Short geodesics in hyperbolic 3-manifolds

0.09 MB·

by  William Breslin

1 Short geodesics in hyperbolic 3-manifolds
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2 WILLIAMBRESLIN

n
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J Foreachg≥2,weproveexistenceofacomputableconstantǫ(g)>0suchthatif

8 S isastronglyirreducibleHeegaardsurfaceofgenus g inacompletehyperbolic
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3-manifoldM and γ isasimplegeodesicoflengthlessthan ǫ(g) in M,then γ is

] isotopicinto S.
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57M50
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1 Introduction
[

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v Let M be a hyperbolic 3-manifold and let S be a surface embedded in M. A simple
6
closed curve γ in M is said to be unknotted with respect to S if γ can be isotoped
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4 into S. A finite collection Γ = {γ ,...,γ } of simple closed curves is unlinked with
1 n
3
respect to S ifthere isa collection of disjoint embedded surfaces S ,...,S which are
. 1 n
2
isotopic to S and with γ ⊂ S forall i. Onecan ask ifshort geodesics areunknotted
1 i i
9 orunlinked withrespecttofibers,Heegaardsurfaces, orleavesofafoliation.
0
: ItfollowsfromworkofOtal[5]thatshortgeodesicsinahyperbolicmappingtorusare
v
i unlinkedwithrespecttothefiber,where“short"dependsonlyonthegenusofthefiber.
X
r
a Theorem 1 (Otal) Forevery g thereisaconstant ǫ > 0 suchthatthefollowing
holds: IfM isaclosed3-dimensionalhyperbolicmappingtoruswithgenus g fiber
andΓisthecollectionofsimpleclosedgeodesicsinM whichareshorterthanǫ,then
Γ isunlinkedwithrespecttoS.
In an unpublished paper [10], Souto proved that short geodesics in hyperbolic com-
pression bodiesareunlinked withrespecttotheboundary ofthecompression body.
Theorem 2 (Souto [10]) IfN¯ isacompressionbodythenthereisaconstantǫ > 0
whichdependsonlyonχ(N¯) suchthatforeverycompletehyperbolicmetriconthe
interiorN ofN¯ wehave:everyfinitecollectionofsimplegeodesicswhichareshorter
thanǫ isunlinkedwithrespectto∂N¯.
2 WilliamBreslin
InthesamepaperSoutosketchedaproofthatshortgeodesicsinhyperbolic3-manifolds
areunlinked withrespecttoastronglyirreducible Heegaardsurface:
Theorem3 (Souto[10]) Foreverygthereisaconstantǫ > 0suchthatthefollowing
holds: ifM isaclosedhyperbolic3-manifold, S isastronglyirreducibleHeegaard
surfaceofgenusginM,andΓisthecollectionofsimpleclosedgeodesicsinMwhich
areshorterthanǫ,thenΓ isunlinkedwithrespecttoS.
Theconstants fromTheorem1andTheorem2areknowntobecomputable. However
SoutoonlysketchesaproofofTheorem3anddoesnotproduceanexplicitconstant. In
thispaper,wedevelopanewapproachtoTheorem3. Ourproofismoretopologicalthan
geometricandismoreelementarythanSouto’sproof. Moreover,weproveexistenceof
acomputableconstant ǫ (dependingonlyonthegenus g)suchthatprimitivegeodesics
of length less than ǫ are unknotted with respect to a strongly irreducible Heegaard
surfaceofgenus g:
Theorem 4 Foreachg ≥ 2 thereexistsacomputableconstantǫ := ǫ(g) > 0 such
thatifSisastronglyirreducibleHeegaardsurfaceofgenusginacompletehyperbolic
3-manifoldM andγ isasimplegeodesicoflengthlessthanǫinM,thenγ isisotopic
intoS.
SeetheremarkaftertheproofofLemma2foradescription oftheconstant ǫ(g).
The proof of Theorem 4 uses three main tools: bounded area sweepouts provided
by work of Pitts and Rubinstein [6], an argument using the Rubinstein-Scharlemann
graphic similar to an argument of Johnson [3] used to prove that spines of strongly
irreducible Heegaard splittings are locally unknotted, and alemma ofSchultens from
[9]. In section 1, we use bounded area sweepouts and the Rubinstein-Scharlemann
graphic to prove existence of an embedded annulus connecting aHeegaard surface to
the boundary ofaMargulis tube around ashort geodesic. First, using abounded area
sweepout and the fact that Margulis tubes around very short geodesics are very fat,
we show that the intersection of one of the sweepout surfaces with the Margulis tube
around a short geodesic contains a simple loop which is homotopic to a power of the
short geodesic. This is the content of Lemma 2 and is the only step which uses a
geometricargument. Next,weusetheRubinstein-Scharlemann graphicandLemma2
to show that there exists an embedded annulus connecting a Heegaard surface to the
boundary ofaMargulis tubearoundtheshortgeodesic. Thisisthecontent ofLemma
1. In section refisotopy, a thin position argument of Schultens is used to show that
Shortgeodesicsinhyperbolic3-manifolds 3
the existence of the annulus provided by Lemma 1 implies that the short geodesic is
isotopicintoaHeegaardsurface.
Definitions. Let M be a closed connected orientable 3-manifold. Let S be a closed
connected orientable surface embedded in M which bounds handlebodies H and H
1 2
on either side. We call (S,H ,H ) a Heegaard splitting of M. A Heegaard splitting
1 2
is weakly reducible if there are properly embedded essential disks in H , H whose
1 2
boundaries aredisjoint. AHeegaardsplitting isstrongly irreducible ifitisnotweakly
reducible.
2 Finding an annulus
Let S be a strongly irreducible Heegaard surface of genus g ≥ 2 in a complete
hyperbolic 3-manifold M. ThegoalofthissectionistoprovethefollowingLemma.
Lemma 1 Thereexistsacomputableconstant ǫ := ǫ(g) suchthatifγ isasimple
closedgeodesicoflengthlessthanǫ inM,thenthereisaregularneighborhoodN of
γ andanembeddedannulusin M \N withboundary α∪ α ,where α isasimple
′
essentialnon-meridinalloopintheboundaryofN,andα iscontainedinasurface
′
isotopictoS.
Sweepouts and Pitts-Rubinstein. A sweepout of of a 3-manifold M with a Heegaard
surface S is a smooth degree one map f : S×[0,1] → M such that S := f(S×{t})
t
is a surface isotopic to S for each t ∈ (0,1) and f(S ×{0}), f(S ×{1}) are spines
of the handlebodies bounded by S . Bywork of Pitts and Rubinstein [6], there is a
1/2
constant A(g) such that if M has a genus g strongly irreducible Heegaard surface S
thenthereexistsasweepout f : S×[0,1] → M of M suchthat area(S×{t}) ≤ A(g)
foreach t ∈[0,1]. Wecanuse A(g)= 4π(g−1)+δ forany δ > 0.
Wewilluseboundedareasweepouts toprovethefollowingLemma.
Lemma2 Thereexistsacomputableconstantǫ := ǫ(g) > 0 suchthatthefollowing
holds:ifγ isasimpleclosedgeodesicoflengthlessthanǫ inM andT isaMargulis
tubeaboutγ,thenSmaybeisotopedinM sothatS∩T containsasimpleloopwhich
isessentialinT.
4 WilliamBreslin
Proof Let f : S×[0,1] → M be asweepout of M with area(S×{t}) ≤ A := A(g)
for each t ∈ [0,1]. Let ǫ = ǫ(g) > 0 be so small that the Margulis tube about any
geodesic in a complete hyperbolic 3-manifold of length at most epsilon has totally
geodesicmeridiandiskswithareaatleast 3A. Notethattheconstant ǫ iscomputable.
Let γ beageodesicoflengthatmost ǫ.
The set M \S is a union of two handlebodies. Let H and W be the closures
1/2 1/2 1/2
ofthesehandlebodies. For t ∈ (0,1),let H betheclosureofthecomponentof M\S
t t
which changes into H as S isotopes to S andlet W be the closure ofthe other
1/2 t 1/2 t
component of M \S . For t ∈ (0,1) near 0, one of the handlebodies, say H , is a
t t
smallneighborhood ofaspine. Since f hasdegree one, thehandlebody W isasmall
t
neighborhood ofaspinefor t ∈ (0,1) near 1.
If S ∩T contains aloopwhichisessential in T forsome t ∈ [0,1],thenwearedone
t
provingLemma2. Assumethat S ∩T doesnotcontainaloopwhichisessentialin T
t
forany t ∈ [0,1].
Claim 1 Foreacht ∈ [0,1] eithertheinteriorofW ∩T ortheinteriorofH ∩T
t t
containsaloopwhichisessentialinT.
Proof. We are assuming that S ∩T does not contain a loop which is essential in T.
t
If S ∩T is empty then we are done with Claim 1, so we will assume that S ∩T is
t t
nonempty. Thuswehavethat S ∩T isnonemptyandthatanyloopin S ∩T istrivial
t t
in T. Let T˜ be a lift of T to the universal cover H3 of M. Since any loop in S ∩T
t
is trivial in T, there is a lift S˜ of S ∩T contained in T˜ which is homeomorphic to
t
S ∩T. Let S˜ beaconnectedcomponentof S˜. Since T˜ isaball, S˜ mustseparate T˜.
t 0 0
Weclaimthat T˜ \S˜ containsacomponentwhichhascompactclosurewhichdoesnot
0
separatetheendsof T˜.
Let D be a totally geodesic meridian disk in T˜ such that D is orthogonal to ∂T˜.
Considertheprojection p : T˜ → Dof T˜ to D alonglinesequidistantfromthegeodesic
core of T˜. The area of S is at least the area of p(S˜). Since the area of S is less
t t
than the area of D, the interior of D contains a point x which is not in p(S˜). The
preimage p 1(x) of x isdisjointfrom S˜ andthereforecontainedinonecomponentof
− 0
T˜ \S˜ . Thus S˜ doesnotseparatetheendsof T˜ Also, S˜ iscompact,hencecontained
0 0 0
in some compact subset K of T˜. The complement T˜ \K of K in T˜ is contained in
thecomponent of T˜ \S˜ whichcontains p 1(x). Thustheothercomponent of T˜ \S˜
0 − 0
iscontained inthecompactset K andtherefore hascompactclosure. Wehaveshown
Shortgeodesicsinhyperbolic3-manifolds 5
that every component of S˜ splits off a connected piece of T˜ which does not separate
theendsof T˜ andwhichhascompactclosure.
Let D and D be distinct meridian disks in T˜ which project to the same disk in T.
1 2
There is one component of T˜ \(D ∪D ) whose closure is compact. Let F be the
1 2
closure ofthiscomponent. Since S iscompact, there arefinitely manylifts S˜ ,...,S˜
t 1 k
of S ∩T which intersect the compact set F. For each i = 1,...,k, S˜ splits a piece
t i
from T˜ which has compact closure so each component of S˜ ∪··· ∪S˜ splits off a
1 k
piece of T˜ which does not separate the ends of T˜ and which has compact closure.
Thustheset T˜ \(S˜ ∪···∪S˜ ) containsaconnectedcomponentwhichintersectsboth
1 k
D and D . Therefore we can find an arc in T˜ \ (S˜ ∪··· ∪S˜ ) with endpoints in
1 2 1 k
D and D . We have shown that the complete pre-image of S ∩T does not separate
1 2 t
the two ends of T˜ and thus some component C˜ of the pre-image of T \ S in T˜ is
t
non-compact. This non-compact component C˜ ⊂ T˜ projects to a set C in T \ S .
t
Since C˜ is non-compact, the set C has nontrivial image in π (T) under the map
1
induced by inclusion. Thus there is a loop contained in C (which is contained in ei-
thertheinteriorofW ∩T ortheinteriorofH ∩T)whichisessentialinT. (cid:3)(Claim1)
t t
ThefollowingClaimwillcompletetheproofofLemma2.
Claim2 TheHeegaardsurfaceS containsasimpleloopinS ∩T whichisessential
β β
inT forsomeβ ∈ (0,1).
Proof. Since H isasmallneighborhoodofaspinefor t ∈ (0,1) near 0,theinteriorof
t
W ∩T contains aloopwhichisessential in T for t ∈(0,1) near 0. Fix δ > 0 sothat
t
the interior of W ∩T contains a loop which is essential in T. If H ∩T contains a
δ δ
loopwhichisessential in T,thenlet β = δ. If H ∩T doesnotcontainaloopwhich
δ
isessential in T,then let σ = inf{t ∈ (δ,1)|H ∩T contains aloop whichisessential
t
in T}. Notethat σ existssince W boundsaverysmallneighborhood ofagraphfor t
t
near 1.
Iftheinteriorof H ∩T contains aloopwhichisessential in T,then H ∩T contains
σ t
a loop which is essential in T for t near σ, contradicting the definition of σ. Thus
the interior of H ∩T does not contain a loop which is essential in T and therefore
σ
Claim1impliesthattheinteriorof W ∩T containsaloopwhichisessentialin T. So
σ
W ∩T containsaloopwhichisessentialin T for t near σ. Thisimpliesthatforsome
t
β > σ near σ, both H ∩T and W ∩T contain loops l ⊂ W and l ⊂ H which
β β W H
are essential in T. For some natural numbers m,n we have (l )m is homotopic in T
W
6 WilliamBreslin
to (l )n. Thusthereisanimmersedannulus A in T withboundary components equal
H
to (l )m and (l )n. If A∩S does not contain any loops which are essential in A,
W H β
then thereisanarcin A withendpoints in l and l which isdisjoint from S . This
W H β
contradicts the factthat l and l are indifferent components of T \S . Thussome
W H β
loop α in A∩S isessential in A. Sincetheboundary components of A areessential
β
in T,theloop α mustbeessential in T. Thus S contains aloopwhichisessentialin
β
T,so S containsasimpleloopwhichisessential in T. (cid:3)(Claim2)
β
Remark. The constant ǫ(g) in Lemma 2 is the constant we will use in Theo-
rem 4. From the proof of Lemma 2, ǫ(g) should be so small that a meridian
disk in a Margulis tube around a closed geodesic of length less than ǫ(g) has area
greater than A(g) = 4π(g − 1). The area of a totally geodesic disk of radius r
in hyperbolic space is 2π(cosh(r) − 1). Thus we want the radius of the Margulis
tube to be greater than arccosh(2g − 1). Meyerhoff [4] proved that the radius r
of a Margulis tube around a closed geodesic of length l less than 0.107 satisfies
sinh2(r) = 1/2(√1 2k −1) where k = cosh( 4πl)−1. Thus choosing ǫ(g) so that
k− q√3
arcsinh(q1/2(√1k-2k-1))> arccosh(2g−1),where k = cosh(q4π√ǫ3(g))−1willsuffice.
By Lemma 2, we may isotope S so that S ∩ T contains a simple loop ω which is
essential in T. Wewillshow thatwemayisotope S sothat S∩∂T contains asimple
loop which is essential in T by using an argument of Johnson used in [3] to prove
thatspines ofstrongly irreducible Heegaardsplittings arelocally unknotted. Theidea
is to use the Rubinstein-Scharlemann graphic to show that some sweepout surface S
′
intersects ∂T inloopswhichareinessentialin S . Since S and S coboundathickened
′ ′
copyof S,thereisanembeddedannuluswithoneboundarycomponentequalto ω and
the other boundary component disjoint from T. Theintersection of this annulus with
∂T mustcontain asimpleloopwhichisessential intheannulus, providing uswithan
embeddedannulus toisotope ω into ∂T.
We will now define a new sweepout of M. Assume that we have isotoped S so that
S∩T containsasimpleloop ω whichisessentialin T. TheHeegaardsurface S splits
M intotwohandlebodies H and H . Let f : M → [−1,1] beasmoothfunctionsuch
1 2
that f 1(−1) is aspine of H , f 1(1) is aspine of H , f 1(t) isasurface isotopic to
− 1 − 2 −
S foreach t ∈ (−1,1), and f 1(0) = S (i.e.,the map f provides asweepout of M by
−
disjoint Heegaard surfaces, one ofwhichis S). Foreach t ∈ (−1,1), let S = f 1(t).
t −
Notethatthesurfaces inthissweepout donotnecessarily havearea bounded interms
Shortgeodesicsinhyperbolic3-manifolds 7
of g.
Lemma3 LetY beasolidtorusinM.Oneofthefollowingholds:
(1)S ∩∂Y containsaloopwhichisessentialin∂Y anddoesnotboundameridian
t
diskinY forsomet ∈ (−1,1),or
(2)Forall t ∈ (−1,1), if S ∩ ∂Y containsaloopwhichisessentialin S , then
t t
S ∩∂Y containsaloopwhichboundsaproperlyembedded,essentialdiskinoneof
t
thehandlebodiesboundedbyS .
t
Proof Assumethat S ∩∂Y doesnotcontainaloopwhichisbothessentialin ∂Y and
t
does not bound ameridian disk in Y forall t ∈ (−1,1). Thus anyloop in S ∩∂Y is
t
either trivial in ∂Y or a meridian for Y. In particular, any loop in S ∩∂Y bounds a
t
properlyembeddeddiskinM. Scharlemann’sNoNestingLemma(see[7])impliesthat
anyloopwhichisessentialin S andboundsadiskin M boundsaproperlyembedded,
t
essential disk in one of the handlebodies bounded by S . Thus if S ∩∂Y contains a
t t
loop which is essential in S , then S ∩∂Y contains a loop which bounds a properly
t t
embedded, essential diskinoneofthehandlebodies bounded by S .
t
The Rubinstein-Scharlemann graphic. Let g : T → [0,1] be a smooth function such
that g 1(0) = γ and g 1(1) = ∂T and g 1(t) isasurfaceisotopicto ∂T foreach t in
− − −
(0,1). Foreach t ∈ [−1,1] let g = g| . Wesayafunctionisnear-Morseifthereis
t St T
∩
asingle degenerate critical point orthere aretwonon-degenerate critical points atthe
samelevel. ByworkofCerf[1],wecanisotope f and g sothat g isaMorsefunction
t
for all but finitely many t and g is near-Morse for the remaining values of t. The
t
Rubinstein-Scharlemann graphic, G,isthesetofpoints (t,s) ∈ [−1,1]×[0,1] such
t
thats isacriticalvalueofthefunction g . RubinsteinandScharlemannoriginallyused
t t
thegraphictocomparetwosweepouts byHeegaardsurfaces (see[8]). Weareusinga
sweepout of the 3-manifold be Heegaard surfaces and a sweepout of a solid torus by
tori. The properties of the graphic we need to use follow from the same arguments
Rubinstein and Scharlemann use intheir paper. Rubinstein and Scharlemann showed
that this set of points is a graph with vertices of valence 2 and 4 in the interior of
[−1,1]×[0,1] and valence 1and 2 inthe edges. Avalence-2 vertex at (t,s) occurs
t
when the map g has a degenerate critical point. A valence-4 vertex at (t,s) occurs
t t
when g has critical points at the same level. We will use the fact that if (t ,s ) and
t 1 1
(t ,s )areinthesamecomponentof[−1,1]×[0,1]\G,thenthesurface S isisotopic
2 2 t1
to S viaanisotopy whichtakestheloops in g 1(s ) totheloopsin g 1(s ). See[8]
t2 −t1 1 −t2 2
or[3]formoreontheRubinstein-Scharlemann graphic.
8 WilliamBreslin
Lemma4 EitherS ∩∂T containsaloopwhichisessentialin∂T anddoesnotbound
t
ameridiandiskinT forsomet ∈ (−1,1) orthereisaσ ∈ (−1,1) suchthatS ∩T
σ
doesnotcontainanessentialloopofS .
σ
Proof AssumethatS ∩∂T doesnotcontainaloopwhichisanessentialnon-meridinal
t
loopinin ∂T forany t ∈ (−1,1). Suppose, forcontradiction, that S ∩T contains an
t
essentialloopofS foreacht ∈ (−1,1). Foreacht ∈ (−1,1),letC beacomponentof
t t
S ∩T thatcontainsaloopwhichisessentialin S . Firstwewillshowthatthisimplies
t t
thatforeach t ∈ (−1,1) suchthat g isaMorsefunction, thereexistsasimpleloopin
t
S ∩∂T whichboundsaproperly embedded,essential diskinoneofthehandlebodies
t
bounded by S . Suppose g is a Morse function and suppose that there is no simple
t t
loopin S ∩∂T whichisessential in S . Inparticular, eachloopin C ∩∂T bounds a
t t t
disk in S . Wemayisotope S toeliminate anyofthese disks which aredisjoint from
t t
the interior of T and we still have that C contains a loop which is essential in S . If
t t
C ∩∂T isstillnon-empty, then C mustbeadisksincewehaveeliminatedanydisks
t t
boundedbyloopsin C ∩∂T whicharedisjointfromtheinteriorof T. Thiscontradicts
t
our assumption that C contains a loop which is essential in S . If C ∩∂T is empty
t t t
after eliminating disks outside T, then wehave that S is isotopic into T, giving us a
t
contradiction. Thus, for each t ∈ (−1,1), if g is a Morse function then there exists
t
a simple loop in S ∩∂T which is essential in S . By Lemma 3 this implies that for
t t
each t ∈ (−1,1) suchthat g isaMorsefunction,thereexistsasimpleloopin S ∩∂T
t t
whichboundsaproperlyembeddedessentialdiskinoneofthehandlebodies bounded
by S .
t
Foreachcomponent A of ([−1,1]×[0,1])\G,chooseapoint (t ,s ) intheinterior
A A
of A. Ifoneoftheloopsin g 1(s ) bounds aproperly embedded essential diskinthe
−tA A
handlebody f 1[−1,t ], then label the component A with a 1. If one of the loops in
− A
g 1(s ) boundsaproperlyembeddedessentialdiskinthehandlebody f 1[t ,1],then
−tA A − A
label the component A with a 2. If some component A of ([−1,1)×[0,1])\G has
morethanonelabel, thensomeloopin g 1(s ) bounds disks inboth f 1[−1,t ] and
−tA A − A
f 1[t ,1] implyingthattheHeegaardsplittingisweaklyreducible.
− A
Wehaveshownthatforeach t ∈ (−1,1) suchthat g isaMorsefunction, thereexists
t
a simple loop in S ∩∂T = g 1(1) which bounds a disk in one of the handlebodies
t −t
bounded by S . Inother words,anycomponent of ([−1,1]×[0,1])\G whichmeets
t
[−1,1]×{1} islabeled.
For t near −1, the vertical line {t} × [0,1] must intersect a component labeled 1,
because S is near the spine of f 1[−1,t]. For t near 1, the vertical line {t}×[0,1]
t −
must intersect a component labeled 2, because S is near the spine of f 1[t,1]. Thus
t −
Shortgeodesicsinhyperbolic3-manifolds 9
theremustbeadjacentsegmentsin[−1,1]×{1}\G whichhavedifferentlabels. Since
G cannotcontainanyverticalsegments,thisimpliesthatsomeverticalline {t}×[0,1]
intersectstwolabeledcomponentswithdifferentlabels. Thenforsome s ,s ∈ [0,1],
1 2
there is a loop in g 1(s ) which bounds a disk in f 1[−1,t ] and there is a loop in
−t 1 − A
g 1(s ) whichboundsadiskin f 1[t ,1]. Since g 1(s ) and g 1(s ) aredisjoint, the
−t 2 − A −t 1 −t 2
Heegaardsplitting isweaklyreducible, givingacontradiction.
We have shown that there is a σ ∈ (−1,1) such that S ∩ T does not contain an
σ
essential curveof S .
σ
Proof of Lemma 1. We will show that we may isotope S so that S∩∂T contains a
loop which is essential and non-meridinal in ∂T. Let f : M → [−1,1] be a smooth
functionsuchthat f 1(−1)isaspineofH , f 1(1) isaspineof H , f 1(t)isasurface
− 1 − 2 −
isotopic to S for each t ∈ (−1,1), and f 1(0) = S. Suppose that S ∩∂T does not
− t
contain a loop which is essential and non-meridinal in ∂T for any t ∈ (−1,1). We
haveisotoped S = S sothat S∩T contains asimpleloop ω whichisessential in T.
0
Wehave also shown that there isa σ ∈ (−1,1) such that S ∩T does not contain an
σ
essential loop of S and therefore each loop in S ∩∂T bounds a disk in S . Let A
σ σ σ
beanannulusembeddedin M withoneboundarycomponentequalto ω andtheother
boundary component l contained in S . Thisannulus exists because S and S bound
σ σ
a surface times interval. Anyloop in S ∩∂T must bound adisk in S since S ∩T
σ σ σ
doesnotcontainaloopwhichisessentialin S . Wemayisotope A sothat l isdisjoint
σ
from S ∩∂T since loops in S ∩∂T bound disks in S . The loop l is now either
σ σ σ
disjoint from T or contained in T since it is disjoint from ∂T. Since l is essential
in the 3-manifold M, l must be essential in S . Since S ∩T does not contain an
σ σ
essentialloopof S ,wemusthavethat l iscontainedin M\T. Wemayisotope ω by
σ
asmallisotopy sothat itiscontained intheinterior of T. Wenowhaveanannulus A
embedded in M suchthat ∂A = ω∪l with ω ⊂ T and l ⊂ (M \T). Thustheremust
beasimpleloop l in A∩∂T whichisessentialin A. Theembeddedannulusbounded
′
by l∪l can be used to isotope S so that S ∩T contains a loop which is essential
′ σ σ
andnon-meridinal in ∂T.
Wehaveshownthatwemayisotope S sothat S∩∂T containsaloopwhichisessential
andnon-meridinalin ∂T. Nowwecanlet N bearegularneighborhoodof γ contained
in T anddisjoint from ∂T. Since S∩∂T contains aloop whichisessential andnon-
meridinalin ∂T,thereisanembeddedannulusin M\N withboundary α∪α ,where
′
α is asimple essential non-meridinal loop in the boundary of N, and α iscontained
′
S. (cid:3)
10 WilliamBreslin
3 Finding an isotopy
Lemma5 Letγbeasimpleloopina3-manifoldM.LetM = H ∪ H beaHeegaard
1 S 2
splittingofM. Letα beasimpleessentialnon-meridinalloopintheboundaryofa
regularneighborhoodN ofγ.IfthereisanembeddedannulusA inM disjointfrom
theinteriorofN withboundaryα∪α whereα ⊂ S,thenγ isisotopicintoS.
′ ′
TheproofofLemma5isathinpositionargumentusedbySchultensin[9]toshowthat
exceptional fibers in Seifert manifolds are isotopic into a Heegaard surface (Lemma
4.1in[9]). SeeJohnson [2]forauseofthisargument toclassify genus-one Heegaard
splittings of lens spaces. Theonly adjustment needed for our result isthat the loop γ
canbeputintothinpositionwhilekeeping theannulus A embedded.
Proof Let f : M → [−1,1] beasmoothfunctionsuchthat f 1(−1) isaspineof H ,
− 1
f 1(1) isaspineof H ,and f 1(t) isasurfaceisotopic to S foreach t ∈ (−1,1). Let
− 2 −
g :A → M beanembeddingofanannulus A into M with ∂A = α∪α suchthat g(α)
′
is an essential non-meridinal loop in the boundary of the regular neighborhood N of
γ, g(α)∩N = ∅,and g(α) ⊂ S. Extend g toanimmersion (also called g)suchthat
′
g(α) = γn and g| isanembedding.
A α
\
If n = 1, then we are done, so assume that n ≥ 2. Let h = f ◦g : A → [−1,1].
Wemay assume that after a small isotopy h| is a Morse function and that h| has
A α α
\
no degenerate critical points. Since α is mapped to the level surface S, wemayalso
′
assumethatthesingular foliation F of A bylevelsetsof h consists ofparallel circles
in a neighborhood of α . Isotope γ and g so that γ is disjoint from the cores of the
′
handlebodies in M\S,whilekeeping g| anembedding.
A α
\
Thesingularfoliation F containsanessentialsaddleifitcontainsasaddlesingularity
x such that the four arcs in the level set containing x emanating from x end on α. If
x is an essential singularity, then x and the arcs emanating from x cut off three disks
from A. If none of the disks cut off by an essential saddle x and the arcs emanating
from x contain an essential saddle, then wecall x an outermost essential singularity.
If β isanoutermostlevelarcinthefoliation F of A,thenitcutsoffadisk D. Call D
anupper disk if h(x) > h(β) forall x ∈ D\β and call D alowerdisk if h(x) < h(β)
forall x ∈ D\β.
Supposethatalevelsetof F containsanoutermostessentialsaddlewhichsplitsoffan
upperdiskD betweentwolowerdisksD andD suchthatg(D ∩α)∩g(D ∩α)= ∅.
u 1 2 1 2
Note that since F consists of parallel circles in a neighborhood of α , we have that
′