In a similar way, we can show that if ∂ − W (say) is empty and ∂ − W is a torus; ∂ − W (say) is empty and ∂ − W consists of two tori; ∂ − W (say) is a torus and ∂ − W consists of two tori; or each of ∂ − W and ∂ − W consists of two tori, we can show that f is standardly extendable over S , a contradiction.When ∂ − W (say) is a torus and ∂ − W is a closed surface of gens two; or ∂ − W (say)consists of two tori and ∂ − W is a closed surface of gens two, a similar argument asabove concludes the assertion. (cid:3) By virtue of Claim 1, in the following we assume that S = V ∪ ι (Σ) M , where V is ahandlebody of genus two, and M is boundary-irreducible. Claim 2.
After changing ι ( and so ˆ f ) if necessary, we can assume that M is atoroidalas well.Proof of Claim . According to the torus decomposition theorem by Jaco-Shalen [12] andJohannson [13], there exists a unique minimal family T = { T , T , . . . , T m } of mutuallydisjoint, mutually non-parallel, essential tori such that each component of the manifoldsobtained by cutting M along ∪ mi =1 T i is either a Seifert fibered space or a simple manifold.By the uniqueness of T , we may assume that ˆ f preserves ∪ mi =1 T i setwise. By the sameargument as in Claim 1, we can eliminate T = { T , T , . . . , T m } by re-embedding Σ into S . Note that if some tori are nested, we can eliminate several tori at once by applyingthe operation in Claim 1 to the outermost one. (cid:3) Due to Claim 2, in the remaining of the proof we assume that S = V ∪ ι (Σ) M , where V is a handlebody of genus two, and M is boundary-irreducible and atoroidal.Suppose that M does not contain essential annuli. Then by Johannson [14], MCG( M )is a finite group. Note that this fact can also be checked as follows. By Thurston’shyperbolization theorem [16] and equivariant torus theorems [10], M admits a hyperbolicstructure with totally geodesic boundary. Thus, due to the well-known fact that theisometry group of a compact hyperbolic manifold is finite, see Kojima [20], MCG( M ) XTENDING AUTOMORPHISMS OF THE GENUS-2 SURFACE OVER THE 3-SPHERE 13 is finite. Consequently, the order of f ∈ MCG(Σ) is finite. Now by Lemma 3.5, f isstandardly extandable over S , a contradiction.Suppose that M contains an essential annulus. By Lemma 1.6 there exists a uniqueW-system W of M . This set W is not empty. To show this, suppose for a contradictionthat W = ∅ . Then the result of performing the W-decomposition of M is M itself. Since M contains an essential annulus, clearly M is not simple. Hence by Lemma 1.7, M iseither a Seifert fibered space or an I -bundle. Since ∂M is a closed surface of genus two, M is not a Seifert fibered space. Thus M is an I -bundle over a compact surface S . If ∂S = ∅ , then M is a handlebody, which is a contradiction. If S is an orientable closedsurface, then ∂M is not connected, a contradiction as well. If S is a non-orientable closedsurface, then H ( M ) contains a non-trivial torsion. This contradicts H ( M ) ∼ = Z ⊕ Z .Since M is atoroidal, W consists of essential annuli. We will use the classification ofthe essential annuli in W introduced in Section 2. Since M is boundary-irreducible andatoroidal, M admits none of Types 1, 3-1 and 4-2 essential annuli. In the following, wewill prove the non-existence of essential annuli of the other Types. Claim 3.
The W-system W does not contain Type - annuli.Proof of Claim . Suppose that W contains a Type 2-2 annulus A . Let a be the com-ponent of ∂A that is the boundary of an essential disk E in V , and let a be the othercomponent. By cutting V along E , we have two solid tori X and X . Without loss ofgenerality we can assume that a ⊂ ∂X . By the definition of a Type 2-2 annulus, a is the preferred longitude of X . Let µ i ( i = 1 ,
2) be the meridian of X i , and λ be thepreferred longitude of X (see Figure 9 (i)). By the uniqueness of W (Lemma 1.6), itholds ˆ f n ( A ) = A for some natural number n . Using 2 n instead of n if necessary, we canassume that ˆ f n | V is orientation-preserving. Since a is separating on ∂V whereas a isnon-separating, ˆ f n does not exchange a and a . Thus ˆ f n preserves each of µ , µ and λ . It follows from Cho [3] that ˆ f n | V is a composition of α , β , β defined as follows.The map α comes from the hyperelliptic involution of ∂V . The maps β and β are thehalf-twists of handles shown in Figure 9 (ii). We note that ( β ◦ β ) | ∂V = T a , where T a is the Dehn twist along a . Then we have β | ∂V = β | ∂V = T a and β ◦ β − = α .Since α , β and β are mutually commutative, we can write ˆ f n | V in the form:ˆ f n | V = α ǫ ◦ β n ◦ β n , ( ǫ ∈ { , } , n i ∈ Z ) . Since ( ˆ f n | V ) = β n ◦ β n , it holds ( ˆ f n | ∂V ) = T a n + n . Suppose that n + n = 0.Sine ( ˆ f n | ∂V ) is extendable over S , a bounds a disk in M by Theorem 3.1. Thiscontradicts the boundary-irreducibility of M . Hence n + n = 0, and thus we haveˆ f n | V = α ◦ β n ◦ β − n . Recalling that β ◦ β − = α , we see that ˆ f n | V is either the identityor α . By Lemma 3.5, f is standardly extendable over S , which is a contradiction. (cid:3) (i) (ii) a a a E Eβ β X X X X µ µ λ µ µ λ Figure 9. (i) The curves a , a , µ , µ and λ on ∂V ; (ii) The maps β and β . Claim 4.
The W-system W does not contain Type - annuli.Proof of Claim . Suppose that W contains a Type 3-3 annulus A . Then there existtwo disjoint solid tori X and X in S , and an arc τ connecting ∂X and ∂X with V = X ∪ X ∪ Nbd( τ ) such that A is properly embedded in E ( X ⊔ X ), and a i := A ∩ X i ( i = 1 ,
2) is a simple closed curve on ∂X i that does not bound a disk in X i . Let E bethe cocore of the 1-handle Nbd( τ ), see Figure 10 (i).(i) (ii) X X X X µ µ λ λ τ γE EA Figure 10. (i) The disk E and the preferred longitudes λ i ( i = 1 ,
2) of X i ; (ii) The map γ .By Lemma 1.3, E is the unique separating disk in V disjoint from a ∪ a . Let µ i and λ i ( i = 1 ,
2) be the meridian and the preferred longitude of X i . By the uniqueness of W (Lemma 1.6), there exists a natural number n with ˆ f n ( A ) = A . Using 2 n instead of n ifnecessary, we can assume that ˆ f n | V is orientation-preserving. Since ˆ f n preserves a ∪ a ,it preserves E as well by Lemma 1.3. Thus µ ∪ µ and λ ∪ λ are also preserved byˆ f n | V . It follows again from Cho [3] that ˆ f n | V is a composition of α , β , β defined inCase 1, and one more element γ shown in Figure 10 (ii). The hyperelliptic involution α iscommutative with the other maps. The maps β , β and γ satisfy β | ∂V = β | ∂V = T ∂E , XTENDING AUTOMORPHISMS OF THE GENUS-2 SURFACE OVER THE 3-SPHERE 15 β ◦ β − = α and γ ◦ β = β ◦ γ . Hence, we can write ˆ f n | V in the formˆ f n | V = α ǫ ◦ β n ◦ β n ◦ γ ǫ ( ǫ i ∈ { , } , n i ∈ Z ) . Then it holds ( ˆ f n | V ) = (cid:26) β n ◦ β n (if ǫ = 0); β n + n ) ◦ β n + n ) (if ǫ = 1) , hence ( ˆ f n | ∂V ) = T ∂E n + n ) . The rest of the proof runs as in Claim 3. (cid:3) Claim 5.
The W-system W does not contain Type - annuli.Proof of Claim . Suppose that W contains a Type 4-1 annulus A . The two circles ∂A are parallel on ∂V , thus cobound an annulus A ′ on ∂V . We note that the union A ∪ A ′ is the boundary of a regular neighborhood Nbd( K ) of an Eudave-Mu˜noz knot K . Theannulus A cuts M into the solid torus Nbd( K ) and a handlebody W of genus two. Thecore k A ′ of the annulus A ′ is an essential simple closed curve on ∂V . Set P := Cl( ∂V \ A ′ ). P is then a two-holed torus.Since P is incompressible in E ( K ), ( V, k A ′ ) is boundary-irreducible. Thus, by Lemma1.4, MCG( V, k A ′ ) / A ( V, k A ′ ) is a finite group. By Lemma 2.7, the relative handlebody( V, k A ′ ) admits no non-separating essential annuli. Since the genus of V is two, theboundary circles of any incompressible, separating annulus B in V is parallel on ∂V .Indeed, each boundary circle of B is essential and separating on ∂V , and two disjoint,separating, essential, simple closed curves on ∂V are parallel on ∂V . Hence the restriction T B | ∂V of the twist T B along B is trivial in MCG( ∂V ). It then follows from the well-known fact that the natural homomorphism MCG( V ) → MCG( ∂V ) taking h ∈ MCG( V )to h | ∂V ∈ MCG( ∂V ) is injective, that MCG( V, k A ′ ) it self is a finite group. By theuniqueness of W (Lemma 1.6), there exists a natural number n with ˆ f n ( A ) = A . Inparticular, ˆ f n | V ( k A ′ ) ∈ MCG(
V, k A ′ ). Therefore, the order of ˆ f n | V (and so the order of f ) is finite, which contradicts Lemma 3.5. (cid:3) Claim 6.
The W-system W does not contain Type - annuli.Proof of Claim . Suppose that W contains a Type 2-1 annulus A . By Lemmas 2.3, 2.4,and Claims 3–5, we have W = { A } .In the following, we show that the order of f is finite, which contradicts Lemma 3.5.Let M be the result of cutting M off by the W-system W = { A } . Then M is alsoatoroidal. Indeed, suppose not, that is, there exists an essential torus T in M . Since M is atoroidal, T admits a compressing disk D in M with D ∩ A = ∅ . We isotope D so that D ∩ A ) is minimal. Note that D ∩ A is non-empty and consists of simple closed curveson D . Let c be a component of D ∩ A that is innermost in D . Then the disk D ⊂ D bounded by c is a compressing disk for A , which is a contradiction. Therefore M isatoroidal. Now by considering the characteristic compression body of M , we see that M is either boundary-irreducible or a handlebody of genus two (otherwise M containsan essential torus, a contradiction). Set ∂A = a ⊔ a , where a bounds a disk in V , and a is a primitive curve on ∂V . Case
1: The case where M is boundary-irreducible. By Lemma 1.7, ( M , ∂ M ) is simple. Thus, by Proposition 27.1 of Johannson [13],MCG( M , ∂ M ) is a finite group, where we recall that ∂ M := ∂M ∩ ∂M (= ∂V \ Nbd( a ∪ a )). Therefore, we have ˆ f n | ∂V \ (Nbd( a ∪ a )) = id for a natural number n . Itfollows that there exist natural numbers m and m satisfying f n = ˆ f n | ∂V = T a m ◦ T a m , where T a i ( i = 1 ,
2) is the Dehn twist along a i . By Theorem 3.1, we have m = m = 0.Indeed, if m = 0 and m = 0, or if m = 0 and m = 0, M is a boundary-reducible,which is a contradiction. If m = 0 and m = 0, either each of a and a bounds a diskin V , or a and a cobound an incompressible annulus in V . Since a is primitive in ∂V ,the former is impossible. The latter is also impossible because as an element of H ( V ),[ a ] = 0 whereas [ a ] = 0. Now we have f n = id, that is, the order of f is finite. Case
2: The case where M is a handlebody of genus two.The union V := V ∪ Nbd( A ) is still a handlebody of genus two with M = E ( V ).That is, S = V ∪ M is a Heegaard splitting. Define ˆ A ⊂ Nbd( A ) = A × I byˆ A = (( S × { / } ) × I ) (see Figure 11).(i) (ii)Nbd( A ) V VA ˆ A Figure 11. (i) The annulus A ; (ii) The annulus ˆ A .Set ∂ ˆ A = ˆ a ⊔ ˆ a . The simple closed curve ˆ a i ( i = 1 ,
2) bounds a non-separating disk D i in V . We note that ( V \ Nbd( ˆ A )) ∼ = ( V \ Nbd( A )) ∼ = V .Suppose that both D and D are primitive disks in V . Since D and D are disjoint,non-parallel, primitive disks in V , there exist disjoint, non-parallel, primitive disks E and E in V such that D i ∩ E j ) = δ ij (see Lemma 2.2 in Cho [3]), where δ ij is theKronecker delta. Thus V can be isotoped as shown in Figure 12 (see also Theorem 4.4 of[18]). It follows that M = E ( V ) is a handlebody of genus two, which is a contradiction.Suppose that at least one of D and D , say D , is not a primitive disk in V . Sinceˆ f | V preserves D , ˆ f is an automorphism of ( S , V , D ). By Proposition 10.1 of Cho-McCullough [5], we have ˆ f | V = id. This implies that ˆ f | ∂V \ (Nbd( a ∪ a )) is the identity.Now, the finiteness of the order of f follows from the same argument as in Case 1. (cid:3) XTENDING AUTOMORPHISMS OF THE GENUS-2 SURFACE OVER THE 3-SPHERE 17 ˆ A ˆ E ˆ E V V V cutting V along ˆ A isotopy Figure 12.
The isotopy of V .The following is our final claim. Claim 7.
The W-system W does not contain Type - annuli.Proof of Claim . Suppose first that W contains a Type 3-2 annulus A satisfying theproperty (ii) in Lemma 2 .
5. To get a contradiction in this case, we will show that theorder of f is finite. Set ∂A = a ⊔ a . By Lemma 2.6, for A , the solid torus X inthe definition of a Type 3-2 annulus is uniquely determined. By the assumption, A isparallel to an annulus A ′ on ∂X . Let Y be the parallelism region between A and A ′ ,which is the solid torus. Note that the simple closed curve a (and so a ) is primitive for Y . Let τ be an arc contained in Y , with endpoints on A ′ , such that V = X ∪ Nbd( τ ).The annulus A separates M into two components M and M , where M ⊂ Y . By thesame argument as in Claim 6, we can check that both M and M are atoroidal. Let E be the cocore of the 1-handle Nbd( τ ). By the uniqueness of W (Lemma 1.6), it holdsˆ f n ( A ) = A , f n ( a ) = a and ˆ f n | A is orientation preserving for some natural number n . By Lemma 1.1, E is the unique non-separating disk in V disjoint from a . Thus wehave f n ( ∂E ) = ∂E . Subclaim.
The image of the natural map
MCG( M , a , ∂E ) → MCG( ∂M , a , ∂E ) thattakes ϕ ∈ MCG( M , a , ∂E ) to ϕ | ∂M is a finite group.Proof of Subclaim. Suppose that there exists an incompressible annulus B in M whoseboundary is disjoint from a ∪ ∂E . Remark that B is separating in Y , thus B is separatingin M as well. Set ∂B = b ⊔ b . Since B is disjoint from ∂E we may assume that b ⊔ b lies in ∂Y . We will show that b and b are parallel on ∂M .Assume that b and b are not parallel on ∂M for a contradiction. Since ∂B is disjointfrom a ⊂ ∂Y , b and b are both essential or both inessential on ∂Y . The attachingregion of the 1-handle Nbd( τ ) consists of two disks D + and D − on ∂X ∪ ∂Y . Both D + and D − are parallel to E . If both b and b are essential on ∂Y , they are parallel on ∂Y . Thus in this case there exists an annulus F on ∂Y with ∂F = b ⊔ b . Note thatthe closure F ′ of ∂Y \ F is also an annulus, and F and F ′ are parallel in Y . Since b and b are not parallel on ∂M by the assumption, we have F ∩ ( D + ∪ D − ) = D + afterchanging the names of F and F ′ , and D + and D − if necessary. See Figure 13 (i). (i) (ii) Y YD + D + D − D − A A∂B ∂B
Figure 13.
The boundary loops of B on ∂Y .This implies that B is non-separating in M , a contradiction. Thus both b and b areinessential on ∂Y . Each simple closed curve b i ( i = 1 ,
2) bounds a disk D i on ∂Y . Since B is incompressible in M , and b and b are not parallel on ∂M , we have D + ⊂ D ⊂ D , D − D and D − ⊂ D after changing the names of b and b , and D + and D − ifnecessary. See Figure 13 (ii). This implies that B is non-separating in M , again acontradiction. In consequence, b and b are parallel on ∂M .Now by exactly the same reason as Lemma 1.4 (see Johannson [13]), the order of thegroup MCG( M , a , ∂E ) / A ( M , a , ∂E ) is finite, where A ( M , a , ∂E ) is the subgroup ofthe mapping class group MCG( M , a , ∂E ) generated by all twists along incompressibleannuli in M disjoint from a ∪ ∂E . Hence the restriction T B | ∂M of the twist T B alongany incompressible annulus B in M disjoint from a ∪ ∂E is trivial in MCG( ∂M ). Thisimplies the assertion. (cid:3) By the above Subclaim, there exists a natural number n such that ˆ f n n | ∂M is theidentity as an element of MCG( ∂M , a , ∂E ). Since MCG + ( A ) is the trivial group (whileMCG + ( A rel ∂A ) = Z ), we may assume that ˆ f n n | ∂M ∩ ∂V is trivial as an element ofMCG( ∂M ∩ ∂V ). Therefore, we have f n n = T mc , where c is the core of the annulusCl( ∂X \ A ), T c is the Dehn twist along c , and m is an integer. If m = 0, the simple closedcurve c bounds a disk D in M by Lemma 3.1, a contradiction. Thus we have m = 0,which implies that the order of f is finite, which is a contradiction.Now the only remaining possibility is that W consists of only Type 3-2 annuli satisfyingthe property (i) in Lemma 2 .
5. Let A ∈ W . By Lemma 2.6, for A , the solid torus X in thedefinition of a Type 3-2 annulus is uniquely determined. Further, X is a neighborhoodof a torus knot or a cable knot, and A is the cabling annulus for X . The number b ofisotopy classes in ∂V of the boundary circles of the annuli in W is at most three.Suppose first that b = 2. Then there exists a Type 3-2 annulus A ′ in W whoseboundary circles are not parallel to those of A . By the uniqueness of W (Lemma 1.6), itholds ˆ f | V ( ∂A ∪ ∂A ′ ) = ∂A ∪ ∂A ′ . Set ∂A = a ⊔ a and ∂A ′ = a ′ ⊔ a ′ . Since a and a ′ arenon-separating and not parallel on ∂V , attaching 2-handles to V along a and a ′ , andthen capping off the resulting boundary by a 3-handle, we obtain a closed 3-manifold N . XTENDING AUTOMORPHISMS OF THE GENUS-2 SURFACE OVER THE 3-SPHERE 19
Suppose first that N is homeomorphic to S . By applying Alexander’s tricks step bystep, ˆ f | V extends to an automorphism of N . This implies that f is standardly extendableover S , which is a contradiction. Thus N is not homeomorphic to S (so N is a lensspace or R P ); we do not know if this is the case. In this case, since both a and a ′ areprimitive with respect to V and N = S , the relative handleobody ( V, k ) is boundary-irreducible, where k = a ∪ a ′ . By Lemma 1.4, MCG( V, k ) / A ( V, k ) is a finite group.We can further show that (
V, k ) admits no non-separating essential annuli. Indeed, ifthere exists a non-separating annulus in (
V, k ), it remains to be non-separating in the3-manifold obtained by attaching a 2-handle to V along a , which is a solid torus. Thisis a contradiction. Now, the same argument as in Claim 5 shows that the order of f isfinite, which contradicts Lemma 3.5.Similar arguments apply to the case of b = 3.In what follows, we suppose that b = 1. Then W consists of one or two annuli, and inthe latter case, the two annuli are parallel in E ( X ) and the simple arc τ in the definitionof a Type 3-2 annulus is contained in the parallelism region. See Figure 14. X XM M τA A A X XM M M M τA A AA ′ A ′ A ′ Figure 14.
The W-decompositions for E ( V ).First, let W = { A, A ′ } . Let M , M , M be the result of cutting M off by theW-system W = { A, A ′ } , where M ∩ Nbd( τ ) = ∅ for the simple arc τ in E ( X ) with V = X ∪ Nbd( τ ) (see the right-hand side in Figure 14). Both of M and M are solidtori, otherwise E ( V ) is toroidal and it contradicts our assumption. We note that ∂M is a closed surface of genus two. By the same argument as in Claim 6, M is atoroidal.Thus by considering the characteristic compression body of M , we see that M is eitherboundary-irreducible or a handlebody of genus two.Suppose that M is boundary-irreducible. By Lemma 1.7, ( M , ∂ M ) is simple. Dueto Proposition 27.1 of Johannson [13], MCG( M , ∂ M ) is a finite group. Thus, thereexists a natural number n with ˆ f n | Cl( ∂V \ ( ∂M ∪ ∂M )) = id Cl( ∂V \ ( ∂M ∪ ∂M )) . Let c i ( i =1 ,
2) be the core of the annulus ∂V ∩ ∂M i (= ∂X ∩ ∂M i ). Then for some m , m we canwrite f n in the form f n = ˆ f n | ∂V = T c m ◦ T c m . If m i = 0, this cannot be extended tothe solid torus M i . Hence m = m = 0. Therefore the order of f is finite, which is acontradiction.Suppose that M is a handlebody of genus two. Let Y be the parallelism region of A and A ′ in E ( X ). Since Y is a solid torus, the arc τ is trivial in Y by Gordon [8]. Let l bethe boundary circle of the cocore of the 1-handle Nbd( τ ). Note that l is primitive withrespect to the handlebody M . Set ∂A = a ⊔ a , ∂A ′ = a ′ ⊔ a ′ and k = a ∪ a ′ ∪ l . Since E ( V ) is boundary-irreducible, the relative handlebody ( M , k ) is boundary-irreducible.If there exists a non-separating annulus in ( V, k ), it remains to be non-separating in the3-manifold obtained by attaching a 2-handle to V along l , which is a solid torus. Thisis a contradiction. Now, the same argument as in Claim 5 shows that the order of f isfinite, which contradicts Lemma 3.5.Finally, let W = { A } . Let M , M be the result of cutting M off by the W-system W = { A } , where M ∩ Nbd( τ ) = ∅ for the simple arc τ in E ( X ) with V = X ∪ Nbd( τ )(see the left-hand side in Figure 14). We note that ∂M is a closed surface of genustwo. By the same argument as in Claim 6, M is atoroidal. Thus by considering thecharacteristic compression body of M , we see that M is either boundary-irreducible ora handlebody of genus two.Suppose that M is boundary-irreducible. By Lemma 1.7, ( M , ∂ M ) is simple. Dueto Proposition 27.1 of Johannson [13], MCG( M , ∂ M ) is a finite group. Thus, thereexists a natural number n with ˆ f n | Cl( ∂V \ ∂M ) = id Cl( ∂V \ ∂M ) . Let c be the core of theannulus ∂V ∩ ∂M (= ∂X ∩ ∂M ). Then for some m we can write f n in the form f n = ˆ f n | ∂V = T cm . If m = 0, then c bounds a disk in M by Theorem 3.1. Thiscontradicts the boundary-irreducibility of M . Hence m = 0. Therefore the order of f isfinite, which is a contradiction.Suppose that M is a handlebody of genus two. Since A is the cabling annulus for X ,and the annulus A separates E ( X ) into M ∪ Nbd( τ ) and M , at least one of M ∪ Nbd( τ )and M is a solid torus. Case
1: The case where M ∪ Nbd( τ ) is a solid torus and M is the exterior of a non-trivialknot.This is impossible because, in this case, T := ∂ ( X ∪ Nbd( A )) ∩ M is an essential torusin M (see Figure 15 (i)), which contradicts the assumption that M is atoroidal.(i) (ii) T V X XA A A A A AD
Figure 15. (i) The essential torus T ; (ii) The handleobody V of genus two. Case
2: The case where M ∪ Nbd( τ ) is the exterior of a non-trivial knot and M is asolid torus.Set V := M ∪ X ∪ Nbd( τ ) (see Figure 15 (ii)). Let D be the cocore of the 1-handleNbd( τ ). We note that now S = V ∪ M is a Heegaard splitting of genus two, and D is XTENDING AUTOMORPHISMS OF THE GENUS-2 SURFACE OVER THE 3-SPHERE 21 a non-separating disk in V . The solid torus Cl( V \ Nbd( D )) (= E ( M ∪ Nbd( A )) is aregular neighborhood of a non-trivial knot in S . The disk D is thus not a primitive diskin V due to Lemma 2.1 of Cho [3]. By the uniqueness of W , X , and A , ˆ f preserves V , M and D . Thus, we can think of ˆ f as an automorphism of ( S , V , D ). By Proposition10.1 of Cho-McCullough [5], MCG( S , V , D ) is a finite group. This implies that for some n , ˆ f n is isotopic to the identity via an isotopy preserving V and D . Set B := X ∩ M . B is an incompressible annulus in V . Since ˆ f preserves V , f preserves B as well. Thenclearly the above isotopy from ˆ f n to the identity also preserves B . Consequently, ˆ f n isisotopic to the identity on ∂V and so the order of f is finite, which is a contradiction. Case
3: The case where both M ∪ Nbd( τ ) and M are solid tori.Set V := M ∪ X ∪ Nbd( τ ) (see again Figure 15 (ii)). Let D be the cocore of the1-handle Nbd( τ ). The solid torus Cl( V \ Nbd( D )) ( ∼ = E ( M ∪ Nbd( A )) is now a regularneighborhood of the trivial knot in S . The disk D is then a primitive disk in V (seeLemma 2.1 of Cho [3]). We can again think of ˆ f as an automorphism of ( S , V , D ). Bythe argument in Lemma 5.5 of [4], ˆ f | V is a composition of α , β , γ shown in Figure 16. β γc D D Figure 16.
The maps β and γ .Here α comes from the hyperelliptic involution of ∂V , β is the half-twist of a handle,and γ exchanges handles. Both α and γ are order two elements, while the order of β isinfinite. We note that β | ∂V is the Dehn twist along a separating simple closed curve c on ∂V . These elements satisfy α ◦ β = β ◦ α , α ◦ γ = γ ◦ α , and β ◦ γ = α ◦ γ ◦ β .Therefore we can write ˆ f | V in the formˆ f | V = α ǫ ◦ γ ǫ ◦ β n ( ǫ , ǫ ∈ { , } , n ∈ Z ) . Thus we have ( ˆ f | ∂V ) = T c n . Since c ∩ ∂A = ∅ , n should be 0. Consequently, forsome n , ˆ f n is isotopic to the identity via an isotopy preserving V and D . The rest ofthe proof runs as in Case 2. (cid:3) By Claims 3–7, M can contain no essential annuli, which is a contradiction.Consequently, f is standardly extendable over S . This completes the proof of Theo-rem 3.3. Acknowledgments
The authors wish to express their gratitude to Shicheng Wang and for helpful com-ments. They are also profoundly grateful to Mario Eudave-Mu˜noz for pointing out anerror in the original draft.
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