Critical Heegaard Surfaces

28 Pages·0.31 MB·

by  David Bachman

CRITICAL HEEGAARD SURFACES
2
0 DAVID BACHMAN
0
2
Abstract. In this paper we introduce critical surfaces, which are described

n viaa1-complexwhosedefinitionisreminiscentofthecurvecomplex. Ourmain
a resultis that ifthe minimalgenus common stabilization ofa pairof strongly
J irreducibleHeegaardsplittingsofa3-manifoldisnotcritical,thenthemanifold

2 contains an incompressible surface. Conversely, we also show that if a non-
2 Haken 3-manifoldadmits at mostone Heegaard splitting of each genus, then
itdoesnotcontainacriticalHeegaardsurface. Inthefinalsectionwediscuss

] how this work leads to a natural metric on the space of strongly irreducible
T Heegaardsplittings,aswellasmanynewandinterestingopenquestions.
G
.
h
t
a
m 1. Introduction.
[
It is a standard exercise in 3-manifold topology to show that every manifold

1 admits Heegaardsplittings of arbitrarilyhighgenus. Hence, a “random”Heegaard
v splitting does notsaymuchaboutthe topologyofthe manifold inwhichit sits. To
3 useHeegaardsplittingstoproveinterestingtheorems,oneneedstomakesomekind
0
ofnon-trivialityassumption. Themostobvioussuchassumptionisthatthesplitting
2
is minimal genus. However, this assumption alone is apparently very difficult to
1
0 use.
2 In [CG87], Casson and Gordon define a new notion of triviality for a Heegaard
0 splitting, called weak reducibility. A Heegaard splitting which is not weakly re-
/
h ducible, then, is said to be strongly irreducible. The assumption that a Heegaard
t splitting is strongly irreducible has proved to be much more useful than the as-
a
m sumption that it is minimal genus. In fact, in [CG87], Casson and Gordon show
that in a non-Haken 3-manifold, minimal genus Heegaard splittings are strongly
:
v irreducible.
Xi The moral here seems to be this: since the assumption of minimal genus is
difficult to make use of, one should pass to a larger class of Heegaard splittings,
r
a which is still restrictive enough that one can prove non-trivial theorems.
Now we switch gears a little. It is a Theorem of Riedemeister and Singer (see
[AM90]) that given two Heegaard splittings, one can always stabilize the higher
genus one some number of times to obtain a stabilization of the lower genus one
(see the next section for definitions of these terms). However, this immediately
impliesthatanytwoHeegaardsplittingshaveacommonstabilizationofarbitrarily
highgenus. Hence, the assumptionthat one has a “random”commonstabilization
cannot be terribly useful. What is of interest, of course, is the minimal genus
Date:December1,2000and,inrevisedform,August6,2001.
Key words and phrases. Incompressible Surface, Heegaard Splitting, Stabilization, Curve
Complex.
1ToappearinTransactions oftheAMS.
1
2 DAVID BACHMAN
common stabilization. As before though, the assumption of minimal genus has
turned out to be very difficult to use.
In this paper, we propose a new class of Heegaard splittings, which we call
critical, and prove that at least in the non-Haken case, this class includes the
minimal genus common stabilizations. As one would hope, the assumption that a
splitting is critical is more useful than the assumption that it is a minimal genus
common stabilization.
We define the term critical via a 1-complex associated with any embedded, sep-
arating surface in a 3-manifold, which is reminiscent of the curve complex. After
the preliminary definitions we present a section of Lemmas which lead up to the
following Theorem:
Theorem 4.6. Suppose M is an irreducible 3-manifold with no closed incompress-
ible surfaces, and at most one Heegaard splitting (up to isotopy) of each genus.
Then M does not contain a critical Heegaard surface.
The remainder of the paper is concerned with the converse of this Theorem.
Thatis,we answerpreciselywhena (non-Haken)3-manifolddoes containacritical
Heegaard surface.
The main technical theorem which starts us off in this direction is:
Theorem 5.1. Let M be a 3-manifold with critical surface, F, and incompress-
ible surface, S. Then there is an incompressible surface, S′, homeomorphic to S,
such that every loop of F ∩S′ is essential on both surfaces. Furthermore, if M is
irreducible, then there is such an S′ which is isotopic to S.
As immediate corollaries to this, we obtain:
Corollary 5.6. If M #M contains a critical surface, then either M or M
1 2 1 2
contains a critical surface.
Corollary5.7. Areducible 3-manifold does notadmit acritical Heegaardsplitting.
Corollary 5.8. Suppose M is a 3-manifold which admits a critical Heegaard split-
ting, such that ∂M 6=∅. Then ∂M is essential in M.
It is this last corollary which we combine with a considerable amount of new
machinery to yield:
Theorem 7.1.′ Suppose F and F′ are distinct strongly irreducible Heegaard split-
tings of some closed 3-manifold, M. If the minimal genus common stabilization of
F and F′ is not critical, then M contains an incompressible surface.
We actually prove a slightly stronger version of this Theorem, that holds for
manifolds with non-empty boundary.
′
Compare Theorem 7.1. to that of Casson and Gordon [CG87]: If the minimal
genus Heegaard splitting of a 3-manifold, M, is not strongly irreducible, then M
contains an incompressible surface. In this light, we see that critical Heegaard
surfaces are a natural follow-up to strongly irreducible Heegaard surfaces.
In the last section, we show how these results lead to a natural metric on the
space of strongly irreducible Heegaard splittings of a non-Haken 3-manifold. It is
the belief of the author that more information about this space would be of great
interest. We also discuss how a better understanding of critical surfaces may help
answer several open questions, including the stabilization conjecture. We end with
CRITICAL HEEGAARD SURFACES 3
a few new questions and conjectures generated by this work which the reader may
find interesting.
The author would like to thank several people for their input during the prepa-
ration of this paper: the referee, for many helpful suggestions regarding earlier
versions; Cameron Gordon, for guidance during the author’s thesis work, from
which this paper grew; Saul Schleimer, for an introduction to the beauty of the
curve complex; and Eric Sedgwick, for helpful conversationsregarding Lemma 4.4.
2. Basic Definitions.
Inthissection,wegivesomeofthestandarddefinitionsthatwillbeusedthrough-
out the paper. The expert in 3-manifold theory can easily skip this.
A2-sphereina 3-manifoldwhichdoesnotbounda3-balloneither sideis called
essential. If amanifolddoesnotcontainanessential2-sphere,thenitis referredto
as irreducible.
A loop on a surface is called essential if it does not bound a disk in the surface.
Givenasurface,F,ina3-manifold,M,acompressing disk forF is adisk,D ⊂M,
such that F ∩D =∂D, and such that ∂D is essentialon F. If we let D×I denote
a thickening of D in M, then to compress F along D is to remove (∂D)×I from
F, and replace it with D×∂I.
Acompression bodyisa3-manifoldwhichcanbeobtainedbystartingwithsome
surface, F, forming the product, F ×I, attaching some number of 2-handles to
F ×{1}, and capping off any remaining 2-sphere boundary components with 3-
balls. The boundary component, F ×{0}, is often referred to as ∂ . The other
+
boundary component is referred to as ∂−. If ∂− =∅, then we say the compression
body is a handlebody.
A surface,F, in a 3-manifold, M,is a Heegaard splitting of M, if F separatesM
into two compression bodies, W, and W′, such that F = ∂ W = ∂ W′. Such a
+ +
splitting is non-trivial if neither W nor W′ are products. A stabilization of F is a
newHeegaardsplittingwhichisthe connectsumofthestandardgenus1Heegaard
splitting of S3 and F. Another way to define a stabilization is by “tunneling” a
1-handle out of W, and attaching it to ∂ W′. If this is done in such a way so as
+
to makethe definitionsymmetric in W andW′,then one arrivesata stabilization.
TheRiedemeister-Singertheorem(see [AM90])statesthatgivenanytwoHeegaard
splittings, F and F′, there is always a stabilization of F which is isotopic to a
stabilization of F′.
3. The isotopy-invariant disk complex.
Foreverysurface,F,wecandefineacomplex,G(F),asfollows: foreachisotopy
class of essential simple closed curve in F, there is a vertex of G(F). There is an
edge connecting two vertices if and only if there are representatives of the corre-
sponding equivalence classes that are disjoint. G(F) is the 1-skeletonof a complex
which is usually referred to as the curve complex of F, and is an object that many
mathematicians have made use of. We now generalize the 1-complex, G(F).
The new 1-complex we define here will be only for embedded, orientable, com-
pact, separating (but not necessarily connected) surfaces in 3-manifolds. Suppose
F is such a surface, separating a 3-manifold, M, into a “red” and a “blue” side. If
D and D′ are compressing disks for F, then we say D is equivalent to D′ if there
is an isotopy of M taking F to F, and D to D′ (we do allow D and D′ to be on
4 DAVID BACHMAN
opposite sides of F). Note that this equivalence relation is stronger than D and
D′ being isotopic in M (rel F). Indeed, if there is an isotopy which takes D to D′,
sweepingouta subsetB ⊂M, thenthere is aisotopy ofM,taking D to D′, which
fixes every point outside of a neighborhood of B.
We now define a 1-complex, Γ(F). For each equivalence class of compressing
disk for F, there is a vertex of Γ(F). Two (not necessarily distinct) vertices are
connected by an edge if there are representatives of the corresponding equivalence
classes on opposite sides of F, which intersect in at most a point. Γ(F) is thus an
“isotopy-invariantdisk complex” for F.
Forexample,ifF isthegenus1HeegaardsplittingofS3,thenthereisanisotopy
ofS3whichtakesF backtoitself,butswitchesthesidesofF. Suchanisotopytakes
a compressing disk on one side of F to a compressing disk on the other. Hence,
Γ(F)hasasinglevertex. However,therearerepresentativesoftheequivalenceclass
that correspondsto this vertex whichare on opposite sides of F, and intersectin a
point. Hence, there is an edge of Γ(F) which connects the vertex to itself.
We can now exploit this terminology to give concise definitions of some of the
standard terms in 3-manifold topology, as well as one important new one.
Definition 3.1. We say F is incompressible if Γ(F) = ∅. If an irreducible 3-
manifold contains no incompressible surfaces, then we say it is non-Haken. Other-
wise, it is Haken.
Definition 3.2. F is reducible if there are compressing disks on opposite sides of
F with the same boundary. The union of these disks is calleda reducing sphere for
F.
It is a standard exercise to show that if a Heegaard splitting of an irreducible
3-manifold is reducible, then there is a lower genus Heegaard splitting.
Definition 3.3. F isstronglyirreducibleiftherearecompressingdisksonopposite
sides of F, but Γ(F) contains no edges.
In [CG87], Casson and Gordon show that if F is a Heegaard splitting of a non-
Haken 3-manifold which is not strongly irreducible, then F is reducible. Hence,
any minimal genusHeegaardsplitting of a non-Haken3-manifoldmust be strongly
irreducible.
Definition 3.4. A vertex of Γ(F) is isolated if it is not the endpoint of any edge.
Definition 3.5. If we remove the isolated vertices from Γ(F) and are left with a
disconnected 1-complex, then we say F is critical.
Equivalently, F is critical if there exist two edges of Γ(F) that can not be con-
nected by a path.
4. Preliminary Lemmas
Before proceeding further, we introduce some notation. Suppose D and E are
compressing disks on opposite sides of F, such that |D∩E| ≤ 1. Then we denote
the edgeofΓ(F)whichconnects the equivalenceclassofD to the equivalenceclass
of E as D−E. If D and D′ are disks in the same equivalence class, then we write
D ∼D′. Hence, a chain of edges in Γ(F) may look something like
D −E ∼E −D ∼D −E −D
1 2 3 4 5 6 7
CRITICAL HEEGAARD SURFACES 5
Many of the proofs of this paper follow by producing such chains. Note that we
will always denote disks on the red side of F (i.e. “red disks”) with the letter “D”
(usually with some subscript), and blue disks with the letter “E”.
We begin with a simple Lemma.
Lemma 4.1. Let F be a critical surface, and suppose D −E and D −E are
1 1 2 2
edges which lie in different components of Γ(F). Then for each red disk, D, there
is an i=1 or 2 such that |D∩E|>1, whenever E is a blue disk equivalent to E .
i
Similarly, for each blue disk, E, there is an i such that |D∩E|>1, whenever D is
a red disk equivalent to D .
i
Proof. If the Lemma is false,then we havethe followingchainof edges: D −E −
1 1
D−E −D , whichclearlycontradictsthe assumptionthatthe edgesD −E and
2 2 1 1
D −E are in different components of Γ(F).
2 2
Note. Lemma 4.1is closelyrelatedto the “3-siteproperty”ofPittsandRubin-
stein (see, for example, [Rub96]). In our language, the 3-site property says that if
D −E andD −E areedgesindifferentcomponentsofΓ(F), thenthere cannot
1 1 2 2
be a third edge, D−E, where D misses E and E , and E misses D and D . It
1 2 1 2
is trivial to show that this is implied by Lemma 4.1.
The next Lemma is a nice self-contained result about the components of a dis-
connected critical surface. It is presented only to help the reader get used to our
definitions, and will not be used in the remainder of the paper.
Lemma 4.2. Suppose F is a critical surface such that every compressing disk for
every component of F is isotopic (rel ∂) to a compressing disk for F, and such that
each component of F is separating. Then either
• exactly one component of F is critical, and the rest are incompressible, or
• exactly two components of F are strongly irreducible, and the rest are incom-
pressible.
Note. It is possible to define the terms strongly irreducible and critical for non-
separatingsurfacesas well, whichmakes the statement ofLemma 4.2a little nicer.
We do not bother here, since we will not need such generality.
Proof. Suppose that D −E and D −E are edges in different components of
1 1 2 2
Γ(F). By Lemma 4.1, ∂D ∩∂E 6= ∅. Hence, D and E are compressing disks
2 1 2 1
for the same component of F, which we refer to as F . Similarly, D and E must
1 1 2
be compressing disks for the same component, F . Any compressing disk for any
2
othercomponentwouldviolateLemma4.1. Hence,anycomponentofF otherthan
F or F is incompressible.
1 2
Now suppose thatF =F . If D −E and D −E are in the same component
1 2 1 1 2 2
ofΓ(F ), thenthereis achainofedgesthatconnectsthem. Buteverycompressing
1
disk for F is a compressing disk for F, so the same chain would connect D −E
1 1 1
and D −E in Γ(F), a contradiction. Hence, F is critical.
2 2 1
If F 6= F , and F is not strongly irreducible, then let D −E be an edge of
1 2 2
Γ(F ). Then we have the chain:
2
D −E −D−E−D −E
1 1 2 2
Sincethisisacontradiction,Γ(F )containsnoedges,andF isstronglyirreducible.
2 2
A symmetric argument shows that F is strongly irreducible.
1
6 DAVID BACHMAN
For the proof of the next Lemma, we will need to define a partial ordering on
embedded surfaces in a 3-manifold. This ordering will play a more crucial role in
the next section.
Definition 4.3. Foranysurface,F,letc(F)= (2−χ(Fn))2,where{Fn}arethe
P
n
componentsofF. IfF andF denotecompact,embeddedsurfacesina3-manifold,
1 2
M, then we say F <F if c(F )<c(F ).
1 2 1 2
Note that this ordering is defined so that if F is obtained from F by a com-
1 2
pression, then F < F . In fact, we could have used any complexity we wished
1 2
(and there are several)which induced this partialordering. The one used here was
introduced to the author by Peter Shalen.
Lemma 4.4. Suppose F is a Heegaard splitting of an irreducible 3-manifold, M,
which does not contain any closed incompressible surfaces. If D and E are red
0 0
and blue disks such that D ∩E = ∅, then there is a reducing sphere for F which
0 0
corresponds to an edge of Γ(F) in the same component as D −E .
0 0
Proof. TheproofisarestatementofCassonandGordon’sprooffrom[CG87],which
says there exists some reducing sphere for F. By being careful, we can guarantee
that this reducing sphere corresponds to an edge of Γ(F) in the same component
as D −E .
0 0
Choose non-empty collections of red and blue disks, D and E, subject to the
following constraints:
1. For all D ∈D and E ∈E, D∩E =∅.
2. For all D ∈ D and E ∈ E, D − E is in the same component of Γ(F) as
D −E .
0 0
3. The surface, F , obtained from F by compressing along all disks of D and
0
E is minimal (in the sense of Definition 4.3), with respect to the above two
constraints.
Note that the existence of D and E is what guarantees that we may find such
0 0
collections, D and E, which are non-empty.
We claim thateachcomponentof the surface,F , is a 2-sphere. First, let F be
0 +
thesurfaceobtainedfromF bycompressingalongallthe disksofD,andletF− be
the surface obtained by compressing along all disks of E. Let M be the closure
+
of the component of M −F0 which contains F+, and let M− be the closure of the
other component. Notice that F+ and F− are Heegaardsplittings of M+ and M−.
If some component of F is not a 2-sphere, then it is compressible into M or
0 +
M− (since M contains no closed incompressible surfaces). Suppose the former.
Let W denote the closure of the component of M −F which contains F , and
+ + 0
let W′ denote the other component. By the Lemma of Haken (see, for example,
[CG87]),thereisacompletecollection,E′,ofcompressingdisksforF inW,anda
+
compressingdisk,Dˆ,forF inM ,suchthatDˆ∩F isasingleloop,andDˆ∩E′ =∅
0 + +
for all E′ ∈E′. Notice that Dˆ contains a compressing disk, Dˆ′, for F in W′. But
+
since F is obtained from F by compression along D, we can consider Dˆ′ and E′
+
to be compressing disks for F, which are disjoint from all D ∈D.
Now, we replace the original collections, D and E, with the collections D′ =
D∪Dˆ′ and E′. By construction, for any D′ ∈ D′ and E′ ∈ E′, D′∩E′ = ∅. For
any E′ ∈E′ and D ∈D (and hence in D′), D−E′ is an edge in Γ(F) which is in
thesamecomponentasD−E,foranyE ∈E,andhenceinthesamecomponentas
CRITICAL HEEGAARD SURFACES 7
D −E . Furthermore,compressingF along alldisks of D′ andE′ yields a surface
0 0
that can be obtained from F by compressing along Dˆ′. But this surface is then
0
smaller than F , contradicting our minimality assumption. We conclude that F is
0 0
a collection of 2-spheres.
TheremainderoftheprooffollowsthatofRubinsteinfrom[Rub96]. Noticethat
F consistsofasubsurface,P,ofF,togetherwithtwocopiesofeachdiskinDand
0
E. These disks are colored red and blue, so we can picture F as a collections of
0
spheres with a bunch of red and blue subdisks. Some sphere in this collection, F′,
0
musthavebothredandbluesubdisks,becauseotherwiseF wouldnotbeconnected.
Letγ be a loopon F′ whichseparatesthe reddisks fromthe blue disks. Note that
0
γ ⊂ P ⊂ F. Let R be the component of F′ −γ which contains the red disks, and
0
let B be the other component. Then we can push P ∩R slightly into the red side,
and P ∩B into the blue side. This turns F′ into a sphere that intersects F in the
0
single essential simple closed curve, γ. Furthermore, R∩E = ∅ for all E ∈ E, so
the edge R−B is in the same component of Γ(F) as D −E .
0 0
Lemma 4.5. Suppose F is a Heegaard splitting of an irreducible 3-manifold, M,
which does not contain any closed incompressible surfaces. For each edge, D−E
of Γ(F), there is some edge, D′−E′, in the same component as D−E, such that
|D′∩E′|=1.
Proof. Since D−E is an edge of Γ(F), either |D∩E| = 1, or D∩E = ∅. In the
former case there is nothing to prove. In the latter case, we may apply Lemma 4.4
to obtain an edge, D′′−E′′, in the same component as D−E, such that D′′∪E′′
is a reducing sphere for F. By the irreducibility of M, D′′∪E′′ bounds a ball, B.
By [Wal68], we know that F ∩B is standard. So inside B we can find a pair of
disks onopposite sides, D′ and E′, such that |D′∩E′|=1. But then D′∩E′′ =∅,
so the chainD′′−E′′−D′−E′ implies D′−E′ is in the same component of Γ(F)
as D−E.
Theorem 4.6. Suppose M is an irreducible 3-manifold with no closed incompress-
ible surfaces, and at most one Heegaard splitting (up to isotopy) of each genus.
Then M does not contain a critical Heegaard surface.
Examplesofsuch3-manifoldsincludeS3andB3[Wal68],L(p,q)[BO83],[Bon83],
and handlebodies [Wal68].
Proof. LetF beaHeegaardsplittingofM,andletD−E andD′−E′ denoteedges
of Γ(F). Our goal is to show that these two edges lie in the same component. By
Lemma 4.5, we may assume that |D∩E|=|D′∩E′|=1. As F cannotbe a torus,
the boundary ofa neighborhoodofD∪E is a sphere whichintersectsF ina single
essentialsimple closed curve. This curve divides the sphere into a red disk, R, and
a blue disk, B, such that ∂R = ∂B. Similarly, D′∪E′ gives rise to disks, R′ and
B′, such that ∂R′ =∂B′.
Now, ∂R separates F into two components, at least one of which, T, is a punc-
tured torus. Furthermore, (F −T)∪R is a Heegaardsurface of lower genus. Simi-
larly,∂R′separatesF intotwocomponents,atleastoneofwhich,T′,isapunctured
torus, and (F −T′)∪R′ is a Heegaard surface of lower genus.
By assumption, (F −T)∪R is isotopic to (F −T′)∪R′. This isotopy can be
realizedasanisotopyofM,whichtakes(F−T)∪Rto(F−T′)∪R′. Furthermore,
8 DAVID BACHMAN
since B and B′ are isotopic to R and R′, we may assume that the isotopy of M
takes B to B′, as long as B and B′ do not end up on opposite sides. Since T and
T′ lieinsidetheballsboundedbyR∪B andR′∪B′ (andinfact,T∪RandT′∪R′
are Heegaard splittings for these balls), we can also arrange the isotopy so that it
takesT toT′. WhatwehavenowproducedisanisotopyofM,takingF toF,and
R to R′. This implies the following chain:
′ ′ ′ ′
E−D−B−R∼R −B −D −E
If, onthe otherhand, the isotopyofM whichtakes(F−T)∪Rto(F−T′)∪R′
“switchessides”,sothatB andB′ enduponoppositesides,thentheremustbe an
isotopywhichtakes(F−T)∪Rto(F−T′)∪B′,and(F−T)∪B to(F−T′)∪R′.
Hence,weconcludethatthereisanisotopyofM takingF toF,andRtoB′. This
implies the chain:
′ ′ ′ ′
E−D−B−R∼B −R −D −E
5. Incompressible Surfaces
Ourgoalinthis sectionisto examinethe interplaybetweencriticalsurfacesand
incompressible surfaces. Our main result is the following:
Theorem 5.1. Let M be a 3-manifold with critical surface, F, and incompress-
ible surface, S. Then there is an incompressible surface, S′, homeomorphic to S,
such that every loop of F ∩S′ is essential on both surfaces. Furthermore, if M is
irreducible, then there is such an S′ which is isotopic to S.
Proof. The proof is in several stages. First, we construct a map, Φ, from S ×D2
into M. We then use Φ to break up D2 into regions, and label them in such a
way so that if any region remains unlabelled, then the conclusion of the theorem
follows. Finally, we construct a map from D2 to a labelled 2-complex, Π, which
has non-trivial first homology, and show that if there is no unlabelled region, then
the induced map on homology is nontrivial, a contradiction. This general strategy
is somewhat similar to that used in [RS96], although the details have very little in
common.
Stage 1: Constructing the map, Φ:S×D2 →M.
Let D −E and D −E denote edges in different components of Γ(F), such
0 0 1 1
that D ∩E = D ∩E = ∅. Such edges are guaranteed to exist in every com-
0 0 1 1
ponent. Indeed, if D−E is an edge where |∂D∩∂E| = 1, then the boundary of
a neighborhood of D ∪E is a sphere, which intersects F in a single loop. This
loop cuts the sphere into disks, D′ and E′, which can be isotoped to be disjoint.
Furthermore,wehavethechainD′−E′−D−E inΓ(F),whichshowsthatD′−E′
is in the same component as D−E.
We now produce a sequence of compressing disks for F, {Di}i=1,...,n−1, such
n
that D ∩D =∅, for all i between 0 and n−1. Begin by isotoping D and D
i i+1 0 1
n n
so that |D ∩D | is minimal. If D ∩D =∅, then we are done. If not, then let γ
0 1 0 1
be some arcofintersectionwhichis outermostonD . So,γ cuts off asubdisk, D′,
1 1
ofD ,whoseinterioris disjointfromD . γ alsocutsD intotwosubdisks. Choose
1 0 0
one to be D′. Since |D ∩D | is assumed to be minimal, the disk D1 = D′ ∪D′
0 0 1 1 0
must be a compressing disk for F. Furthermore, D1 can be isotoped so that is
CRITICAL HEEGAARD SURFACES 9
misses D , and so that |D1 ∩D | is strictly less than |D ∩D |. Continue now
0 1 0 1
in the same fashion, using D1 and D to construct D2, where D1∩D2 = ∅, and
1
|D2∩D |<|D1∩D |. Eventually,wecometoadisk,Dn−1,whereDn−1∩D =∅.
1 1 1
Now, for all i between 1 and n−1, let D =Di, and we are done.
i
n
Wecanapplyasymmetricargumenttoproduceasequenceofdisks,{Ei }i=1,...,m−1,
m
such that E ∩E =∅, for all i between 0 and m−1.
i i+1
m m
Our goalnow is to use allofthese disks to define Φ. The definition we givemay
seem overly technical, but there are certain features that are worth pointing out
before-hand. If Φ is any mapfromS×D2 into M, then letΦ (s)=Φ(s,x). If x is
x
thecenterofD2,wewouldlikeΦ tobetheidentityonS. Thatis,atthecenterof
x
D2,wedonothingtoS. ForxneartheboundaryofD2,wewouldlikeΦ (S)tobe
x
disjoint fromat least one of the disks, {D }, or {E }. In particular,there should
i i
n m
be two points, θ and θ on ∂D2 such that for all x∈D near θ , Φ (S) is disjoint
0 1 i x
from both D and E , for i = 0,1. In fact, any map Φ which fits this description
i i
will work for the remainder of our proof. We give an explicit construction of one
here for concreteness.
Let {U } and {V } denote neighborhoods of the disks, {D } and
i i=0,...,n i i=0,...,m i
n
{E }, such that
i
m
1. U ∩U =∅, for 0≤i<n
i i+1
2. V ∩V =∅, for 0≤i<m
i i+1
3. U ∩V =∅
0 0
4. U ∩V =∅
n m
Foreachibetween0andn,letγi :M×I →M beanisotopy,suchthatγi(x)=x
0
forallx∈M,γi(x)=xforallt,andallxoutsideofU ,andγi(S)∩D =∅(where
t i 1 i
n
γi(x) is short-hand for γi(x,t)). In other words, γi is an isotopy which pushes S
t
off of D , inside U . Similarly, for each i between 0 and m, let δi be an isotopy
i i
n
which pushes S off of E , inside V .
i i
m
Now choosen+m+2points onS1, whicharelabelled andcyclicallyorderedas
follows: x ,...,x ,y ,...,y . Let {f } and {g } be sets of continuous
0 n m 0 i i=0,...,n i i=0,...,m
functions from S1 to I, such that
1. supp(fi)=(xi−1,xi+1), for 0<i<n,
2. supp(gi)=(yi+1,yi−1), for 0<i<m,
3. supp(f0)=(y0,x1), supp(fn)=(xn−1,ym),
4. supp(g0)=(y1,x0), supp(gm)=(xn,ym−1),
5. for each θ ∈S1, there is an i such that f (θ)=1 or g (θ)=1,
i i
6. there are points, θ ∈(y ,x ) and θ ∈(x ,y ), such that f (θ )=g (θ )=
0 0 0 1 n m 0 0 0 0
f (θ )=g (θ )=1.
n 1 m 1
The existence of such functions is easily verified by the reader.
Finally, we define Φ:S×D2 →M. For each x∈S and (r,θ)∈D2, let
n m
Φ(r,θ)(x)=Yγrifi(θ)(x)Yδrigi(θ)(x)
i=0 i=0
Note that it is a consequence of how we defined the functions {f } and {g }
i i
that for each value of θ there are at most two functions that are not the identity.
Since these two functions will have support on disjoint subsets of M, the order of
the above product does not matter. For example, suppose n = 7. Let θ′ be some
10 DAVID BACHMAN
value of S1 between x and x . Then f and f are the only functions that can be
3 4 3 4
non-zero at θ′. Hence, the above product simplifies to:
Φ(r,θ′)(x)=γr3f3(θ′)(x)·γr4f4(θ′)(x)
But γ3 is the identity outside U , γ4 is the identity outside U , and U ∩U = ∅.
3 4 3 4
Hence, γ3 and γ4 commute.
We now perturb Φ slightly so that it is in general position with respect to S,
and again denote the new function as Φ. Consider the set Σ = {x ∈ D2|Φ (S) is
x
nottransversetoF}. IfΦisingeneralpositionwithrespecttoS,thenCerftheory
(see[Cer68])tellsusthatΣishomeomorphictoagraph,andthemaximumvalence
of each vertex of this graph is 4. We will use these facts later.
Stage 2: Labelling D2.
Let C and C denote sets of components of Γ(F) such that
0 1
1. any isolated vertex is in both sets,
2. acomponentwhichisnotanisolatedvertex(i.e. itcontainsatleastoneedge)
is in exactly one of the sets, and
3. the component containing the edge D −E is an element of C , for i=0,1.
i i i
Note: The salient feature of these sets is that if v is a vertex of Γ(F) in a compo-
nent, A ∈ C , and v′ is some other vertex, in a component, B ∈ C , then A 6= B.
0 1
Hence, v−v′ cannot be an edge of Γ(F). We may conclude then that if D is a red
disk representing v, and E is a blue disk representing v′, then D∩E 6=∅. We will
make extensive use of these facts.
As at the end of Stage 1, let Σ = {x ∈ D2|Φ (S) is not transverse to F}. A
x
region of D2 is a component of D2−Σ. Let x be any point in the interior of some
region. Labelthisregionwitha“D ”(“E ”)ifthereisadisk,∆,inΦ (S)suchthat
i i x
∂∆ is an essential loop on F, int(∆)∩F is a (possibly empty) union of inessential
loops on F, and ∆ is isotopic rel ∂ to a red (blue) compressing disk for F which
corresponds to a vertex of Γ(F) in C . It is possible that some regions will have
i
morethanonelabel. Forexample,ifthereisareddiskinΦ (S)whichcorresponds
x
to an isolated vertex of Γ(F), then it will have both of the labels “D ” and “D ”.
0 1
Note: We willbe consistentaboutusingquotationmarksto denote labels. Hence,
the label “D ”, for example, should not be confused with the disk, D .
0 0
ThepointofourlabellingisthatifthereisanycompressingdiskforF contained
inΦ (S),thentheregioncontainingxwillbeassignedalabel. Hence,theexistence
x
of an unlabelled region would imply that all components of intersection of Φ (S)
x
andF are either essentialonboth surfaces,orinessentialonboth. Fromthere it is
easy to prove the Theorem. We now proceed under the assumption that there are
no unlabelled regions.
Claim 5.2. No region can have both of the labels “Di” and “E1−i”.
Proof. Let x be a point in the interior of a region with the labels “D ” and “E ”.
0 1
Thenthereisareddisk,D,andabluedisk,E,inΦ (S). Furthermore,accordingto
x
ourlabelling,theequivalenceclassofD representsavertexofΓ(F)inacomponent
A ∈ C . Similarly, [E] ∈ B, for some B ∈ C . But since D and E are both
0 1
subsets of Φ (S) they must be disjoint, a contradiction (see the above note after
x
the definitions of C and C ).
0 1