the disjoint curve property and genus 2 manifolds

Topology and its Applications 97 (1999) 273–279

The disjoint curve property and genus 2 manifolds

Abigail Thompson1,2

Department of Mathematics, University of California, Davis, CA 95616, USA

Received 5 January 1998; received in revised form 9 February 1998

Abstract

Genus 2 manifolds are a convenient and accessible place to introduce an interesting condition onHeegaard splittings, called thedisjoint curve property. This paper will describe the disjoint curveproperty and its ramifications for understanding genus 2 manifolds, and use it to find a necessarycondition for a genus 2 manifold to be hyperbolic. 1999 Elsevier Science B.V. All rights reserved.

Keywords:Heegaard splitting; Disjoint curve; Genus 2

AMS classification: 57N10

Definition. Let (H1,H2,F ) be a Heegaard splitting of a 3-manifoldM, whereH1 andH2

are handlebodies andF = ∂H1= ∂H2. Thegenusof the Heegaard splitting is the genus ofthe surfaceF . The Heegaard splitting isreducibleif there exists an essential simple closedcurvec onF such thatc bounds (imbedded) disksD1 in H1 andD2 in H2. The splitting isstabilizedif there exist essential simple closed curvesc1 andc2 onF such thatci bounds an(imbedded) diskDi in Hi andc1 andc2 intersect in a single point. A stabilized splitting ofgenus at least 2 is reducible. The splitting isweakly reducibleif there exist essential simpleclosed curvesc1 andc2 on F such thatci bounds an (imbedded) diskDi in Hi andc1

andc2 are disjoint. A splitting that is not weakly reducible isstrongly irreducible. As thispaper will deal largely with genus 2 Heegaard splittings, it is worth noting the followingwell-known proposition:

Proposition 1. A weakly reducible genus2 Heegaard splitting is reducible.

Proof. Let (H1,H2,F ) be a genus 2 Heegaard splitting of a 3-manifoldM. Let c1 andc2

be essential simple closed curves onF such thatci bounds an (imbedded) diskDi in Hi

1 The author is grateful to MSRI and the National Science Foundation for their support.2 E-mail: [email protected].

0166-8641/99/$ – see front matter 1999 Elsevier Science B.V. All rights reserved.PII: S0166-8641(98)00063-7

274 A. Thompson / Topology and its Applications 97 (1999) 273–279

andc1 andc2 are disjoint. Assumec1 andc2 are non-separating (the argument is similar,but easier, if one is separating). LetF ′ = F − (c1 ∪ c2). F ′ is a 4-punctured sphere. Letc be a curve onF ′ separating the copies ofc1 from the copies ofc2. c can be obtainedby banding together two copies ofci . Hencec bounds a disk inHi . So (H1,H2,F ) isreducible. 2

The notion of weak reducibility was introduced by Casson and Gordon [2]. It has proveda very useful tool. They showed that ifM has a weakly reducible Heegaard splitting, eitherM is Haken or the splitting is reducible. Subsequent to their results, most attention hasconcentrated on the strongly irreducible case.

We introduce a ‘weaker’ form of reducibility:

Definition. Let (H1,H2,F ) be a Heegaard splitting of a 3-manifoldM. The Heegaardsplitting has thedisjoint curve propertyif there exist essential simple closed curvesc, aandb onF such thatc is disjoint froma andb, a bounds a disk inH1, andb bounds a diskin H2.

Notes.(1) If the splitting is assumed to be strongly irreducible,a and b must necessarily

intersect.(2) If a strongly irreducible Heegaard splitting fails to have the disjoint curve property,

then every pair of curves such asa andb must cut the splitting surface into disks,i.e., they mustfill the surface. In particular, any such pair of curvesa andb mustintersect in many points.

Theorem 2. Let (H1,H2,F ) be a strongly irreducible Heegaard splitting of a closedorientable3-manifoldM. If M contains an essential torusT , then(H1,H2,F ) has thedisjoint curve property.

Proof. Let T be an essential torus inM. By Kobayashi [5],T can be isotoped to intersectF in a collection of (at least 2) simple closed curves which are essential onT . These curvessplit T into annuli. Further, none of the annuli are boundary parallel inH1 orH2. LetA1

andA2 be two adjacent annuli onT , sharing the curvec on their boundaries. SinceT isessential,A1 andA2 are incompressible. Since they are imbedded in the handlebodiesH1

andH2 they are boundary compressible. LetD1 andD2 be the disks obtained by boundarycompressions onA1 andA2, respectively.D1 andD2 are essential sinceA1 andA2 are notboundary parallel.D1 andD2 are both disjoint fromc, hencec is a disjoint curve for theHeegaard splitting. 2Corollary 3. Let (H1,H2,F ) be a Heegaard splitting of a closed orientable3-manifoldM. If (H1,H2,F ) does not have the disjoint curve property, thenM is atoroidal.

So there is a simple condition on a Heegaard splitting ofM which guarantees thatM isatoroidal.

A. Thompson / Topology and its Applications 97 (1999) 273–279 275

We now focus on the genus 2 case, to understand genus 2 splittings with the disjointcurve property. We will assume all curves on the splitting surface intersect transversely.

Definition. Let T be a separating torus in a 3-manifoldM (= a Lens space, S3, orS1 × S2), bounding solid tori,T1 andT2, on both sides. LetK be a knot inM. K is a1-bridge braid in M if K can be isotoped to intersectT transversely in two points, suchthatK intersects eachTi in a boundary parallel arc.

Theorem 4. Let (H1,H2,F ) be a genus2 Heegaard splitting of a3-manifoldM. Suppose(H1,H2,F ) has the disjoint curve property. Then either:

(0) (H1,H2,F ) is a stabilization;(1) M is reducible(hence a connect sum of Lens spaces);(2) M is a small Seifert fibered space;(3) M contains an essential torus—this case was analyzed by Kobayashi in[6];(4) M is obtained by surgery on a1-bridge braid on an unknotted torus in a Lens space,

S3 or S1× S2.

Proof. Let c, a andb be essential simple closed curves onF such thatc is disjoint fromaandb, a bounds a diskD1 in H1, andb bounds a diskD2 in H2.

Definition. A simple closed curvec on ∂Hi is acoreof Hi if there exists a simple closedcurvec′ on∂Hi such thatc intersectsc′ in a single point andc′ bounds a diskE in Hi .

Case1. c is not a core inH1 andc is not a core inH2. CutHi alongDi . If Di is non-separating, letTi be the resulting solid torus. IfDi is separating, letTi be the solid toruscontainingc in its boundary.

Let M∗ be the manifold obtained by gluingT1 and T2 together along an annularneighborhood ofc. M∗ is contained inM. Sincec is not a core inH1 or H2, M∗ is aSeifert-fibered space, with base space a disk, with two singular fibers. The boundary ofM∗ is a torusT . LetM∗∗ =M −M∗.

If T is incompressible inM∗∗, condition (3) holds.If M∗∗ is a solid torus, condition (0), (1) or (2) hold.SupposeT is compressible inM∗∗, butM∗∗ is not a solid torus. LetS be the 2-sphere

obtained by compressingT in M∗∗. If S is essential inM, then (1) holds. IfS is notessential inM, then it bounds a 3-ball containingM∗, and hencec′, a core of the solidtorusT1. c′ is also a core ofH1. It follows from an argument due to Frohman [3] that theHeegaard splitting is weakly reducible, and so reducible.

Case2. c is a core inH1. LetK be a core ofH2, disjoint fromb. LetG be an essentialdisk inH2, disjoint fromb, intersectingK once, such that[

algebraic intersection #(∂G, c)]= [geometric intersection #(∂G, c)

].

If b is separating, chooseK in the component ofH2 which containsc. PushK slightlyinto the interior ofH2. LetA be an annulus with one boundary component equal toc on


∂H2 and one boundary componentr on the boundary of a neighborhood ofK. Noticethat r surgery onK yields a manifoldN with a natural genus 2 Heegaard splitting,(H1,H

′2,F )whereH ′2 is obtained fromH2 by r-surgery onK. This new Heegaard splitting

is stabilized;c now bounds a diskE in H ′2, andc intersects the boundaryc′ of a diskE′ inH1 in a single point. ThusN is either a Lens space,S3 or S1× S2.

CompressingH ′2 along the diskE yields an unknotted solid torusW in N . ∂W ispunctured twice byK ′, the dual ofK, in N . The arc ofK ′ exterior toW is boundaryparallel to∂W via the remnants of the diskE′. The arc ofK ′ interior toW is boundaryparallel to∂W via the remnants of the diskG. SoK ′ is a 1-bridge braid on the unknottedsolid torusW , and condition (4) holds.2Note. Hempel has obtained results similar to Theorems 2–4 independently [4].

Theorem 5. Let (H1,H2,F ) be a genus2 Heegaard splitting of a3-manifoldM. Supposethere exist essential disksD1 andD2 in H1 andH2, respectively such that∂D1 intersects∂D2 in at most three points. Then

(1) (H1,H2,F ) has the disjoint curve property, and(2) M is not hyperbolic.

Proof.(I) If ∂D1 intersects∂D2 in one point, the Heegaard splitting is stabilized.

(II) Suppose∂D1 intersects∂D2 in exactly two points. ThenD1 andD2 cannot bothbe separating. There exists an essential simple closed curvec on F disjoint from∂D1 and∂D2. If c is not a core on eitherH1 orH2, we are done by the argument inCase 1 of Theorem 4.

Case1.D1 is separating andc is a core onH1. Then there exists a compressing diskE for H1 which intersects∂D2 in at most one point, so the splitting is either reducible orstabilized.

Case2.D1 is non-separating andc is a core onH1. Recall that∂D2 is disjoint from thecorec. Consider the graphG on the annulusA= (F − (∂D1 ∪ c)) formed by contractingthe two copies ofD1, E1 andE2, to points, and taking the arcs of∂D2 as edges. Thetwo vertices ofG are both order two. ExaminingG, one can see that either there exists acompressing diskE forH1 which intersects∂D2 in at most one point, orG consists of twoessential loops inA. If G is two essential loops, letc′ be an essential curve inA separatingthe two loops ofG. Then there exists a properly imbedded annulusX inH1, with boundaryc ∪ c′, disjoint from∂D1 and∂D2, which is incompressible, non-boundary parallel, andnon-separating inH1. Note thatc andc′ are not parallel onF .

Subcase(a).D2 is non-separating inH2. Thenc ∪ c′ bound an essential non-separatingannulusY in H2. X ∪ Y is a non-separating torus inM.

Subcase(b). D2 is separating inH2. Sincec andc′ are not parallel onF , ∂D2 mustseparatec andc′. If eitherc or c′ is a core ofH2, the arguments from case 1 apply. Supposeneither is a core. LetT1 andT2 be the solid tori obtained by cuttingH2 open alongD2. Let


M∗ be the manifold[T1 ∪ T2 ∪ (nhd(X))]. Apply the arguments from Theorem 4 case 1toM∗.

(III) Suppose∂D1 intersects∂D2 in exactly three points. Then bothD1 andD2 arenon-separating.

Claim. There exists an essential simple closed curvec onF disjoint from∂D1 and∂D2.

Proof. LetR = F − ∂D1. ∂D2 intersectsR in three arcs,x, y andz. Suppose there are nosimple closed curves disjoint from∂D1 and∂D2. Thenx, y andz must cutR into a disk.At least one of the arcs, sayx, must connect the two boundary components ofR. CutRopen alongx to obtainR′, a once-punctured torus. The remaining two arcs of∂D2, y andz, cutR′ into a disk. Thus the endpoints of these arcs are linked on∂R′. Since the endpointsof y andz are linked,y andz must also connect the two boundary components ofR, and,further, the order of the endpoints ofx, y andz on the two boundary components is thesame, that is, the endpoints ofx, y andz occur in clockwise (or counterclockwise) order onboth boundary components ofR. WhenF is reconstructed fromR by re-attaching the twoboundary components, exactly one of the three arcs will form a single simple closed curvewhile the other two form a second simple closed curve. This contradicts the connectivityof ∂D2. Hence there is a simple closed curvec disjoint from∂D1 and∂D2.

Case1. c is not a core inH1 orH2. The argument is the same as for Theorem 4, Case 1.Case2. c is a core inH1. CutH1 open alongD1, to obtain a solid torusQ. c is a core

of Q, and the image ofD1 on ∂Q is two disksE1 andE2. ∂D2 is a collection of at mostthree arcs imbedded in∂Q with endpoints onE1 andE2, and∂D2 is disjoint fromc. 2Claim (with thanks to Y.Q. Wu).There exists an essential diskJ for Q such that∂Jintersects∂D2 in at most two points.

Proof. Recall that∂D2 is disjoint from the corec, and consider the graphG on∂Q formedby contractingE1 andE2 to points and taking the arcs of∂D2 as edges. Since the twovertices ofG are both order three, there are only three possibilities forG, and in all casesthere exists a meridian diskJ ofQ intersecting∂D2 in at most two points. 2

We can use this idea to examine tunnel number one knots in the 3-sphere:Let K be a hyperbolic tunnel number one knot inS3 with tunnel t . Let H1 be the

compression body obtained by taking a(2δ)-neighborhood ofK ∪ t and removing a(δ)-neighborhood ofK from it. ∂H1 = F separatesS3 into two pieces. LetH2 be the piecewhich does not containK. H2 is a handlebody by construction.

Corollary 6. Let m ⊂ ∂H1 be a meridian oft . Then every compressing diskD2 for H2

crossesm at least4 times.

Proof. Consider the generalized Heegaard splitting(H1,H2,F ) of the complement ofK.m is the boundary of a compressing diskD1 in H1. LetK(r) be the 3-manifold obtainedby r-Dehn surgery onK. By Thurston’s hyperbolic surgery theorem,K(r) is hyperbolic


for all but a finite number of slopesr. Chooses such thatK(s) is hyperbolic. ApplyingTheorem 5 toK(s), it follows that every compressing diskD2 for H2 crossesm= ∂D1 atleast 4 times. 2Note. Adams obtained Corollary 6 in the case that the unknotting tunnel is a verticalgeodesic [1, Corollary 5.4].

Theorem 7. Let (H1,H2,F ) be a genus2 Heegaard splitting of a3-manifoldM. Suppose(H1,H2,F ) has the disjoint curve property with curvesa, b andc. If a intersectsb in atmost four points, thenM is not hyperbolic.

Proof. If a intersectsb in at most three points, we are done by Theorem 5. So assumea

intersectsb in exactly four points.If c is not a core inH1 and it is not a core inH2 we are done.Assumec is a core inH1.Subcase(a). SupposeD1 is separating. CutH1 open alongD1. LetQ be the solid torus

component with corec. The image ofD1 on ∂Q is a diskE. ∂D2 is a collection of twoarcs imbedded in∂Q with endpoints onE, and∂D2 is disjoint fromc. Consider the graphG on ∂Q formed by contractingE to a point and taking the arcs of∂D2 as edges. For allpossibilities forG, there exists a compressing diskJ for H1 intersecting∂D2 in at mosttwo points. This case is covered by Theorem 5.

Subcase(b). SupposeD1 is non-separating. CutH1 open alongD1 to obtain a solidtorusQ. c is a core ofQ, and the image ofD1 on ∂Q is two disksE1 andE2. ∂D2 is acollection of four arcs imbedded in∂Q with endpoints onE1 andE2, and∂D2 is disjointfrom c. Consider the graphG on∂Q formed by contractingE1 andE2 to points and takingthe arcs of∂D2 as edges.

If G contains at most one loop at each vertex, then there exists a compressing diskJ forH1 intersecting∂D2 in at most three points.

SupposeG consists of two loops at each vertex. If any of the loops are not essential in∂Q− c, then there exists a compressing diskJ for H1 intersecting∂D2 in at most threepoints. SupposeG consists of two loops at each vertex, all essential in∂Q− c. Then theargument proceeds exactly as in Theorem 5 (II), Case 2.2

Acknowledgement

I would like to thank the referee for many helpful comments, and particularly for thesimplification of the proof of Corollary 6.

References

[1] C. Adams, Unknotting tunnels in hyperbolic 3-manifolds, Math. Ann. 302 (1995) 177–195.[2] A. Casson and C.McA. Gordon, Reducing Heegaard splittings, Topology Appl. 27 (1987) 275–

283.


[3] C. Frohman, Heegaard splittings of the three-ball, Topology ’90, Ohio State Univ. Math. Res.Inst. Publ. 1 (de Gruyter, Berlin, 1992) 123–131.

[4] J. Hempel, Three manifolds as viewed from the curve complex, Preprint.[5] T. Kobayashi, Casson–Gordon’s rectangle condition of Heegaard diagrams and incompressible

tori in 3-manifolds, Osaka J. Math. 25 (3) (1988) 553–573.[6] T. Kobayashi, Structures of Haken manifolds with Heegaard splittings of genus two, Osaka J.

Math. 21 (1984) 581–598.[7] W. Thurston, The geometry and topology of 3-manifolds, Notes, 1977–1978.

the disjoint curve property and genus 2 manifolds

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