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DATE DUE DATE DUE I DATE DUE
JEN—”73W 4W.
3 I 1 0 8

 

act 33 :1 "2085?

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1M W14

A DECOMPOSITION OF SMOOTH SIMPLY-CONNECTED h—COBORDANT
FOUR-MANIFOLDS
by

Rostislav Matveyev

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for degree of

DOCTOR OF PHILOSOPHY
Department of Mathematics

1997

ABSTRACT

A DECOMPOSITION OF SMOOTH SIMPLY-CONNECTED h-COBORDANT
FOUR-MANIFOLDS
by

Rostislav Matveyev

In [A] S. Akbulut found the simplest possible example of a non-product h—co-
bordism. It is relative to the boundary cobordism A of two compact contractible
4-manifolds, which are diffeomorphic to each other (not relative to the boundary).
Akbulut found his example as a subcobordism in a bigger non-product h-cobordism
W of two closed manifolds homotopy equivalent to K3 surface blown up once. Closure
of the complement of A in W smoothly has a product structure.

In my dissertation I prove a theorem, which generalizes example of Akbulut. It
states that given a (4+1)-dimensional smooth simply-connected h-cobordism, one can
find a subcobordism between two compact, contractible, diffeomorphic (not relative
to the boundary) manifolds, so that there is a product structure on the closure of
the complement of this subcobordism. This, for example, implies that, given two
homotopy equivalent, smooth, compact, simply-connected, not diffeomorphic mani-
folds, one can cut off a compact contractible submanifold from one of them and reglue
it back via some diffeomorphism of the boundary to get another one. Another im-
plication is that any four-dimensional homotopy sphere is a twisted double of some

compact contractible manifold.

ACKNOWLEDGMENTS

I would like to express my deep gratitude to my teacher Selman Akbulut who
guided me in the wonderful world of 4-dimensional t0pology. During the last three
years I have experienced his continuous guidance and encouragement. In the process
of writing my dissertation I have received many helpful hints and ideas from Professor
Akbulut. I’m grateful for his generosity in sharing these with me.

While working on my dissertation I had the opportunity to meet a number of
mathematicians. Many interesting and stimulating conversations took place. I would
like to thank Weimin Chen, Tedi Draghici and Gregory Mikhalkin for their interest

and helpful suggestions.

iii

TABLE OF CONTENTS

List of Figures .................................... v
List of Tables .................................... vi
Introduction ..................................... 1
1. Statement of the Main Theorem ........................ 3
2. Proof of the First Part of the Theorem ..................... 5
3. Proof of the Second Part of the Theorem ................... 11
4. Proof of Lemma 1 ................................ 17

iv

LIST OF FIGURES

Disk K may intersect both S... and P... .................. 8
Boundaries of discs X,- and Y]- are not necessarily disjoint ....... 8
Making contractible parts in decompositions diffeomorphic ...... 11
Handlebody of the neighborhood of wedges of spheres ......... 12
Finger move ................................ 12
Cusp move ................................. 13
Handlebody of W1 ............................ 14

LIST OF TABLES

1 Handles of W1 ............................... 15

vi

Introduction

One of the main tools of classification up to diffeomorphism of higher—dimensional
smooth manifolds is an s-cobordism theorem, which states that smooth s-cobordism
W between two smooth closed manifolds M1 and M2 of dimension grater then or
equal to 5 is diffeomorphic to a product cobordism M1 x I and hence manifolds M1
and M2 are diffeomorphic. If we restrict our attention to simply-connected manifolds
then corresponding theorem is called h-cobordism theorem. It turns out that both
s-cobordism and h-cobordism theorems fail for manifolds of dimension 4. This failure
is one of the reasons of mysterious behavior of smooth 4-manifolds.

First examples of s-cobordant and, in fact, homeomorphic, but not diffeomorphic
manifolds of dimension 4 were discovered by Cappel and Shaneson in [CS], see also [F]
for the proof that exotic manifold found by Cappel and Shaneson is actually homeo-
morphic to RP“; first simply-connected examples were found by S. Donaldson in [D].
Since then great many pairs of simply-connected h-cobordant but not diffeomorphic
manifolds were found using invariants defined by S. Donaldson and Seiberg-Witten
invariants.

Even before first counterexample to h-cobordism theorem was found, it was known
that such counterexample would imply existence of exotic R4, i.e. open smooth 4-
manifold which is homeomorphic but not diffeomorphic to Euclidean space of dimen-
sion 4. In fact, it was shown that any simply-connected h-cobordism can be split
into two sub-cobordisms between open manifolds, one has a product structure, and
another isan h-cobordism between two fake R4’s. R. Gompf and Z. Bizaca build very
simple (in terms of handlebody) R4(see [BGD starting with a non-trivial h-cobordism

between two K3 surfaces.

In [A] S. Akbulut obtained an example of simplest possible non-trivial h—cobor-
dism. It is a relative to the boundary cobordism between two compact contractible
manifolds and it was found as a subcobordism in a bigger cobordism between two
closed manifolds homotopy equivalent to K3 surface blown up once. In Akbulut’s
example that bigger cobordism splits into two parts: one is a product cobordism,
another is a non-trivial cobordism relative to the boundary between two compact
contractible 4-manifolds. Moreover, these two manifolds are diffeomorphic to each
other, but they are not diffeomorphic relative to the boundary.

In my thesis I prove that any h-cobordism is a union of two compact sub-cobor-
disms, one of which has a product structure and another is an h-cobordism between
two compact contractible manifolds. In addition, I show that this two manifolds are
diffeomorphic. This result generalizes the example of Akbulut.

Similar result has been obtained by C.L. Curtis, M.H. Freedman, W.C. Hsiang
and R. Stong in [CH0, CH1, CHFS]. Namely, they have shown that any two smooth
simply-connected h-cobordant four-manifolds become diffeomorphic after removing
compact contractible submanifold from each of them; moreover, remaining pieces are
also simply-connected. It is not hard to see that their argument to show simply-
connectedness can be merged with the proof below, thus proving that fundamental

group of manifold M (see main theorem below) is trivial.

1. Statement of the Main Theorem

Throughout these notes all maps and manifolds are smooth and immersions are
in general position (or do their best if they have to obey some extra conditions). We
also make the convention that if a star appears in place of a subindex, we consider a
union of all objects in the family, where the index substituted by the star runs over
its range. For example, D. (i—Sf U, 0,.

Following is the statement of decomposition theorem we prove.

Theorem. Let U be a smooth, 5-dimensional, simply-connected h-cobordism with

EU = M1 1.1 (—Mg). Let f : M1 —+ M2 be the homotopy equivalence induced by U.
1. There are decompositions
M1 = M#3W1, M2 = M#2W2

such that ing,l o in1. 2 f, : H2(M1) —> H2(M2). Here in2,, inl. are the
maps induced in the second homology by embeddings of M into M1 and M2

respectively, and W1, W2 are smooth, compact, contractible 4-manifolds, and

2. These decompositions may be chosen so that W1 is difieomorphic to W2.

In fact, it can be seen from the proof that the whole cobordism can be decomposed
into two subcobordisms, one is a product cobordism and one is diffeomorphic to D5
(as a smooth manifold, without any additional structure).

We will also need the following

Definition. We say that two collections {Si}?=1, {13,-}?=1 of oriented 2-spheres im-

mersed in oriented 4-manifold are algebraically dual if ([S,], [P,-]) = 6,]; Here, [S] is a
3

homology class of immersed sphere S and ( , ) is the intersection form in the second
homology of 4-manifold.
They are geometrically dual if they are algebraically dual and, moreover, card(S,~ 0

P1) = 51'1-

Note that is {5,} is a collection of immersed spheres in simply-connected manifold
X, such that it has geometrically dual collection, then the complement of the first set

of spheres X \ S. is also simply-connected.

2. Proof of the First Part of the Theorem

We start with studying a handlebody decomposition of cobordism U. Observe
that U has a handlebody with no 1- and 4-handles. Let N be the middle level of U
between 2- and 3-handles. Then we have two diffeomorphisms

$01 : M1#Sf1 x Sf2# . . . #5311 x 332 ——> N
902 : ngtyéSfl x 5122#...#S,2u x 8,2,2 —> N.

By enriching the handlebody of U by 2—3 canceling pairs of handles and choosing
cpl and 902, we can assume that (cpgl o 301).|H2(M,) = f. and («pgl o (p1).[SfJ-] == [33,-],
i = 1,... ,n, j := 1,2. See [W1, W2]. Then the two embeddings c1 : U331 V 5,22 ——>
N, c2 : U531 V SE2 —> N of disjoint unions of wedges of 2-spheres, representing
cores of products of spheres in decompositions «p1 and 992, are homologous and hence
homotopic, for 7r1 (N) = {1}. We have two algebraically dual collections of embedded

2-spheres {8,- = c1(S,-21) C N}?:1, {P.- = c2(S,-22) C N}?=1, and surgery along {R} gives

 

M1; along {3,} gives M2. Put V0 2 NdN(S. U R), the closed regular neighborhood
of S. U P. in N.
Here we give a sketch of the rest of the construction for the proof of the first part

of the theorem and than work out the details.

1. Manifold V0 has a free fundamental group and its group of second homology
is generated by classes of spheres S.- and 13,-, i = 1,... ,n. We enlarge V0
to obtain the bigger manifold V1 so that collections of spheres {3,} and {P,-}
have geometrically dual collections of immersed spheres inside of V1 and the

properties mentioned above are satisfied.

2. We add 1-handles and essential 2-handles to V1 so that the fundamental group

vanishes and no new two dimensional homology classes appear.

5

3. Surgery of the manifold obtained in step 2 along collections {5,} and {P,-} gives
two contractible submanifolds W1 and W2 of M1 and M2, respectively, and

M1\W1§M2\W2.

The intersection points of embedded spheres S, and P,- can be grouped in pairs
of points of opposite sign with one extra point of positive sign when i = j, which we
refer to as wedge points. Consider a disjoint collection of Whitney circles l1, l2, . . . ,lm
in S. U P., one for each pair of points considered above. Push these circles to the
boundary of the neighborhood of S. U P. in N and call this new circles 1’1, l'2, . . . ,l;
as in. Each of l: is contractible in N \ S., as well as in N \ P., since the latter two are
simply-connected manifolds. Thus, for each I,- we can find two immersed disks D,- and
E,- with 6D,- : 6E, = li, coinciding along the collar of the boundary, and satisfying

the following condition:
13.08.:0, E.flP.—~=0 (1)

were [0) stands for interior of disc D.

Now we shall show that it is possible to find disks D; and E,’ so that they are
homotopic rel(collar of the boundary) and the condition similar to ( 1) is satisfied.

The union of D.- and E.- with appropriate orientations gives us a class [D.- U E,]
in the group of the second homology of N, which splits into the direct sum H2(N) =
H2(M1)€B < [8,], [P1] >. So, we can write [Di UEi] = a+ Zflk[Sk] + 27,.[Pk], where
a E H2(M1). Using the diffeomorphism (p1 and the fact that 7r1(M1) = {1} we can
realize class a, considered as a class in H2(N), by an immersed sphere A in N disjoint
from S... Classes 2 [3,.[Sk] and Zyk[Pk] can be realized by the immersed spheres B
and C in N disjoint from S. and P., respectively. Now taking the connected sums
D; = D.#(—A)#(—B), E: = E,#(—C) (‘—’ here denotes reversal of orientation)

ambiently along carefully chosen paths, we obtain discs D; and E,’ satisfying property

similar to (1) and homotopic rel(collar of the boundary). Here we again use 7r1(N) =
{ 1}.

Consider homotopy Ft : L] D? —-> N rel(collar of the boundary), where F0([_] D?) =
D; and F1(|_] D?) 2 E. It can be also viewed as a homotopy of the union of the

disks and spheres S. and P., where the spheres stay fixed during the homotopy.

Lemma 1. Homotopy F can be perturbed, firing ends, S. U P. and the collar of
the boundary of disks in the disks, to homotopy F’ satisfying the following property:
F [[0, % 1 can be decomposed in a sequence of simple homotopies each of which is either
a cusp or finger move; analogously, F |[ % ,1] can be decomposed in a sequence of inverse

cusp moves and Whitney tricks.

For definitions of cusp, finger move, inverse cusp move and Whitney trick see [FQ,
K, GM].

Proof of Lemma 1 is given in the last section.

Assume, now, that F has the property provided by Lemma 1 above. Consider
K. = U. K,- = F% (|_] DE). Each of K,- may intersect both S. and P., but it is ho-
motopic rel(collar of the boundary) to the disk with interior disjoint from 8., via a
homotopy decreasing number of intersection points. This means that the intersection
points of K. and S. can be grouped in pairs so that for each pair there is an embed-
ded Whitney disk X). with interior disjoint from the rest of the picture. The same
argument shows that there are embedded disks Y. for intersection points of K. and
R, see Figure 1.

Using embedded disks from collection {X k} we can push all disks K,- off S., pro-
ducing the new disks K]. Each K: has an interior disjoint from S. and its boundary is

a Whitney circle for a pair of points of opposite sign in S. flP.. Applying an immersed

 

Figure 1: Disk K may intersect both S. and P.

Whitney trick to P. along K L we eliminate all the intersections with 5., except those
required by algebraic conditions and produce the collection {Sf} geometrically dual
to {3.}. Note that, since disks K ,’ are, in general, immersed and may have non-trivial
relative normal bundle in N, we may have to introduce (self-)intersection points to
spheres {Sf} during this process. In the same way, using disks from collection {Y,-}
we can obtain the immersed Whitney disks K,” disjoint from P... They allow us to
create a collection {Pf} of immersed spheres dual to {P,}. Note, that all described

homotopies are supported in a regular neighborhood of K. U X. U Y...

 

 

Figure 2: Boundaries of discs X.- and Y]- are not necessarily disjoint

 

Thus, one can see that the manifold V1 = VoUNdN(K. U X. U Y.) has the following

properties:

1. H2(V1) is generated by homology classes of embedded spheres S,, P,.

2. The collection of spheres {3,} has the geometrically dual collection {Sf} of
immersed spheres in V1. Similarly, collection of spheres {P,-} has geometrically

dual collection {Pf}.
3. 7r1(V1) is a free group.

Consider a handlebody of N starting from V1. Put V2 2 V1 U (union of l-handles).
Manifold V2 still satisfies properties similar to (1), (2), (3) for V1.

Let g1, . .. ,g, be free generators of 1r1(V2). If we fix paths from a basepoint
to attaching circles of 2-handles in the handlebody of N, then these circles rep-
resent elements of «1(8V2), say h1,... ,hL. Since V2 U (all 2-handles) is a sim-
ply connected manifold, each g. has a lift g.- in 7r1(8V2), such that it belongs to
the normal subgroup of 7r1(6V2) generated by h1,... ,hL. In other words g.- =
(alhiillafl)(agh:1a;1)...(akhfila;l). Duplicating 2—handles, equipping them with
appropriate orientations and choosing paths joining attaching circles of 2-handles to
the basepoint we may write g,- = h’lh’2 . .. 3,, where h’l, ’2, . .. ,h]c are elements of
«1(0V2) obtained from attaching spheres of new handles with new paths to the base-
point. Now, handles in the decomposition of g",- are all distinct and we can slide the
first handle over all others along paths joining feet of the handles to the basepoint.
The attaching circle of the resulting handle, call it G., is freely homotopic to g,- in V2.
Adding such Gi’s to V2 for each generator of 7r1(V2) we obtain the simply-connected
manifold V3. Since g1, . . . , g, are free generators of the fundamental group, we do not
create any additional two-dimensional homology classes.

Surgery of V3 along collections of embedded spheres {Si} and {R} gives two con-

tractible sub-manifolds W1 and W2 of M1 and M2, respectively. Put M 9—3! N \ V3 C—‘_-’

10

 

M1 \ W1 ’—_‘-’ M2 \ W2. So we have the decompositions:

M1 = M#2W1, M2 = M#2W2

The property of induced maps in the homology stated in Theorem is obvious. The

first part of Theorem is proved.

 

3. Proof of the Second Part of the Theorem

Proof of the second part is based on the following Lemma 2.

Lemma 2. If W1, W2 are homotopy balls built in the proof of the first part of

Theorem and S4 is a 4-dimensional sphere with standard smooth structure, then

W1#2W1 '5 54, W1#2W2 9% 34-

Denote the boundary connected sum by ‘h’. Then we can write
M1 = M#2W1 '5 (M#2W1)#(W1#2W2) '5’ (MUWI)#2#2(W1UW2)
and
M2 = M#2W2 95 (M#2W2)#(W1#2W1) g (MUW1)#2#2(W2UW1)-

Decompositions above are shown on Figure 3.

 

 

 

 

 

\Ezfiéfl/

Figure 3: Making contractible parts in decompositions diffeomorphic

In order to proof Lemma 2 we will use the art of Kirby calculus.
First, we build a handlebody of V3.
Consider Vo’, a closed neighborhood of S. U P. U (arcs joining wedge points in

S,- 0 P.- and S.“ (1 PM)- If there were no intersections between S. and P., except a

11

12

wedge point for each pair 5., P., then the handlebody would look like as shown on

DOC. COO...

Figure 4: Handlebody of the neighborhood of wedges of spheres

Figure 4.

Introducing a pair of intersection points of opposite signs corresponds to the move

in Kirby calculus shown in Figure 5.

 

Accessory
:c‘\| circle
One of One of "'"'1
S ’8 P’s or K’s
/'\/, [-— 7 Whitney
J] circle

 

 

Pair of intersection

point. of oppooite sign J

 

 

 

Figure 5: Finger move

Here, we introduce two 1-handles, one corresponding to a Whitney circle, another
to an accessory circle of a newly introduced pair of intersections. The handlebody
of V}; is obtained by applying several moves shown in Figure 5 to the picture in
Figure 4. Again, if disks K,- were embedded and disjoint from S. U P., then to obtain
the handlebody of V}, U Nd_(K.—) one has to attach 2—handles to Whitney circles for
each pair of intersection points of S. and P.. Then we have to introduce intersections
between K . and S. UP. and self-intersections of K .. Corresponding moves are shown
on Figures 5 and 6.

Introducing a self-intersection corresponds to adding one l-handle to the picture,

as shown on Figure 6. Addition of disks from collections {X,}, {Y,} corresponds

13

One of
S’s

   
  

’— ‘1"

‘ Whitney
~-‘ circle

/\/

Selfintersection
point

Figure 6: Cusp move

to attaching 2—handles to the circles linked once to the 1—handles corresponding to
the Whitney circles, and unlinked from other 1-handles. They may link each other
according to the fact that boundaries of X’s and Y’s are not necessarily disjoint. This
phenomenon is illustrated on Figure 2. Then, as in previous steps, add 1-hand1es to
the picture reflecting intersections of X. and K, as in Figure 7.

To obtain the handlebody of V2 we have to attach several 1-handles. The link on
Figure 7 shows all possible phenomena which can occur. Thus, the handlebody of V2

has 1-handles of six types coming from

1. Whitney circles of intersections of S. and P.;

N

. accessory circles of intersections of S. and P..;

00

. Whitney circles of intersections of K. and S. U P.;

A

. accessory circles of intersections of K. and S. U P..;

5. Whitney and accessory circles of self-intersections of K .;

6. Extra 1-handles.

For each 1-handle of type 1 and 3 there is a 2-handle attached to the circle linked to

this 1-handle geometrically once and algebraically zero times to other 1-handles.

14

Attaching circles of
extra 2-handles
up to homotopy in 6V2

 

 

 

 

  

Attaching circles of
extra 2—handles
up to homotOpy in V2

 

Figure 7: Handlebody of W1

The attaching circles hl, . . . ,h; of other 2-handles are homotopic to free generators
of 1r1(V2). As generators we may choose the cores 91,. .. , g, of 1- handles of type 2,
4, 5 and 6.

Consider the homotopy F: S1 x I —+ V2, F(-,0) = g., F(-, 1) = h.. We can make
it disjoint from S. U K. U X. U K.. First, intersections of the image of the homotopy
with X. and Y. can be turned into intersections with K. by pushing them toward the
boundary of X. U Y... Then, in the same way, we can avoid intersections with K. by
cost of new intersections with P.. Finally, intersections with S. can be removed using
the geometrically dual collection of immersed spheres {Sf}. Call this new homotopy
F’ and let {2:1, 3:2, . . . ,n.} = F’ "1(P.). If we remove a small neighborhood of a union
of disjoint arcs joining each x,- to the point on S1 x {0} from S1 x I, then restriction
of F’ on this set is a homotopy of h,- to the curve g’i which is a band connected sum
of g.- and meridians of P..

This homotopy is disjoint from S. U P. U K. U X. U Y. and can be pushed to the

boundary of V2.

 

15

We obtain the handlebody of W1 by attaching 2—handles to hl, . . . ,h, and perform-
ing surgery along 3., which corresponds to putting dots on the circles representing
S. in Kirby calculus.

We summarize all the information about the handlebody of W2 in Table 1. For

2-handles we use same notation as for the corresponding object in construction of W1

above.

1-handles 2-handles
P.; attaching circles go geometrically

once through corresponding 1-handles
S..

1. Surgery of S..

K .; attaching circles go geometrically

Whitney “Ides Of intersections Of 5‘ once through corresponding l-handles

and P" from the left entry of the table.
X., Y.; attaching circles go geomet-
3 Whitney circles of intersections of K. rically once through corresponding 1-

and S. U P.. handles from the left entry of the ta-

ble.

Extra 2—handles H.; attaching circles

 

Accessory circles of all intersections,
Whitney circles of (self-)intersections
of K ., Whitney circles of intersections

go geometrically once through corre-
sponding l-handles from the left en-
try of the table and homotopic to

of X .. and Y“ extra 1-handles. band connected sum of 1-handles with

meridians of P..

Table 1: Handles of W1

Taking a double of W1 corresponds to attaching zero-framed 2-handles to the
meridians of existent 2-handles, and as many 3-handles as there are l-handles in the
handlebody of W1.

Recall that attaching circles h,- of 2-handles H,- are homotopic to g]. Sliding H.
over dual handles H: we can obtain handles H1 attached to g1. Now g’ 1, . . . , g’ i may

be linked to the meridians of handles P., K ., X. and Y., but situation can be improved

16

by sliding handles dual to P., K ., X. and Y. over handles dual to H1. Further we may
slide H; over handles dual to P. to obtain handles attached to meridians of 1-handles
from the fourth row of the table, so they can be canceled. The framings of H1 result
in a twist of attaching circles of remaining 2-handles.

We undo this twist using dual handles.

With the next step we unlink handles X. and Y. from each other using dual 2-
handles and cancel them with 1-handles of type 3. Now K.- are attached to meridians
of 1-handles of type 1 and also can be canceled. After canceling P. with 1-handles
from surgery of S. we end up with unknotted, unlinked, zero framed handles and the
same number of 3—handles, which is clearly 8".

We need only small changes in our construction to prove the second diffeomor-
phism in Lemma 2. The handlebody of W2 differs from W1 by putting dots on the
attaching circles of P. rather than S.. Thus, the handlebody of W1#2W2 is obtained
from W1 by attaching 2-handles to meridians of l-handles, coming from surgered Si’s,
and to meridians of all 2-handles, except R’s. Using the same trick we can make the
homotopy of h. to g. disjoint from P. U K. U X. U K, rather than S. U K. U X. U Y...
Applying the same procedure to H. gives us .2-handles Gf,’ attached to the band
connected sum of meridians of 1-handles corresponding to generators of n1(I/'2) and
meridians of S.. But now 1-handles corresponding S. have dual 2-handles, so sliding
H. over them gives handles dual to 1-handles generating 7r1(V2). So we may apply
the same procedure to simplify the handlebody and end up with 8“. This finishes the

proof of Lemma 2 and the second part of Theorem.

 

4. Proof of Lemma 1

It is a simple consequence of singularity theory that a homotopy of a surface
in 4-manifold can be decomposed (after small perturbation) in a sequence of finger
moves, cusps, Whitney tricks and inverse cusp moves. In our case we have homotopy
of 2-complex consisting of the union of surfaces with double points and collection of
discs attached to this union along some embedded 100ps. Thus, one has to consider,
in addition, a finger move along a path in one of the disks joining (self-)intersection
of the disks with a point on its boundary. The inverse of this move is a Whitney trick
with a Whitney circle intersecting the boundary of the disk. We have to show that
it is possible to reorder these simple homotopies so that those increasing number of
intersections come first. This is obvious after the following consideration: cusp birth
happens in a small neighborhood of the point in the surface and we can assume that
part of the surface in this neighborhood stays fixed during the part of homotopy pre-
ceding this cusp birth, so we can push this cusp up to the beginning of the homotopy.
A finger move can be localized in the neighborhood of an embedded arc joining two
points in the surface, thus we may apply the same argument and this finishes the

proof of Lemma 1.

17

 

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[cam

[CHM

[cars

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[FQ]

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